ACM模板

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清华大学 代码册 2
1、几 何
25
1.1 注意 ................................................................................................................. 25
1.2 几何公式 ......................................................................................................... 25
1.3 多边形 ............................................................................................................. 27
1.4 多边形切割 ..................................................................................................... 30
1.5 浮点函数 ......................................................................................................... 31
1.6 面积 ................................................................................................................. 36
1.7 球面 ................................................................................................................. 37
1.8 三角形 ............................................................................................................. 38
1.9 三维几何 ......................................................................................................... 40
1.10 凸包 ................................................................................................................. 47
1.11 网格 ................................................................................................................. 49
1.12 圆 ..................................................................................................................... 49
1.13 整数函数 ......................................................................................................... 51 3
2、组 合
54
2.1 组合公式 .................................................................................................................. 54
2.2 排列组合生成 .......................................................................................................... 54
2.3 生成 gray码 ............................................................................................................. 56
2.4 置换(polya) .............................................................................................................. 56
2.5 字典序全排列 .......................................................................................................... 57
2.6 字典序组合 .............................................................................................................. 57 4
3、结 构
58
3.1 并查集 ...................................................................................................................... 58
3.2 堆 .............................................................................................................................. 59
3.3 线段树 ...................................................................................................................... 60
3.4 子段和 ...................................................................................................................... 65
3.5 子阵和 ...................................................................................................................... 65 5
4、数 论
66
4.1 阶乘最后非0 位 ...................................................................................................... 66
4.2 模线性方程组 .......................................................................................................... 67
4.3 素数 .......................................................................................................................... 68
4.4 欧拉函数 .................................................................................................................. 69 6
5、数 值 计 算
70
5.1 定积分计算(Romberg) ............................................................................................ 70
5.2 多项式求根(牛顿法) ............................................................................................... 72
5.3 周期性方程(追赶法) ............................................................................................... 73 7
6、图论—NP 搜索
74
6.1 最大团 ...................................................................................................................... 74
6.2 最大团(n<64)(faster) ............................................................................................... 75 8
7、图 论 — 连 通 性
77
7.1 无向图关键点(dfs邻接阵) ..................................................................................... 77
7.2 无向图关键边(dfs邻接阵) ..................................................................................... 78
7.3 无向图的块(bfs邻接阵) ......................................................................................... 79
7.4 无向图连通分支(dfs/bfs邻接阵) ........................................................................... 80
7.5 有向图强连通分支(dfs/bfs邻接阵) ....................................................................... 81
7.6 有向图最小点基(邻接阵) ....................................................................................... 82 9
8、图 论 — 匹 配
83
8.1 二分图最大匹配(hungary邻接表) ......................................................................... 83
8.2 二分图最大匹配(hungary邻接阵) ......................................................................... 84
8.3 二分图最大匹配(hungary正向表) ......................................................................... 84
8.4 二分图最佳匹配(kuhn_munkras邻接阵)................................................................ 85
8.5 一般图匹配(邻接表) ............................................................................................... 86
8.6 一般图匹配(邻接阵) ............................................................................................... 87
8.7 一般图匹配(正向表) ............................................................................................... 87 10
9、图 论 — 网 络 流
88
9.1 最大流(邻接阵) ....................................................................................................... 88
9.2 上下界最大流(邻接阵) ........................................................................................... 89
9.3 上下界最小流(邻接阵) ........................................................................................... 90
9.4 最大流无流量(邻接阵) ........................................................................................... 91
9.5 最小费用最大流(邻接阵) ....................................................................................... 91 11
10、图 论 — 应 用
92 12
10.1 欧拉回路(邻接
阵) ........................... 92 13
10.2 树的前序表转化
93 14
10.3 树的优化算法 94 15
10.4 拓扑排序(邻接
阵) ........................... 95 16
10.5 最佳边割集 ... 96 17
10.6 最佳点割集 ... 97 18
10.7 最小边割集 ... 98 19
10.8 最小点割集 ... 99 20
10.9 最小路径覆盖
101 21
11、图 论 — 支 撑 树
101
11.1 最小生成树(kruskal 邻接表)............................................................................... 101
11.2 最小生成树(kruskal 正向表)............................................................................... 103
11.3 最小生成树(prim+binary_heap邻接表) ............................................................. 104
11.4 最小生成树(prim+binary_heap正向表) ............................................................. 105
11.5 最小生成树(prim+mapped_heap邻接表) .......................................................... 106
11.6 最小生成树(prim+mapped_heap正向表) .......................................................... 108
11.7 最小生成树(prim邻接阵) ................................................................................... 109
11.8 最小树形图(邻接阵) ........................................................................................... 109 22
12、图论—最短路径
111
12.1 最短路径(单源bellman_ford邻接阵) ................................................................ 111
12.2 最短路径(单源dijkstra+bfs邻接表) .................................................................. 111
12.3 最短路径(单源dijkstra+bfs正向表) .................................................................. 112
12.4 最短路径(单源dijkstra+binary_heap邻接表) ................................................... 113
12.5 最短路径(单源dijkstra+binary_heap正向表) ................................................... 114
12.6 最短路径(单源dijkstra+mapped_heap邻接表) ................................................. 115
12.7 最短路径(单源dijkstra+mapped_heap正向表) ................................................. 116
12.8 最短路径(单源dijkstra 邻接阵) ......................................................................... 117
12.9 最短路径(多源floyd_warshall 邻接阵) ............................................................. 118 23
13、应 用
118
13.1 Joseph问题 ........................................................................................................... 118
13.2 N 皇后构造解 ....................................................................................................... 119
13.3 布尔母函数 .......................................................................................................... 120
13.4 第k元素 .............................................................................................................. 120
13.5 幻方构造 .............................................................................................................. 121
13.6 模式匹配(kmp) .................................................................................................... 122
13.7 逆序对数 .............................................................................................................. 123
13.8 字符串最小表示 .................................................................................................. 123
13.9 最长公共单调子序列 .......................................................................................... 124
13.10 最长子序列 ........................................................................................................ 125
13.11 最大子串匹配 .................................................................................................... 126
13.12 最大子段和 ........................................................................................................ 127
13.13 最大子阵和 ........................................................................................................ 127 24
14、其 它
128
14.1 大数(只能处理正数) ........................................................................................... 128
14.2 分数 ...................................................................................................................... 134
14.3 矩阵 ...................................................................................................................... 136
14.4 线性方程组 .......................................................................................................... 138
14.5 线性相关 .............................................................................................................. 140
14.6 日期 ...................................................................................................................... 140
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1、 几何
1.1 注意
1. 注意舍入方式(0.5的舍入方向);防止输出-0.

2. 几何题注意多测试不对称数据.

3. 整数几何注意xmult和 dmult是否会出界;
符点几何注意eps的使用.

4. 避免使用斜率;注意除数是否会为 0.

5. 公式一定要化简后再代入.

6. 判断同一个2*PI域内两角度差应该是
abs(a1-a2)pi+pi-beta;
相等应该是
abs(a1-a2)pi+pi-eps;

7. 需要的话尽量使用atan2,注意:atan2(0,0)=0,
atan2(1,0)=pi/2,atan2(-1,0)=-pi/2,atan2(0,1)=0,atan2(0,-1)=pi.

8. cross product = |u|*|v|*sin(a)
dot product = |u|*|v|*cos(a)

9. (P1-P0)x(P2-P0)结果的意义:
正: 在顺时针(0,pi)内
负: 在逆时针(0,pi)内
0 : ,共线,夹角为0 或pi

10. 误差限缺省使用1e-8!
1.2 几何公式
三角形:
1. 半周长 P=(a+b+c)/2
2. 面积 S=aHa/2=absin(C)/2=sqrt(P(P-a)(P-b)(P-c))
3. 中线 Ma=sqrt(2(b^2+c^2)-a^2)/2=sqrt(b^2+c^2+2bccos(A))/2
4. 角平分线 Ta=sqrt(bc((b+c)^2-a^2))/(b+c)=2bccos(A/2)/(b+c)
5. 高线 Ha=bsin(C)=csin(B)=sqrt(b^2-((a^2+b^2-c^2)/(2a))^2) 26
6. 内切圆半径 r=S/P=asin(B/2)sin(C/2)/sin((B+C)/2)
=4Rsin(A/2)sin(B/2)sin(C/2)=sqrt((P-a)(P-b)(P-c)/P)
=Ptan(A/2)tan(B/2)tan(C/2)
7. 外接圆半径 R=abc/(4S)=a/(2sin(A))=b/(2sin(B))=c/(2sin(C))

四边形:
D1,D2 为对角线,M对角线中点连线,A为对角线夹角
1. a^2+b^2+c^2+d^2=D1^2+D2^2+4M^2
2. S=D1D2sin(A)/2
(以下对圆的内接四边形)
3. ac+bd=D1D2
4. S=sqrt((P-a)(P-b)(P-c)(P-d)),P为半周长

正 n边形:
R为外接圆半径,r 为内切圆半径
1. 中心角 A=2PI/n
2. 内角 C=(n-2)PI/n
3. 边长 a=2sqrt(R^2-r^2)=2Rsin(A/2)=2rtan(A/2)
4. 面积 S=nar/2=nr^2tan(A/2)=nR^2sin(A)/2=na^2/(4tan(A/2))

圆:
1. 弧长 l=rA
2. 弦长 a=2sqrt(2hr-h^2)=2rsin(A/2)
3. 弓形高 h=r-sqrt(r^2-a^2/4)=r(1-cos(A/2))=atan(A/4)/2
4. 扇形面积 S1=rl/2=r^2A/2
5. 弓形面积 S2=(rl-a(r-h))/2=r^2(A-sin(A))/2

棱柱:
1. 体积 V=Ah,A为底面积,h为高
2. 侧面积 S=lp,l为棱长,p为直截面周长
3. 全面积 T=S+2A

棱锥:
1. 体积 V=Ah/3,A为底面积,h为高
(以下对正棱锥)
2. 侧面积 S=lp/2,l为斜高,p为底面周长
3. 全面积 T=S+A

棱台:
1. 体积 V=(A1+A2+sqrt(A1A2))h/3,A1.A2为上下底面积,h为高
(以下为正棱台)
2. 侧面积 S=(p1+p2)l/2,p1.p2为上下底面周长,l为斜高
3. 全面积 T=S+A1+A2
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圆柱:
1. 侧面积 S=2PIrh
2. 全面积 T=2PIr(h+r)
3. 体积 V=PIr^2h

圆锥:
1. 母线 l=sqrt(h^2+r^2)
2. 侧面积 S=PIrl
3. 全面积 T=PIr(l+r)
4. 体积 V=PIr^2h/3

圆台:
1. 母线 l=sqrt(h^2+(r1-r2)^2)
2. 侧面积 S=PI(r1+r2)l
3. 全面积 T=PIr1(l+r1)+PIr2(l+r2)
4. 体积 V=PI(r1^2+r2^2+r1r2)h/3

球:
1. 全面积 T=4PIr^2
2. 体积 V=4PIr^3/3

球台:
1. 侧面积 S=2PIrh
2. 全面积 T=PI(2rh+r1^2+r2^2)
3. 体积 V=PIh(3(r1^2+r2^2)+h^2)/6

球扇形:
1. 全面积 T=PIr(2h+r0),h为球冠高,r0为球冠底面半径
2. 体积 V=2PIr^2h/3
1.3 多边形
#include
#include
#define MAXN 1000
#define offset 10000
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x)) #define _sign(x) ((x)>eps?1:((x)<-eps?2:0))
struct point{double x,y;};
struct line{point a,b;};

double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y); 28
}

//判定凸多边形,顶点按顺时针或逆时针给出,允许相邻边共线
int is_convex(int n,point* p){
int i,s[3]={1,1,1};
for (i=0;i s[_sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))]=0;
return s[1]|s[2];
}

//判定凸多边形,顶点按顺时针或逆时针给出,不允许相邻边共线
int is_convex_v2(int n,point* p){
int i,s[3]={1,1,1};
for (i=0;i s[_sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))]=0;
return s[0]&&s[1]|s[2];
}

//判点在凸多边形内或多边形边上,顶点按顺时针或逆时针给出
int inside_convex(point q,int n,point* p){
int i,s[3]={1,1,1};
for (i=0;i s[_sign(xmult(p[(i+1)%n],q,p[i]))]=0;
return s[1]|s[2];
}

//判点在凸多边形内,顶点按顺时针或逆时针给出,在多边形边上返回 0
int inside_convex_v2(point q,int n,point* p){
int i,s[3]={1,1,1};
for (i=0;i s[_sign(xmult(p[(i+1)%n],q,p[i]))]=0;
return s[0]&&s[1]|s[2];
}

//判点在任意多边形内,顶点按顺时针或逆时针给出
//on_edge表示点在多边形边上时的返回值,offset 为多边形坐标上限
int inside_polygon(point q,int n,point* p,int on_edge=1){
point q2;
int i=0,count;
while (i for (count=i=0,q2.x=rand()+offset,q2.y=rand()+offset;i if
(zero(xmult(q,p[i],p[(i+1)%n]))&&(p[i].x-q.x)*(p[(i+1)%n].x-q.x) n].y-q.y) return on_edge;
else if (zero(xmult(q,q2,p[i])))
break;
else if
(xmult(q,p[i],q2)*xmult(q,p[(i+1)%n],q2)<-eps&&xmult(p[i],q,p[(i+1)%n])*xmult(p[i],q2,p[(i+1)
%n])<-eps)
count++;
return count&1;
}

inline int opposite_side(point p1,point p2,point l1,point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}

inline int dot_online_in(point p,point l1,point l2){
return zero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x) }

//判线段在任意多边形内,顶点按顺时针或逆时针给出,与边界相交返回1
int inside_polygon(point l1,point l2,int n,point* p){
point t[MAXN],tt;
int i,j,k=0;
if (!inside_polygon(l1,n,p)||!inside_polygon(l2,n,p))
return 0;
for (i=0;i if (opposite_side(l1,l2,p[i],p[(i+1)%n])&&opposite_side(p[i],p[(i+1)%n],l1,l2))
return 0;
else if (dot_online_in(l1,p[i],p[(i+1)%n]))
t[k++]=l1;
else if (dot_online_in(l2,p[i],p[(i+1)%n]))
t[k++]=l2;
else if (dot_online_in(p[i],l1,l2))
t[k++]=p[i];
for (i=0;i for (j=i+1;j tt.x=(t[i].x+t[j].x)/2;
tt.y=(t[i].y+t[j].y)/2;
if (!inside_polygon(tt,n,p))
return 0;
}
return 1;
}

point intersection(line u,line v){ 30
point ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}

point barycenter(point a,point b,point c){
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b=c;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b=b;
return intersection(u,v);
}

//多边形重心
point barycenter(int n,point* p){
point ret,t;
double t1=0,t2;
int i;
ret.x=ret.y=0;
for (i=1;i if (fabs(t2=xmult(p[0],p[i],p[i+1]))>eps){
t=barycenter(p[0],p[i],p[i+1]);
ret.x+=t.x*t2;
ret.y+=t.y*t2;
t1+=t2;
}
if (fabs(t1)>eps)
ret.x/=t1,ret.y/=t1;
return ret;
}
1.4 多边形切割
//多边形切割
//可用于半平面交
#define MAXN 100
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x)) struct point{double x,y;};

double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}

int same_side(point p1,point p2,point l1,point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}

point intersection(point u1,point u2,point v1,point v2){
point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret;
}

//将多边形沿 l1,l2确定的直线切割在 side侧切割,保证l1,l2,side不共线
void polygon_cut(int& n,point* p,point l1,point l2,point side){
point pp[100];
int m=0,i;
for (i=0;i if (same_side(p[i],side,l1,l2))
pp[m++]=p[i];
if
(!same_side(p[i],p[(i+1)%n],l1,l2)&&!(zero(xmult(p[i],l1,l2))&&zero(xmult(p[(i+1)%n],l1,l2))))
pp[m++]=intersection(p[i],p[(i+1)%n],l1,l2);
}
for (n=i=0;i if (!i||!zero(pp[i].x-pp[i-1].x)||!zero(pp[i].y-pp[i-1].y))
p[n++]=pp[i];
if (zero(p[n-1].x-p[0].x)&&zero(p[n-1].y-p[0].y))
n--;
if (n<3)
n=0;
}
1.5 浮点函数
//浮点几何函数库
#include
#define eps 1e-8 32
#define zero(x) (((x)>0?(x):-(x)) struct point{double x,y;};
struct line{point a,b;};

//计算 cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
double xmult(double x1,double y1,double x2,double y2,double x0,double y0){
return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}

//计算 dot product (P1-P0).(P2-P0)
double dmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
double dmult(double x1,double y1,double x2,double y2,double x0,double y0){
return (x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}

//两点距离
double distance(point p1,point p2){
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
double distance(double x1,double y1,double x2,double y2){
return sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
}

//判三点共线
int dots_inline(point p1,point p2,point p3){
return zero(xmult(p1,p2,p3));
}
int dots_inline(double x1,double y1,double x2,double y2,double x3,double y3){
return zero(xmult(x1,y1,x2,y2,x3,y3));
}

//判点是否在线段上,包括端点
int dot_online_in(point p,line l){
return zero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x) }
int dot_online_in(point p,point l1,point l2){
return zero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x) }
int dot_online_in(double x,double y,double x1,double y1,double x2,double y2){ 33
return zero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x) }

//判点是否在线段上,不包括端点
int dot_online_ex(point p,line l){
return
dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y));
}
int dot_online_ex(point p,point l1,point l2){
return
dot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y))&&(!zero(p.x-l2.x)||!zero(p.y-l2.y));
}
int dot_online_ex(double x,double y,double x1,double y1,double x2,double y2){
return
dot_online_in(x,y,x1,y1,x2,y2)&&(!zero(x-x1)||!zero(y-y1))&&(!zero(x-x2)||!zero(y-y2));
}

//判两点在线段同侧,点在线段上返回 0
int same_side(point p1,point p2,line l){
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
int same_side(point p1,point p2,point l1,point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}

//判两点在线段异侧,点在线段上返回 0
int opposite_side(point p1,point p2,line l){
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
int opposite_side(point p1,point p2,point l1,point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}

//判两直线平行
int parallel(line u,line v){
return zero((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y));
}
int parallel(point u1,point u2,point v1,point v2){
return zero((u1.x-u2.x)*(v1.y-v2.y)-(v1.x-v2.x)*(u1.y-u2.y));
}

//判两直线垂直
int perpendicular(line u,line v){
return zero((u.a.x-u.b.x)*(v.a.x-v.b.x)+(u.a.y-u.b.y)*(v.a.y-v.b.y)); 34
}
int perpendicular(point u1,point u2,point v1,point v2){
return zero((u1.x-u2.x)*(v1.x-v2.x)+(u1.y-u2.y)*(v1.y-v2.y));
}

//判两线段相交,包括端点和部分重合
int intersect_in(line u,line v){
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point u1,point u2,point v1,point v2){
if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
return
dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u
2);
}

//判两线段相交,不包括端点和部分重合
int intersect_ex(line u,line v){
return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point u1,point u2,point v1,point v2){
return opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}

//计算两直线交点,注意事先判断直线是否平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
point intersection(line u,line v){
point ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}
point intersection(point u1,point u2,point v1,point v2){
point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret; 35
}

//点到直线上的最近点
point ptoline(point p,line l){
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
return intersection(p,t,l.a,l.b);
}
point ptoline(point p,point l1,point l2){
point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
return intersection(p,t,l1,l2);
}

//点到直线距离
double disptoline(point p,line l){
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoline(point p,point l1,point l2){
return fabs(xmult(p,l1,l2))/distance(l1,l2);
}
double disptoline(double x,double y,double x1,double y1,double x2,double y2){
return fabs(xmult(x,y,x1,y1,x2,y2))/distance(x1,y1,x2,y2);
}

//点到线段上的最近点
point ptoseg(point p,line l){
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a) return intersection(p,t,l.a,l.b);
}
point ptoseg(point p,point l1,point l2){
point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
return distance(p,l1) return intersection(p,t,l1,l2);
}

//点到线段距离
double disptoseg(point p,line l){
point t=p; 36
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a) return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoseg(point p,point l1,point l2){
point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
return distance(p,l1) return fabs(xmult(p,l1,l2))/distance(l1,l2);
}

//矢量 V以P为顶点逆时针旋转angle并放大 scale倍
point rotate(point v,point p,double angle,double scale){
point ret=p;
v.x-=p.x,v.y-=p.y;
p.x=scale*cos(angle);
p.y=scale*sin(angle);
ret.x+=v.x*p.x-v.y*p.y;
ret.y+=v.x*p.y+v.y*p.x;
return ret;
}
1.6 面积
#include
struct point{double x,y;};

//计算 cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
double xmult(double x1,double y1,double x2,double y2,double x0,double y0){
return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}

//计算三角形面积,输入三顶点
double area_triangle(point p1,point p2,point p3){
return fabs(xmult(p1,p2,p3))/2;
}
double area_triangle(double x1,double y1,double x2,double y2,double x3,double y3){
return fabs(xmult(x1,y1,x2,y2,x3,y3))/2;
} 37

//计算三角形面积,输入三边长
double area_triangle(double a,double b,double c){
double s=(a+b+c)/2;
return sqrt(s*(s-a)*(s-b)*(s-c));
}

//计算多边形面积,顶点按顺时针或逆时针给出
double area_polygon(int n,point* p){
double s1=0,s2=0;
int i;
for (i=0;i s1+=p[(i+1)%n].y*p[i].x,s2+=p[(i+1)%n].y*p[(i+2)%n].x;
return fabs(s1-s2)/2;
}
1.7 球面
#include
const double pi=acos(-1);

//计算圆心角 lat表示纬度,-90<=w<=90,lng表示经度
//返回两点所在大圆劣弧对应圆心角,0<=angle<=pi
double angle(double lng1,double lat1,double lng2,double lat2){
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return acos(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2));
}

//计算距离,r为球半径
double line_dist(double r,double lng1,double lat1,double lng2,double lat2){
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return r*sqrt(2-2*(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2)));
}
38
//计算球面距离,r 为球半径
inline double sphere_dist(double r,double lng1,double lat1,double lng2,double lat2){
return r*angle(lng1,lat1,lng2,lat2);
}
1.8 三角形
#include
struct point{double x,y;};
struct line{point a,b;};

double distance(point p1,point p2){
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}

point intersection(line u,line v){
point ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}

//外心
point circumcenter(point a,point b,point c){
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}

//内心
point incenter(point a,point b,point c){
line u,v;
double m,n;
u.a=a;
m=atan2(b.y-a.y,b.x-a.x); 39
n=atan2(c.y-a.y,c.x-a.x);
u.b.x=u.a.x+cos((m+n)/2);
u.b.y=u.a.y+sin((m+n)/2);
v.a=b;
m=atan2(a.y-b.y,a.x-b.x);
n=atan2(c.y-b.y,c.x-b.x);
v.b.x=v.a.x+cos((m+n)/2);
v.b.y=v.a.y+sin((m+n)/2);
return intersection(u,v);
}

//垂心
point perpencenter(point a,point b,point c){
line u,v;
u.a=c;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a=b;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}

//重心
//到三角形三顶点距离的平方和最小的点
//三角形内到三边距离之积最大的点
point barycenter(point a,point b,point c){
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b=c;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b=b;
return intersection(u,v);
}

//费马点
//到三角形三顶点距离之和最小的点
point fermentpoint(point a,point b,point c){
point u,v;
double step=fabs(a.x)+fabs(a.y)+fabs(b.x)+fabs(b.y)+fabs(c.x)+fabs(c.y);
int i,j,k;
u.x=(a.x+b.x+c.x)/3; 40
u.y=(a.y+b.y+c.y)/3;
while (step>1e-10)
for (k=0;k<10;step/=2,k++)
for (i=-1;i<=1;i++)
for (j=-1;j<=1;j++){
v.x=u.x+step*i;
v.y=u.y+step*j;
if
(distance(u,a)+distance(u,b)+distance(u,c)>distance(v,a)+distance(v,b)+distance(v,c))
u=v;
}
return u;
}
1.9 三维几何
//三维几何函数库
#include
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x)) struct point3{double x,y,z;};
struct line3{point3 a,b;};
struct plane3{point3 a,b,c;};

//计算 cross product U x V
point3 xmult(point3 u,point3 v){
point3 ret;
ret.x=u.y*v.z-v.y*u.z;
ret.y=u.z*v.x-u.x*v.z;
ret.z=u.x*v.y-u.y*v.x;
return ret;
}

//计算 dot product U . V
double dmult(point3 u,point3 v){
return u.x*v.x+u.y*v.y+u.z*v.z;
}

//矢量差 U - V
point3 subt(point3 u,point3 v){
point3 ret;
ret.x=u.x-v.x;
ret.y=u.y-v.y;
ret.z=u.z-v.z; 41
return ret;
}

//取平面法向量
point3 pvec(plane3 s){
return xmult(subt(s.a,s.b),subt(s.b,s.c));
}
point3 pvec(point3 s1,point3 s2,point3 s3){
return xmult(subt(s1,s2),subt(s2,s3));
}

//两点距离,单参数取向量大小
double distance(point3 p1,point3 p2){
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y)+(p1.z-p2.z)*(p1.z-p2.z));
}

//向量大小
double vlen(point3 p){
return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
}

//判三点共线
int dots_inline(point3 p1,point3 p2,point3 p3){
return vlen(xmult(subt(p1,p2),subt(p2,p3))) }

//判四点共面
int dots_onplane(point3 a,point3 b,point3 c,point3 d){
return zero(dmult(pvec(a,b,c),subt(d,a)));
}

//判点是否在线段上,包括端点和共线
int dot_online_in(point3 p,line3 l){
return zero(vlen(xmult(subt(p,l.a),subt(p,l.b))))&&(l.a.x-p.x)*(l.b.x-p.x) (l.a.y-p.y)*(l.b.y-p.y) }
int dot_online_in(point3 p,point3 l1,point3 l2){
return zero(vlen(xmult(subt(p,l1),subt(p,l2))))&&(l1.x-p.x)*(l2.x-p.x) (l1.y-p.y)*(l2.y-p.y) }

//判点是否在线段上,不包括端点
int dot_online_ex(point3 p,line3 l){
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y)||!zero(p.z-l.a.z))&& 42
(!zero(p.x-l.b.x)||!zero(p.y-l.b.y)||!zero(p.z-l.b.z));
}
int dot_online_ex(point3 p,point3 l1,point3 l2){
return dot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y)||!zero(p.z-l1.z))&&
(!zero(p.x-l2.x)||!zero(p.y-l2.y)||!zero(p.z-l2.z));
}

//判点是否在空间三角形上,包括边界,三点共线无意义
int dot_inplane_in(point3 p,plane3 s){
return zero(vlen(xmult(subt(s.a,s.b),subt(s.a,s.c)))-vlen(xmult(subt(p,s.a),subt(p,s.b)))-
vlen(xmult(subt(p,s.b),subt(p,s.c)))-vlen(xmult(subt(p,s.c),subt(p,s.a))));
}
int dot_inplane_in(point3 p,point3 s1,point3 s2,point3 s3){
return zero(vlen(xmult(subt(s1,s2),subt(s1,s3)))-vlen(xmult(subt(p,s1),subt(p,s2)))-
vlen(xmult(subt(p,s2),subt(p,s3)))-vlen(xmult(subt(p,s3),subt(p,s1))));
}

//判点是否在空间三角形上,不包括边界,三点共线无意义
int dot_inplane_ex(point3 p,plane3 s){
return dot_inplane_in(p,s)&&vlen(xmult(subt(p,s.a),subt(p,s.b)))>eps&&
vlen(xmult(subt(p,s.b),subt(p,s.c)))>eps&&vlen(xmult(subt(p,s.c),subt(p,s.a)))>eps;
}
int dot_inplane_ex(point3 p,point3 s1,point3 s2,point3 s3){
return dot_inplane_in(p,s1,s2,s3)&&vlen(xmult(subt(p,s1),subt(p,s2)))>eps&&
vlen(xmult(subt(p,s2),subt(p,s3)))>eps&&vlen(xmult(subt(p,s3),subt(p,s1)))>eps;
}

//判两点在线段同侧,点在线段上返回 0,不共面无意义
int same_side(point3 p1,point3 p2,line3 l){
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))>eps;
}
int same_side(point3 p1,point3 p2,point3 l1,point3 l2){
return dmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))>eps;
}

//判两点在线段异侧,点在线段上返回 0,不共面无意义
int opposite_side(point3 p1,point3 p2,line3 l){
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))<-eps;
}
int opposite_side(point3 p1,point3 p2,point3 l1,point3 l2){
return dmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))<-eps;
}

//判两点在平面同侧,点在平面上返回 0 43
int same_side(point3 p1,point3 p2,plane3 s){
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))>eps;
}
int same_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3){
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))>eps;
}

//判两点在平面异侧,点在平面上返回 0
int opposite_side(point3 p1,point3 p2,plane3 s){
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))<-eps;
}
int opposite_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3){
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))<-eps;
}

//判两直线平行
int parallel(line3 u,line3 v){
return vlen(xmult(subt(u.a,u.b),subt(v.a,v.b))) }
int parallel(point3 u1,point3 u2,point3 v1,point3 v2){
return vlen(xmult(subt(u1,u2),subt(v1,v2))) }

//判两平面平行
int parallel(plane3 u,plane3 v){
return vlen(xmult(pvec(u),pvec(v))) }
int parallel(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
return vlen(xmult(pvec(u1,u2,u3),pvec(v1,v2,v3))) }

//判直线与平面平行
int parallel(line3 l,plane3 s){
return zero(dmult(subt(l.a,l.b),pvec(s)));
}
int parallel(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
return zero(dmult(subt(l1,l2),pvec(s1,s2,s3)));
}

//判两直线垂直
int perpendicular(line3 u,line3 v){
return zero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}
int perpendicular(point3 u1,point3 u2,point3 v1,point3 v2){ 44
return zero(dmult(subt(u1,u2),subt(v1,v2)));
}

//判两平面垂直
int perpendicular(plane3 u,plane3 v){
return zero(dmult(pvec(u),pvec(v)));
}
int perpendicular(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
return zero(dmult(pvec(u1,u2,u3),pvec(v1,v2,v3)));
}

//判直线与平面平行
int perpendicular(line3 l,plane3 s){
return vlen(xmult(subt(l.a,l.b),pvec(s))) }
int perpendicular(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
return vlen(xmult(subt(l1,l2),pvec(s1,s2,s3))) }

//判两线段相交,包括端点和部分重合
int intersect_in(line3 u,line3 v){
if (!dots_onplane(u.a,u.b,v.a,v.b))
return 0;
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point3 u1,point3 u2,point3 v1,point3 v2){
if (!dots_onplane(u1,u2,v1,v2))
return 0;
if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
return
dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u
2);
}

//判两线段相交,不包括端点和部分重合
int intersect_ex(line3 u,line3 v){
return dots_onplane(u.a,u.b,v.a,v.b)&&opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point3 u1,point3 u2,point3 v1,point3 v2){
return
dots_onplane(u1,u2,v1,v2)&&opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2); 45
}

//判线段与空间三角形相交,包括交于边界和(部分)包含
int intersect_in(line3 l,plane3 s){
return !same_side(l.a,l.b,s)&&!same_side(s.a,s.b,l.a,l.b,s.c)&&
!same_side(s.b,s.c,l.a,l.b,s.a)&&!same_side(s.c,s.a,l.a,l.b,s.b);
}
int intersect_in(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
return !same_side(l1,l2,s1,s2,s3)&&!same_side(s1,s2,l1,l2,s3)&&
!same_side(s2,s3,l1,l2,s1)&&!same_side(s3,s1,l1,l2,s2);
}

//判线段与空间三角形相交,不包括交于边界和(部分)包含
int intersect_ex(line3 l,plane3 s){
return opposite_side(l.a,l.b,s)&&opposite_side(s.a,s.b,l.a,l.b,s.c)&&
opposite_side(s.b,s.c,l.a,l.b,s.a)&&opposite_side(s.c,s.a,l.a,l.b,s.b);
}
int intersect_ex(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
return opposite_side(l1,l2,s1,s2,s3)&&opposite_side(s1,s2,l1,l2,s3)&&
opposite_side(s2,s3,l1,l2,s1)&&opposite_side(s3,s1,l1,l2,s2);
}

//计算两直线交点,注意事先判断直线是否共面和平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
point3 intersection(line3 u,line3 v){
point3 ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
ret.z+=(u.b.z-u.a.z)*t;
return ret;
}
point3 intersection(point3 u1,point3 u2,point3 v1,point3 v2){
point3 ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
ret.z+=(u2.z-u1.z)*t;
return ret;
}

//计算直线与平面交点,注意事先判断是否平行,并保证三点不共线! 46
//线段和空间三角形交点请另外判断
point3 intersection(line3 l,plane3 s){
point3 ret=pvec(s);
double t=(ret.x*(s.a.x-l.a.x)+ret.y*(s.a.y-l.a.y)+ret.z*(s.a.z-l.a.z))/
(ret.x*(l.b.x-l.a.x)+ret.y*(l.b.y-l.a.y)+ret.z*(l.b.z-l.a.z));
ret.x=l.a.x+(l.b.x-l.a.x)*t;
ret.y=l.a.y+(l.b.y-l.a.y)*t;
ret.z=l.a.z+(l.b.z-l.a.z)*t;
return ret;
}
point3 intersection(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
point3 ret=pvec(s1,s2,s3);
double t=(ret.x*(s1.x-l1.x)+ret.y*(s1.y-l1.y)+ret.z*(s1.z-l1.z))/
(ret.x*(l2.x-l1.x)+ret.y*(l2.y-l1.y)+ret.z*(l2.z-l1.z));
ret.x=l1.x+(l2.x-l1.x)*t;
ret.y=l1.y+(l2.y-l1.y)*t;
ret.z=l1.z+(l2.z-l1.z)*t;
return ret;
}

//计算两平面交线,注意事先判断是否平行,并保证三点不共线!
line3 intersection(plane3 u,plane3 v){
line3 ret;
ret.a=parallel(v.a,v.b,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.a,v.b,u.a,u.b,u.
c);
ret.b=parallel(v.c,v.a,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.c,v.a,u.a,u.b,u.
c);
return ret;
}
line3 intersection(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
line3 ret;
ret.a=parallel(v1,v2,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v1,v2,u1,u2,u3);
ret.b=parallel(v3,v1,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v3,v1,u1,u2,u3);
return ret;
}

//点到直线距离
double ptoline(point3 p,line3 l){
return vlen(xmult(subt(p,l.a),subt(l.b,l.a)))/distance(l.a,l.b);
}
double ptoline(point3 p,point3 l1,point3 l2){
return vlen(xmult(subt(p,l1),subt(l2,l1)))/distance(l1,l2);
}
47
//点到平面距离
double ptoplane(point3 p,plane3 s){
return fabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}
double ptoplane(point3 p,point3 s1,point3 s2,point3 s3){
return fabs(dmult(pvec(s1,s2,s3),subt(p,s1)))/vlen(pvec(s1,s2,s3));
}

//直线到直线距离
double linetoline(line3 u,line3 v){
point3 n=xmult(subt(u.a,u.b),subt(v.a,v.b));
return fabs(dmult(subt(u.a,v.a),n))/vlen(n);
}
double linetoline(point3 u1,point3 u2,point3 v1,point3 v2){
point3 n=xmult(subt(u1,u2),subt(v1,v2));
return fabs(dmult(subt(u1,v1),n))/vlen(n);
}

//两直线夹角cos值
double angle_cos(line3 u,line3 v){
return dmult(subt(u.a,u.b),subt(v.a,v.b))/vlen(subt(u.a,u.b))/vlen(subt(v.a,v.b));
}
double angle_cos(point3 u1,point3 u2,point3 v1,point3 v2){
return dmult(subt(u1,u2),subt(v1,v2))/vlen(subt(u1,u2))/vlen(subt(v1,v2));
}

//两平面夹角 cos值
double angle_cos(plane3 u,plane3 v){
return dmult(pvec(u),pvec(v))/vlen(pvec(u))/vlen(pvec(v));
}
double angle_cos(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
return dmult(pvec(u1,u2,u3),pvec(v1,v2,v3))/vlen(pvec(u1,u2,u3))/vlen(pvec(v1,v2,v3));
}

//直线平面夹角 sin值
double angle_sin(line3 l,plane3 s){
return dmult(subt(l.a,l.b),pvec(s))/vlen(subt(l.a,l.b))/vlen(pvec(s));
}
double angle_sin(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
return dmult(subt(l1,l2),pvec(s1,s2,s3))/vlen(subt(l1,l2))/vlen(pvec(s1,s2,s3));
}
1.10 凸包 48
#include
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x)) struct point{double x,y;};

//计算 cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
//graham算法顺时针构造包含所有共线点的凸包,O(nlogn)
point p1,p2;
int graham_cp(const void* a,const void* b){
double ret=xmult(*((point*)a),*((point*)b),p1);
return zero(ret)?(xmult(*((point*)a),*((point*)b),p2)>0?1:-1):(ret>0?1:-1);
}
void _graham(int n,point* p,int& s,point* ch){
int i,k=0;
for (p1=p2=p[0],i=1;i if (p1.y-p[i].y>eps||(zero(p1.y-p[i].y)&&p1.x>p[i].x))
p1=p[k=i];
p2.x/=n,p2.y/=n;
p[k]=p[0],p[0]=p1;
qsort(p+1,n-1,sizeof(point),graham_cp);
for (ch[0]=p[0],ch[1]=p[1],ch[2]=p[2],s=i=3;i for (;s>2&&xmult(ch[s-2],p[i],ch[s-1])<-eps;s--);
}

//构造凸包接口函数,传入原始点集大小 n,点集p(p原有顺序被打乱!)
//返回凸包大小,凸包的点在convex 中
//参数 maxsize为 1包含共线点,为 0不包含共线点,缺省为1
//参数 clockwise为 1顺时针构造,为 0逆时针构造,缺省为1
//在输入仅有若干共线点时算法不稳定,可能有此类情况请另行处理!
//不能去掉点集中重合的点
int graham(int n,point* p,point* convex,int maxsize=1,int dir=1){
point* temp=new point[n];
int s,i;
_graham(n,p,s,temp);
for (convex[0]=temp[0],n=1,i=(dir?1:(s-1));dir?(i if (maxsize||!zero(xmult(temp[i-1],temp[i],temp[(i+1)%s])))
convex[n++]=temp[i];
delete []temp;
return n;
} 49
1.11 网格
#define abs(x) ((x)>0?(x):-(x))
struct point{int x,y;};

int gcd(int a,int b){
return b?gcd(b,a%b):a;
}

//多边形上的网格点个数
int grid_onedge(int n,point* p){
int i,ret=0;
for (i=0;i ret+=gcd(abs(p[i].x-p[(i+1)%n].x),abs(p[i].y-p[(i+1)%n].y));
return ret;
}

//多边形内的网格点个数
int grid_inside(int n,point* p){
int i,ret=0;
for (i=0;i ret+=p[(i+1)%n].y*(p[i].x-p[(i+2)%n].x);
return (abs(ret)-grid_onedge(n,p))/2+1;
}
1.12 圆
#include
#define eps 1e-8
struct point{double x,y;};

double xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}

double distance(point p1,point p2){
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}

double disptoline(point p,point l1,point l2){
return fabs(xmult(p,l1,l2))/distance(l1,l2);
}
50
point intersection(point u1,point u2,point v1,point v2){
point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret;
}

//判直线和圆相交,包括相切
int intersect_line_circle(point c,double r,point l1,point l2){
return disptoline(c,l1,l2) }

//判线段和圆相交,包括端点和相切
int intersect_seg_circle(point c,double r,point l1,point l2){
double t1=distance(c,l1)-r,t2=distance(c,l2)-r;
point t=c;
if (t1 return t1>-eps||t2>-eps;
t.x+=l1.y-l2.y;
t.y+=l2.x-l1.x;
return xmult(l1,c,t)*xmult(l2,c,t) }

//判圆和圆相交,包括相切
int intersect_circle_circle(point c1,double r1,point c2,double r2){
return distance(c1,c2)fabs(r1-r2)-eps;
}

//计算圆上到点 p最近点,如 p与圆心重合,返回p本身
point dot_to_circle(point c,double r,point p){
point u,v;
if (distance(p,c) return p;
u.x=c.x+r*fabs(c.x-p.x)/distance(c,p);
u.y=c.y+r*fabs(c.y-p.y)/distance(c,p)*((c.x-p.x)*(c.y-p.y)<0?-1:1);
v.x=c.x-r*fabs(c.x-p.x)/distance(c,p);
v.y=c.y-r*fabs(c.y-p.y)/distance(c,p)*((c.x-p.x)*(c.y-p.y)<0?-1:1);
return distance(u,p) }

//计算直线与圆的交点,保证直线与圆有交点
//计算线段与圆的交点可用这个函数后判点是否在线段上 51
void intersection_line_circle(point c,double r,point l1,point l2,point& p1,point& p2){
point p=c;
double t;
p.x+=l1.y-l2.y;
p.y+=l2.x-l1.x;
p=intersection(p,c,l1,l2);
t=sqrt(r*r-distance(p,c)*distance(p,c))/distance(l1,l2);
p1.x=p.x+(l2.x-l1.x)*t;
p1.y=p.y+(l2.y-l1.y)*t;
p2.x=p.x-(l2.x-l1.x)*t;
p2.y=p.y-(l2.y-l1.y)*t;
}

//计算圆与圆的交点,保证圆与圆有交点,圆心不重合
void intersection_circle_circle(point c1,double r1,point c2,double r2,point& p1,point& p2){
point u,v;
double t;
t=(1+(r1*r1-r2*r2)/distance(c1,c2)/distance(c1,c2))/2;
u.x=c1.x+(c2.x-c1.x)*t;
u.y=c1.y+(c2.y-c1.y)*t;
v.x=u.x+c1.y-c2.y;
v.y=u.y-c1.x+c2.x;
intersection_line_circle(c1,r1,u,v,p1,p2);
}
1.13 整数函数
//整数几何函数库
//注意某些情况下整数运算会出界!
#define sign(a) ((a)>0?1:(((a)<0?-1:0)))
struct point{int x,y;};
struct line{point a,b;};

//计算 cross product (P1-P0)x(P2-P0)
int xmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
int xmult(int x1,int y1,int x2,int y2,int x0,int y0){
return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}

//计算 dot product (P1-P0).(P2-P0)
int dmult(point p1,point p2,point p0){
return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y); 52
}
int dmult(int x1,int y1,int x2,int y2,int x0,int y0){
return (x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}

//判三点共线
int dots_inline(point p1,point p2,point p3){
return !xmult(p1,p2,p3);
}
int dots_inline(int x1,int y1,int x2,int y2,int x3,int y3){
return !xmult(x1,y1,x2,y2,x3,y3);
}

//判点是否在线段上,包括端点和部分重合
int dot_online_in(point p,line l){
return !xmult(p,l.a,l.b)&&(l.a.x-p.x)*(l.b.x-p.x)<=0&&(l.a.y-p.y)*(l.b.y-p.y)<=0;
}
int dot_online_in(point p,point l1,point l2){
return !xmult(p,l1,l2)&&(l1.x-p.x)*(l2.x-p.x)<=0&&(l1.y-p.y)*(l2.y-p.y)<=0;
}
int dot_online_in(int x,int y,int x1,int y1,int x2,int y2){
return !xmult(x,y,x1,y1,x2,y2)&&(x1-x)*(x2-x)<=0&&(y1-y)*(y2-y)<=0;
}

//判点是否在线段上,不包括端点
int dot_online_ex(point p,line l){
return dot_online_in(p,l)&&(p.x!=l.a.x||p.y!=l.a.y)&&(p.x!=l.b.x||p.y!=l.b.y);
}
int dot_online_ex(point p,point l1,point l2){
return dot_online_in(p,l1,l2)&&(p.x!=l1.x||p.y!=l1.y)&&(p.x!=l2.x||p.y!=l2.y);
}
int dot_online_ex(int x,int y,int x1,int y1,int x2,int y2){
return dot_online_in(x,y,x1,y1,x2,y2)&&(x!=x1||y!=y1)&&(x!=x2||y!=y2);
}

//判两点在直线同侧,点在直线上返回 0
int same_side(point p1,point p2,line l){
return sign(xmult(l.a,p1,l.b))*xmult(l.a,p2,l.b)>0;
}
int same_side(point p1,point p2,point l1,point l2){
return sign(xmult(l1,p1,l2))*xmult(l1,p2,l2)>0;
}

//判两点在直线异侧,点在直线上返回 0 53
int opposite_side(point p1,point p2,line l){
return sign(xmult(l.a,p1,l.b))*xmult(l.a,p2,l.b)<0;
}
int opposite_side(point p1,point p2,point l1,point l2){
return sign(xmult(l1,p1,l2))*xmult(l1,p2,l2)<0;
}

//判两直线平行
int parallel(line u,line v){
return (u.a.x-u.b.x)*(v.a.y-v.b.y)==(v.a.x-v.b.x)*(u.a.y-u.b.y);
}
int parallel(point u1,point u2,point v1,point v2){
return (u1.x-u2.x)*(v1.y-v2.y)==(v1.x-v2.x)*(u1.y-u2.y);
}

//判两直线垂直
int perpendicular(line u,line v){
return (u.a.x-u.b.x)*(v.a.x-v.b.x)==-(u.a.y-u.b.y)*(v.a.y-v.b.y);
}
int perpendicular(point u1,point u2,point v1,point v2){
return (u1.x-u2.x)*(v1.x-v2.x)==-(u1.y-u2.y)*(v1.y-v2.y);
}

//判两线段相交,包括端点和部分重合
int intersect_in(line u,line v){
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(point u1,point u2,point v1,point v2){
if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
return
dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u
2);
}

//判两线段相交,不包括端点和部分重合
int intersect_ex(line u,line v){
return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(point u1,point u2,point v1,point v2){
return opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
} 54
2、 组合
2.1 组合公式
1. C(m,n)=C(m,m-n)
2. C(m,n)=C(m-1,n)+C(m-1,n-1)

derangement D(n) = n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)
= (n-1)(D(n-2) - D(n-1))
Q(n) = D(n) + D(n-1)

求和公式,k = 1..n
1. sum( k ) = n(n+1)/2
2. sum( 2k-1 ) = n^2
3. sum( k^2 ) = n(n+1)(2n+1)/6
4. sum( (2k-1)^2 ) = n(4n^2-1)/3
5. sum( k^3 ) = (n(n+1)/2)^2
6. sum( (2k-1)^3 ) = n^2(2n^2-1)
7. sum( k^4 ) = n(n+1)(2n+1)(3n^2+3n-1)/30
8. sum( k^5 ) = n^2(n+1)^2(2n^2+2n-1)/12
9. sum( k(k+1) ) = n(n+1)(n+2)/3
10. sum( k(k+1)(k+2) ) = n(n+1)(n+2)(n+3)/4
12. sum( k(k+1)(k+2)(k+3) ) = n(n+1)(n+2)(n+3)(n+4)/5
2.2 排列组合生成
//gen_perm产生字典序排列P(n,m)
//gen_comb产生字典序组合 C(n,m)
//gen_perm_swap产生相邻位对换全排列P(n,n)
//产生元素用 1..n表示
//dummy为产生后调用的函数,传入a[]和 n,a[0]..a[n-1]为一次产生的结果
#define MAXN 100
int count;

#include
void dummy(int* a,int n){
int i;
cout< for (i=0;i cout< cout< } 55

void _gen_perm(int* a,int n,int m,int l,int* temp,int* tag){
int i;
if (l==m)
dummy(temp,m);
else
for (i=0;i if (!tag[i]){
temp[l]=a[i],tag[i]=1;
_gen_perm(a,n,m,l+1,temp,tag);
tag[i]=0;
}
}

void gen_perm(int n,int m){
int a[MAXN],temp[MAXN],tag[MAXN]={0},i;
for (i=0;i a[i]=i+1;
_gen_perm(a,n,m,0,temp,tag);
}

void _gen_comb(int* a,int s,int e,int m,int& count,int* temp){
int i;
if (!m)
dummy(temp,count);
else
for (i=s;i<=e-m+1;i++){
temp[count++]=a[i];
_gen_comb(a,i+1,e,m-1,count,temp);
count--;
}
}

void gen_comb(int n,int m){
int a[MAXN],temp[MAXN],count=0,i;
for (i=0;i a[i]=i+1;
_gen_comb(a,0,n-1,m,count,temp);
}

void _gen_perm_swap(int* a,int n,int l,int* pos,int* dir){
int i,p1,p2,t;
if (l==n)
dummy(a,n); 56
else{
_gen_perm_swap(a,n,l+1,pos,dir);
for (i=0;i p2=(p1=pos[l])+dir[l];
t=a[p1],a[p1]=a[p2],a[p2]=t;
pos[a[p1]-1]=p1,pos[a[p2]-1]=p2;
_gen_perm_swap(a,n,l+1,pos,dir);
}
dir[l]=-dir[l];
}
}

void gen_perm_swap(int n){
int a[MAXN],pos[MAXN],dir[MAXN],i;
for (i=0;i a[i]=i+1,pos[i]=i,dir[i]=-1;
_gen_perm_swap(a,n,0,pos,dir);
}
2.3 生成 gray码
//生成 reflected gray code
//每次调用gray取得下一个码
//000...000是第一个码,100...000是最后一个码
void gray(int n,int *code){
int t=0,i;
for (i=0;i if (t&1)
for (n--;!code[n];n--);
code[n-1]=1-code[n-1];
}
2.4 置换(polya)
//求置换的循环节,polya 原理
//perm[0..n-1]为 0..n-1 的一个置换(排列)
//返回置换最小周期,num返回循环节个数
#define MAXN 1000

int gcd(int a,int b){
return b?gcd(b,a%b):a;
}
57
int polya(int* perm,int n,int& num){
int i,j,p,v[MAXN]={0},ret=1;
for (num=i=0;i if (!v[i]){
for (num++,j=0,p=i;!v[p=perm[p]];j++)
v[p]=1;
ret*=j/gcd(ret,j);
}
return ret;
}
2.5 字典序全排列
//字典序全排列与序号的转换
int perm2num(int n,int *p){
int i,j,ret=0,k=1;
for (i=n-2;i>=0;k*=n-(i--))
for (j=i+1;j if (p[j] ret+=k;
return ret;
}

void num2perm(int n,int *p,int t){
int i,j;
for (i=n-1;i>=0;i--)
p[i]=t%(n-i),t/=n-i;
for (i=n-1;i;i--)
for (j=i-1;j>=0;j--)
if (p[j]<=p[i])
p[i]++;
}
2.6 字典序组合
//字典序组合与序号的转换
//comb为组合数 C(n,m),必要时换成大数,注意处理C(n,m)=0|n int comb(int n,int m){
int ret=1,i;
m=m<(n-m)?m:(n-m);
for (i=n-m+1;i<=n;ret*=(i++));
for (i=1;i<=m;ret/=(i++));
return m<0?0:ret; 58
}

int comb2num(int n,int m,int *c){
int ret=comb(n,m),i;
for (i=0;i ret-=comb(n-c[i],m-i);
return ret;
}

void num2comb(int n,int m,int* c,int t){
int i,j=1,k;
for (i=0;i for (;t>(k=comb(n-j,m-i-1));t-=k,j++);
}
3、 结构
3.1 并查集
//带路径压缩的并查集,用于动态维护查询等价类
//图论算法中动态判点集连通常用
//维护和查询复杂度略大于 O(1)
//集合元素取值 1..MAXN-1(注意0 不能用!),默认不等价
#include
#define MAXN 100000
#define _ufind_run(x) for(;p[t=x];x=p[x],p[t]=(p[x]?p[x]:x))
#define _run_both _ufind_run(i);_ufind_run(j)

struct ufind{
int p[MAXN],t;
void init(){memset(p,0,sizeof(p));}
void set_friend(int i,int j){_run_both;p[i]=(i==j?0:j);}
int is_friend(int i,int j){_run_both;return i==j&&i;}
};

//带路径压缩的并查集扩展形式
//用于动态维护查询friend-enemy型等价类
//维护和查询复杂度略大于 O(1)
//集合元素取值 1..MAXN-1(注意0不能用!),默认无关
#include

#define MAXN 100000
#define sig(x) ((x)>0?1:-1) 59
#define abs(x) ((x)>0?(x):-(x))
#define _ufind_run(x)
for(;p[t=abs(x)];x=sig(x)*p[abs(x)],p[t]=sig(p[t])*(p[abs(x)]?p[abs(x)]:abs(p[t])))
#define _run_both _ufind_run(i);_ufind_run(j)
#define _set_side(x) p[abs(i)]=sig(i)*(abs(i)==abs(j)?0:(x)*j)
#define _judge_side(x) (i==(x)*j&&i)

struct ufind{
int p[MAXN],t;
void init(){memset(p,0,sizeof(p));}
int set_friend(int i,int j){_run_both;_set_side(1);return !_judge_side(-1);}
int set_enemy(int i,int j){_run_both;_set_side(-1);return !_judge_side(1);}
int is_friend(int i,int j){_run_both;return _judge_side(1);}
int is_enemy(int i,int j){_run_both;return _judge_side(-1);}
};
3.2 堆
//二分堆(binary)
//可插入,获取并删除最小(最大)元素,复杂度均O(logn)
//可更改元素类型,修改比较符号或换成比较函数
#define MAXN 10000
#define _cp(a,b) ((a)<(b))
typedef int elem_t;

struct heap{
elem_t h[MAXN];
int n,p,c;
void init(){n=0;}
void ins(elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(elem_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

//映射二分堆(mapped)
//可插入,获取并删除任意元素,复杂度均O(logn)
//插入时提供一个索引值,删除时按该索引删除,获取并删除最小元素时一起获得该索引 60
//索引值范围 0..MAXN-1,不能重复,不负责维护索引的唯一性,不在此返回请另外映射
//主要用于图论算法,该索引值可以是节点的下标
//可更改元素类型,修改比较符号或换成比较函数
#define MAXN 10000
#define _cp(a,b) ((a)<(b))
typedef int elem_t;

struct heap{
elem_t h[MAXN];
int ind[MAXN],map[MAXN],n,p,c;
void init(){n=0;}
void ins(int i,elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
h[map[ind[p]=i]=p]=e;
}
int del(int i,elem_t& e){
i=map[i];if (i<1||i>n) return 0;
for (e=h[p=i];p>1;h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
for
(c=2;c =1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
int delmin(int& i,elem_t& e){
if (n<1) return 0;i=ind[1];
for
(e=h[p=1],c=2;c ],p=c,c<<=1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
};
3.3 线段树
线段树应用：
求面积:
1) 坐标离散化
2) 垂直边按x坐标排序
3) 从左往右用线段树处理垂直边
累计每个离散x 区间长度和线段树长度的乘积
求周长:
1) 坐标离散化
2) 垂直边按x 坐标排序, 第二关键字为入边优于出边
3) 从左往右用线段树处理垂直边 61
在每个离散点上先加入所有入边, 累计线段树长度变化值
再删除所有出边, 累计线段树长度变化值
4) 水平边按y坐标排序, 第二关键字为入边优于出边
5) 从上往下用线段树处理水平边
在每个离散点上先加入所有入边, 累计线段树长度变化值
再删除所有出边, 累计线段树长度变化值

//线段树
//可以处理加入边和删除边不同的情况
//inc_seg和 dec_seg用于加入边
//seg_len求长度
//t传根节点(一律为1)
//l0,r0 传树的节点范围(一律为1..t)
//l,r 传线段(端点)
#define MAXN 10000
struct segtree{
int n,cnt[MAXN],len[MAXN];
segtree(int t):n(t){
for (int i=1;i<=t;i++)
cnt[i]=len[i]=0;
};
void update(int t,int l,int r);
void inc_seg(int t,int l0,int r0,int l,int r);
void dec_seg(int t,int l0,int r0,int l,int r);
int seg_len(int t,int l0,int r0,int l,int r);
};

int length(int l,int r){
return r-l;
}

void segtree::update(int t,int l,int r){
if (cnt[t]||r-l==1)
len[t]=length(l,r);
else
len[t]=len[t+t]+len[t+t+1];
}

void segtree::inc_seg(int t,int l0,int r0,int l,int r){
if (l0==l&&r0==r)
cnt[t]++;
else{
int m0=(l0+r0)>>1;
if (l inc_seg(t+t,l0,m0,l,m0 if (r>m0)
inc_seg(t+t+1,m0,r0,m0>l?m0:l,r);
if (cnt[t+t]&&cnt[t+t+1]){
cnt[t+t]--;
update(t+t,l0,m0);
cnt[t+t+1]--;
update(t+t+1,m0,r0);
cnt[t]++;
}
}
update(t,l0,r0);
}

void segtree::dec_seg(int t,int l0,int r0,int l,int r){
if (l0==l&&r0==r)
cnt[t]--;
else if (cnt[t]){
cnt[t]--;
if (l>l0)
inc_seg(t,l0,r0,l0,l);
if (r inc_seg(t,l0,r0,r,r0);
}
else{
int m0=(l0+r0)>>1;
if (l dec_seg(t+t,l0,m0,l,m0 if (r>m0)
dec_seg(t+t+1,m0,r0,m0>l?m0:l,r);
}
update(t,l0,r0);
}

int segtree::seg_len(int t,int l0,int r0,int l,int r){
if (cnt[t]||(l0==l&&r0==r))
return len[t];
else{
int m0=(l0+r0)>>1,ret=0;
if (l ret+=seg_len(t+t,l0,m0,l,m0 if (r>m0)
ret+=seg_len(t+t+1,m0,r0,m0>l?m0:l,r);
return ret; 63
}
}

//线段树扩展
//可以计算长度和线段数
//可以处理加入边和删除边不同的情况
//inc_seg和 dec_seg用于加入边
//seg_len求长度,seg_cut求线段数
//t传根节点(一律为1)
//l0,r0 传树的节点范围(一律为1..t)
//l,r 传线段(端点)
#define MAXN 10000
struct segtree{
int n,cnt[MAXN],len[MAXN],cut[MAXN],bl[MAXN],br[MAXN];
segtree(int t):n(t){
for (int i=1;i<=t;i++)
cnt[i]=len[i]=cut[i]=bl[i]=br[i]=0;
};
void update(int t,int l,int r);
void inc_seg(int t,int l0,int r0,int l,int r);
void dec_seg(int t,int l0,int r0,int l,int r);
int seg_len(int t,int l0,int r0,int l,int r);
int seg_cut(int t,int l0,int r0,int l,int r);
};

int length(int l,int r){
return r-l;
}

void segtree::update(int t,int l,int r){
if (cnt[t]||r-l==1)
len[t]=length(l,r),cut[t]=bl[t]=br[t]=1;
else{
len[t]=len[t+t]+len[t+t+1];
cut[t]=cut[t+t]+cut[t+t+1];
if (br[t+t]&&bl[t+t+1])
cut[t]--;
bl[t]=bl[t+t],br[t]=br[t+t+1];
}
}

void segtree::inc_seg(int t,int l0,int r0,int l,int r){
if (l0==l&&r0==r)
cnt[t]++; 64
else{
int m0=(l0+r0)>>1;
if (l inc_seg(t+t,l0,m0,l,m0 if (r>m0)
inc_seg(t+t+1,m0,r0,m0>l?m0:l,r);
if (cnt[t+t]&&cnt[t+t+1]){
cnt[t+t]--;
update(t+t,l0,m0);
cnt[t+t+1]--;
update(t+t+1,m0,r0);
cnt[t]++;
}
}
update(t,l0,r0);
}

void segtree::dec_seg(int t,int l0,int r0,int l,int r){
if (l0==l&&r0==r)
cnt[t]--;
else if (cnt[t]){
cnt[t]--;
if (l>l0)
inc_seg(t,l0,r0,l0,l);
if (r inc_seg(t,l0,r0,r,r0);
}
else{
int m0=(l0+r0)>>1;
if (l dec_seg(t+t,l0,m0,l,m0 if (r>m0)
dec_seg(t+t+1,m0,r0,m0>l?m0:l,r);
}
update(t,l0,r0);
}

int segtree::seg_len(int t,int l0,int r0,int l,int r){
if (cnt[t]||(l0==l&&r0==r))
return len[t];
else{
int m0=(l0+r0)>>1,ret=0;
if (l ret+=seg_len(t+t,l0,m0,l,m0 if (r>m0)
ret+=seg_len(t+t+1,m0,r0,m0>l?m0:l,r);
return ret;
}
}

int segtree::seg_cut(int t,int l0,int r0,int l,int r){
if (cnt[t])
return 1;
if (l0==l&&r0==r)
return cut[t];
else{
int m0=(l0+r0)>>1,ret=0;
if (l ret+=seg_cut(t+t,l0,m0,l,m0 if (r>m0)
ret+=seg_cut(t+t+1,m0,r0,m0>l?m0:l,r);
if (lm0&&br[t+t]&&bl[t+t+1])
ret--;
return ret;
}
}
3.4 子段和
//求 sum{[0..n-1]}
//维护和查询复杂度均为O(logn)
//用于动态求子段和,数组内容保存在 sum.a[]中
//可以改成其他数据类型
#include
#define lowbit(x) ((x)&((x)^((x)-1)))
#define MAXN 10000
typedef int elem_t;

struct sum{
elem_t a[MAXN],c[MAXN],ret;
int n;
void init(int i){memset(a,0,sizeof(a));memset(c,0,sizeof(c));n=i;}
void update(int i,elem_t v){for (v-=a[i],a[i++]+=v;i<=n;c[i-1]+=v,i+=lowbit(i));}
elem_t query(int i){for (ret=0;i;ret+=c[i-1],i^=lowbit(i));return ret;}
};
3.5 子阵和 66
//求 sum{a[0..m-1][0..n-1]}
//维护和查询复杂度均为O(logm*logn)
//用于动态求子阵和,数组内容保存在 sum.a[][]中
//可以改成其他数据类型
#include
#define lowbit(x) ((x)&((x)^((x)-1)))
#define MAXN 100
typedef int elem_t;

struct sum{
elem_t a[MAXN][MAXN],c[MAXN][MAXN],ret;
int m,n,t;
void init(int i,int j){memset(a,0,sizeof(a));memset(c,0,sizeof(c));m=i,n=j;}
void update(int i,int j,elem_t v){
for (v-=a[i][j],a[i++][j++]+=v,t=j;i<=m;i+=lowbit(i))
for (j=t;j<=n;c[i-1][j-1]+=v,j+=lowbit(j));
}
elem_t query(int i,int j){
for (ret=0,t=j;i;i^=lowbit(i))
for (j=t;j;ret+=c[i-1][j-1],j^=lowbit(j));
return ret;
}
};
4、 数论
4.1 阶乘最后非 0位
//求阶乘最后非零位,复杂度O(nlogn)
//返回该位,n以字符串方式传入
#include
#define MAXN 10000

int lastdigit(char* buf){
const int mod[20]={1,1,2,6,4,2,2,4,2,8,4,4,8,4,6,8,8,6,8,2};
int len=strlen(buf),a[MAXN],i,c,ret=1;
if (len==1)
return mod[buf[0]-'0'];
for (i=0;i a[i]=buf[len-1-i]-'0';
for (;len;len-=!a[len-1]){
ret=ret*mod[a[1]%2*10+a[0]]%5;
for (c=0,i=len-1;i>=0;i--) 67
c=c*10+a[i],a[i]=c/5,c%=5;
}
return ret+ret%2*5;
}
4.2 模线性方程组
#ifdef WIN32
typedef __int64 i64;
#else
typedef long long i64;
#endif
//扩展 Euclid求解gcd(a,b)=ax+by
int ext_gcd(int a,int b,int& x,int& y){
int t,ret;
if (!b){
x=1,y=0;
return a;
}
ret=ext_gcd(b,a%b,x,y);
t=x,x=y,y=t-a/b*y;
return ret;
}

//计算 m^a, O(loga), 本身没什么用, 注意这个按位处理的方法 :-P
int exponent(int m,int a){
int ret=1;
for (;a;a>>=1,m*=m)
if (a&1)
ret*=m;
return ret;
}

//计算幂取模a^b mod n, O(logb)
int modular_exponent(int a,int b,int n){ //a^b mod n
int ret=1;
for (;b;b>>=1,a=(int)((i64)a)*a%n)
if (b&1)
ret=(int)((i64)ret)*a%n;
return ret;
}

//求解模线性方程ax=b (mod n)
//返回解的个数,解保存在 sol[]中 68
//要求 n>0,解的范围0..n-1
int modular_linear(int a,int b,int n,int* sol){
int d,e,x,y,i;
d=ext_gcd(a,n,x,y);
if (b%d)
return 0;
e=(x*(b/d)%n+n)%n;
for (i=0;i sol[i]=(e+i*(n/d))%n;
return d;
}

//求解模线性方程组(中国余数定理)
// x = b[0] (mod w[0])
// x = b[1] (mod w[1])
// ...
// x = b[k-1] (mod w[k-1])
//要求 w[i]>0,w[i]与w[j]互质,解的范围1..n,n=w[0]*w[1]*...*w[k-1]
int modular_linear_system(int b[],int w[],int k){
int d,x,y,a=0,m,n=1,i;
for (i=0;i n*=w[i];
for (i=0;i m=n/w[i];
d=ext_gcd(w[i],m,x,y);
a=(a+y*m*b[i])%n;
}
return (a+n)%n;
}
4.3 素数
//用素数表判定素数,先调用initprime
int plist[10000],pcount=0;

int prime(int n){
int i;
if ((n!=2&&!(n%2))||(n!=3&&!(n%3))||(n!=5&&!(n%5))||(n!=7&&!(n%7)))
return 0;
for (i=0;plist[i]*plist[i]<=n;i++)
if (!(n%plist[i]))
return 0;
return n>1;
} 69

void initprime(){
int i;
for (plist[pcount++]=2,i=3;i<50000;i++)
if (prime(i))
plist[pcount++]=i;
}

//miller rabin
//判断自然数 n是否为素数
//time越高失败概率越低,一般取10到 50
#include
#ifdef WIN32
typedef __int64 i64;
#else
typedef long long i64;
#endif

int modular_exponent(int a,int b,int n){ //a^b mod n
int ret;
for (;b;b>>=1,a=(int)((i64)a)*a%n)
if (b&1)
ret=(int)((i64)ret)*a%n;
return ret;
}

// Carmicheal number: 561,41041,825265,321197185
int miller_rabin(int n,int time=10){
if (n==1||(n!=2&&!(n%2))||(n!=3&&!(n%3))||(n!=5&&!(n%5))||(n!=7&&!(n%7)))
return 0;
while (time--)
if
(modular_exponent(((rand()&0x7fff<<16)+rand()&0x7fff+rand()&0x7fff)%(n-1)+1,n-1,n)!=1)
return 0;
return 1;
}
4.4 欧拉函数
int gcd(int a,int b){
return b?gcd(b,a%b):a;
}

inline int lcm(int a,int b){ 70
return a/gcd(a,b)*b;
}

//求 1..n-1 中与 n互质的数的个数
int eular(int n){
int ret=1,i;
for (i=2;i*i<=n;i++)
if (n%i==0){
n/=i,ret*=i-1;
while (n%i==0)
n/=i,ret*=i;
}
if (n>1)
ret*=n-1;
return ret;
}
5、 数值计算
5.1 定积分计算(Romberg)
/* Romberg求定积分
输入：积分区间[a,b]，被积函数f(x,y,z)
输出：积分结果

f(x,y,z)示例：
double f0( double x, double l, double t )
{
return sqrt(1.0+l*l*t*t*cos(t*x)*cos(t*x));
}
*/

double Integral(double a, double b, double (*f)(double x, double y, double z), double eps,
double l, double t)

double Romberg (double a, double b, double (*f)(double x, double y, double z), double eps,
double l, double t)
{
#define MAX_N 1000

int i, j, temp2, min;
double h, R[2][MAX_N], temp4;
71
for (i=0; i R[0][i] = 0.0;
R[1][i] = 0.0;
}
h = b-a;
min = (int)(log(h*10.0)/log(2.0)); //h should be at most 0.1
R[0][0] = ((*f)(a, l, t)+(*f)(b, l, t))*h*0.50;
i = 1;
temp2 = 1;
while (i i++;
R[1][0] = 0.0;
for (j=1; j<=temp2; j++)
R[1][0] += (*f)(a+h*((double)j-0.50), l, t);
R[1][0] = (R[0][0] + h*R[1][0])*0.50;
temp4 = 4.0;
for (j=1; j R[1][j] = R[1][j-1] + (R[1][j-1]-R[0][j-1])/(temp4-1.0);
temp4 *= 4.0;
}
if ((fabs(R[1][i-1]-R[0][i-2])min))
return R[1][i-1];
h *= 0.50;
temp2 *= 2;
for (j=0; j R[0][j] = R[1][j];
}
return R[1][MAX_N-1];
}

double Integral(double a, double b, double (*f)(double x, double y, double z), double eps,
double l, double t)
{
#define pi 3.1415926535897932

int n;
double R, p, res;

n = (int)(floor)(b * t * 0.50 / pi);
p = 2.0 * pi / t;
res = b - (double)n * p;
if (n)
R = Romberg (a, p, f0, eps/(double)n, l, t);
R = R * (double)n + Romberg( 0.0, res, f0, eps, l, t ); 72

return R/100.0;
}
5.2 多项式求根(牛顿法)
/* 牛顿法解多项式的根
输入：多项式系数 c[]，多项式度数 n，求在[a,b]间的根
输出：根
要求保证[a,b]间有根
*/

double fabs( double x )
{
return (x<0)? -x : x;
}

double f(int m, double c[], double x)
{
int i;
double p = c[m];

for (i=m; i>0; i--)
p = p*x + c[i-1];
return p;
}

int newton(double x0, double *r,
double c[], double cp[], int n,
double a, double b, double eps)
{
int MAX_ITERATION = 1000;
int i = 1;
double x1, x2, fp, eps2 = eps/10.0;

x1 = x0;
while (i < MAX_ITERATION) {
x2 = f(n, c, x1);
fp = f(n-1, cp, x1);
if ((fabs(fp)<0.000000001) && (fabs(x2)>1.0))
return 0;
x2 = x1 - x2/fp;
if (fabs(x1-x2) if (x2b) 73
return 0;
*r = x2;
return 1;
}
x1 = x2;
i++;
}
return 0;
}

double Polynomial_Root(double c[], int n, double a, double b, double eps)
{
double *cp;
int i;
double root;

cp = (double *)calloc(n, sizeof(double));
for (i=n-1; i>=0; i--) {
cp[i] = (i+1)*c[i+1];
}
if (a>b) {
root = a; a = b; b = root;
}
if ((!newton(a, &root, c, cp, n, a, b, eps)) &&
(!newton(b, &root, c, cp, n, a, b, eps)))
newton((a+b)*0.5, &root, c, cp, n, a, b, eps);
free(cp);
if (fabs(root) return fabs(root);
else
return root;
}
5.3 周期性方程(追赶法)
/* 追赶法解周期性方程
周期性方程定义：| a1 b1 c1 ... | = x1
| a2 b2 c2 ... | = x2
| ... | * X = ...
| cn-1 ... an-1 bn-1 | = xn-1
| bn cn an | = xn
输入：a[],b[],c[],x[]
输出：求解结果X 在 x[]中
*/ 74

void run()
{
c[0] /= b[0]; a[0] /= b[0]; x[0] /= b[0];
for (int i = 1; i < N - 1; i ++) {
double temp = b[i] - a[i] * c[i - 1];
c[i] /= temp;
x[i] = (x[i] - a[i] * x[i - 1]) / temp;
a[i] = -a[i] * a[i - 1] / temp;
}
a[N - 2] = -a[N - 2] - c[N - 2];
for (int i = N - 3; i >= 0; i --) {
a[i] = -a[i] - c[i] * a[i + 1];
x[i] -= c[i] * x[i + 1];
}
x[N - 1] -= (c[N - 1] * x[0] + a[N - 1] * x[N - 2]);
x[N - 1] /= (c[N - 1] * a[0] + a[N - 1] * a[N - 2] + b[N - 1]);
for (int i = N - 2; i >= 0; i --)
x[i] += a[i] * x[N - 1];
}
6、 图论—NP搜索
6.1 最大团
//最大团
//返回最大团大小和一个方案,传入图的大小n和邻接阵 mat
//mat[i][j]为布尔量

#define MAXN 60

void clique(int n, int* u, int mat[][MAXN], int size, int& max, int& bb, int* res, int* rr, int* c) {
int i, j, vn, v[MAXN];
if (n) {
if (size + c[u[0]] <= max) return;
for (i = 0; i < n + size - max && i < n; ++ i) {
for (j = i + 1, vn = 0; j < n; ++ j)
if (mat[u[i]][u[j]])
v[vn ++] = u[j];
rr[size] = u[i];
clique(vn, v, mat, size + 1, max, bb, res, rr, c);
if (bb) return;
} 75
} else if (size > max) {
max = size;
for (i = 0; i < size; ++ i)
res[i] = rr[i];
bb = 1;
}
}

int maxclique(int n, int mat[][MAXN], int *ret) {
int max = 0, bb, c[MAXN], i, j;
int vn, v[MAXN], rr[MAXN];
for (c[i = n - 1] = 0; i >= 0; -- i) {
for (vn = 0, j = i + 1; j < n; ++ j)
if (mat[i][j])
v[vn ++] = j;
bb = 0;
rr[0] = i;
clique(vn, v, mat, 1, max, bb, ret, rr, c);
c[i] = max;
}
return max;
}
6.2 最大团(n<64)(faster)
/**
* WishingBone's ACM/ICPC Routine Library
*
* maximum clique solver
*/

#include

using std::vector;

// clique solver calculates both size and consitution of maximum clique
// uses bit operation to accelerate searching
// graph size limit is 63, the graph should be undirected
// can optimize to calculate on each component, and sort on vertex degrees
// can be used to solve maximum independent set
class clique {
public:
static const long long ONE = 1;
static const long long MASK = (1 << 21) - 1; 76
char* bits;
int n, size, cmax[63];
long long mask[63], cons;
// initiate lookup table
clique() {
bits = new char[1 << 21];
bits[0] = 0;
for (int i = 1; i < 1 << 21; ++i) bits[i] = bits[i >> 1] + (i & 1);
}
~clique() {
delete bits;
}
// search routine
bool search(int step, int size, long long more, long long con);
// solve maximum clique and return size
int sizeClique(vector >& mat);
// solve maximum clique and return constitution
vector consClique(vector >& mat);
};

// search routine
// step is node id, size is current solution, more is available mask, cons is
constitution mask
bool clique::search(int step, int size, long long more, long long cons) {
if (step >= n) {
// a new solution reached
this->size = size;
this->cons = cons;
return true;
}
long long now = ONE << step;
if ((now & more) > 0) {
long long next = more & mask[step];
if (size + bits[next & MASK] + bits[(next >> 21) & MASK] + bits[next >>
42] >= this->size
&& size + cmax[step] > this->size) {
// the current node is in the clique
if (search(step + 1, size + 1, next, cons | now)) return true;
}
}
long long next = more & ~now;
if (size + bits[next & MASK] + bits[(next >> 21) & MASK] + bits[next >> 42]
> this->size) {
// the current node is not in the clique 77
if (search(step + 1, size, next, cons)) return true;
}
return false;
}

// solve maximum clique and return size
int clique::sizeClique(vector >& mat) {
n = mat.size();
// generate mask vectors
for (int i = 0; i < n; ++i) {
mask[i] = 0;
for (int j = 0; j < n; ++j) if (mat[i][j] > 0) mask[i] |= ONE << j;
}
size = 0;
for (int i = n - 1; i >= 0; --i) {
search(i + 1, 1, mask[i], ONE << i);
cmax[i] = size;
}
return size;
}

// solve maximum clique and return constitution
// calls sizeClique and restore cons
vector clique::consClique(vector >& mat) {
sizeClique(mat);
vector ret;
for (int i = 0; i < n; ++i) if ((cons & (ONE << i)) > 0) ret.push_back(i);
return ret;
}
7、 图论—连通性
7.1 无向图关键点(dfs邻接阵)
//无向图的关键点,dfs邻接阵形式,O(n^2)
//返回关键点个数,key[]返回点集
//传入图的大小 n和邻接阵mat,不相邻点边权 0
#define MAXN 110

void search(int n,int mat[][MAXN],int* dfn,int* low,int now,int& ret,int* key,int& cnt,int
root,int& rd,int* bb){
int i;
dfn[now]=low[now]=++cnt; 78
for (i=0;i if (mat[now][i]){
if (!dfn[i]){
search(n,mat,dfn,low,i,ret,key,cnt,root,rd,bb);
if (low[i] low[now]=low[i];
if (low[i]>=dfn[now]){
if (now!=root&&!bb[now])
key[ret++]=now,bb[now]=1;
else if(now==root)
rd++;
}
}
else if (dfn[i] low[now]=dfn[i];
}
}

int key_vertex(int n,int mat[][MAXN],int* key){
int ret=0,i,cnt,rd,dfn[MAXN],low[MAXN],bb[MAXN];
for (i=0;i for (cnt=i=0;i if (!dfn[i]){
rd=0;
search(n,mat,dfn,low,i,ret,key,cnt,i,rd,bb);
if (rd>1&&!bb[i])
key[ret++]=i,bb[i]=1;
}
return ret;
}
7.2 无向图关键边(dfs邻接阵)
//无向图的关键边,dfs邻接阵形式,O(n^2)
//返回关键边条数,key[][2]返回边集
//传入图的大小 n和邻接阵mat,不相邻点边权 0
#define MAXN 100

void search(int n,int mat[][MAXN],int* dfn,int* low,int now,int& cnt,int key[][2]){
int i;
for (low[now]=dfn[now],i=0;i if (mat[now][i]){
if (!dfn[i]){
dfn[i]=dfn[now]+1; 79
search(n,mat,dfn,low,i,cnt,key);
if (low[i]>dfn[now])
key[cnt][0]=i,key[cnt++][1]=now;
if (low[i] low[now]=low[i];
}
else if (dfn[i] low[now]=lev[i];
}
}

int key_edge(int n,int mat[][MAXN],int key[][2]){
int ret=0,i,dfn[MAXN],low[MAXN];
for (i=0;i for (i=0;i if (!dfn[i])
dfn[i]=1,bridge(n,mat,dfn,low,i,ret,key);
return ret;
}
7.3 无向图的块(bfs邻接阵)
//无向图的块,dfs邻接阵形式,O(n^2)
//每产生一个块调用dummy
//传入图的大小n和邻接阵mat,不相邻点边权0
#define MAXN 100

#include
void dummy(int n,int* a){
for (int i=0;i cout< cout< }

void search(int n,int mat[][MAXN],int* dfn,int* low,int now,int& cnt,int* st,int& sp){
int i,m,a[MAXN];
dfn[st[sp++]=now]=low[now]=++cnt;
for (i=0;i if (mat[now][i]){
if (!dfn[i]){
search(n,mat,dfn,low,i,cnt,st,sp);
if (low[i] low[now]=low[i];
if (low[i]>=dfn[now]){ 80
for (st[sp]=-1,a[0]=now,m=1;st[sp]!=i;a[m++]=st[--sp]);
dummy(m,a);
}
}
else if (dfn[i] low[now]=dfn[i];
}
}

void block(int n,int mat[][MAXN]){
int i,cnt,dfn[MAXN],low[MAXN],st[MAXN],sp=0;
for (i=0;i for (cnt=i=0;i if (!dfn[i])
search(n,mat,dfn,low,i,cnt,st,sp);
}
7.4 无向图连通分支(dfs/bfs邻接阵)
//无向图连通分支,dfs邻接阵形式,O(n^2)
//返回分支数,id返回1..分支数的值
//传入图的大小 n和邻接阵mat,不相邻点边权0
#define MAXN 100

void floodfill(int n,int mat[][MAXN],int* id,int now,int tag){
int i;
for (id[now]=tag,i=0;i if (!id[i]&&mat[now][i])
floodfill(n,mat,id,i,tag);
}

int find_components(int n,int mat[][MAXN],int* id){
int ret,i;
for (i=0;i for (ret=i=0;i if (!id[i])
floodfill(n,mat,id,i,++ret);
return ret;
}

//无向图连通分支,bfs邻接阵形式,O(n^2)
//返回分支数,id返回 1..分支数的值
//传入图的大小 n和邻接阵mat,不相邻点边权0
#define MAXN 100 81

int find_components(int n,int mat[][MAXN],int* id){
int ret,k,i,j,m;
for (k=0;k for (ret=k=0;k if (!id[k])
for (id[k]=-1,ret++,m=1;m;)
for (m=i=0;i if (id[i]==-1)
for (m++,id[i]=ret,j=0;j if (!id[j]&&mat[i][j])
id[j]=-1;
return ret;
}
7.5 有向图强连通分支(dfs/bfs邻接阵)
//有向图强连通分支,dfs邻接阵形式,O(n^2)
//返回分支数,id返回1..分支数的值
//传入图的大小 n和邻接阵mat,不相邻点边权0
#define MAXN 100

void search(int n,int mat[][MAXN],int* dfn,int* low,int now,int& cnt,int& tag,int* id,int* st,int&
sp){
int i,j;
dfn[st[sp++]=now]=low[now]=++cnt;
for (i=0;i if (mat[now][i]){
if (!dfn[i]){
ssearch(n,mat,dfn,low,i,cnt,tag,id,st,sp);
if (low[i] low[now]=low[i];
}
else if (dfn[i] for (j=0;j if (j low[now]=dfn[i];
}
}
if (low[now]==dfn[now])
for (tag++;st[sp]!=now;id[st[--sp]]=tag);
}
int find_components(int n,int mat[][MAXN],int* id){
int ret=0,i,cnt,sp,st[MAXN],dfn[MAXN],low[MAXN]; 82
for (i=0;i for (sp=cnt=i=0;i if (!dfn[i])
search(n,mat,dfn,low,i,cnt,ret,id,st,sp);
return ret;
}
//有向图强连通分支,bfs邻接阵形式,O(n^2)
//返回分支数,id返回 1..分支数的值
//传入图的大小 n和邻接阵mat,不相邻点边权0
#define MAXN 100

int find_components(int n,int mat[][MAXN],int* id){
int ret=0,a[MAXN],b[MAXN],c[MAXN],d[MAXN],i,j,k,t;
for (k=0;k for (k=0;k if (!id[k]){
for (i=0;i a[i]=b[i]=c[i]=d[i]=0;
a[k]=b[k]=1;
for (t=1;t;)
for (t=i=0;i if (a[i]&&!c[i])
for (c[i]=t=1,j=0;j if (mat[i][j]&&!a[j])
a[j]=1;
if (b[i]&&!d[i])
for (d[i]=t=1,j=0;j if (mat[j][i]&&!b[j])
b[j]=1;
}
for (ret++,i=0;i if (a[i]&b[i])
id[i]=ret;
}
return ret;
}
7.6 有向图最小点基(邻接阵)
//有向图最小点基,邻接阵形式,O(n^2)
//返回电集大小和点集
//传入图的大小 n和邻接阵mat,不相邻点边权0
//需要调用强连通分支
#define MAXN 100 83

int base_vertex(int n,int mat[][MAXN],int* sets){
int ret=0,id[MAXN],v[MAXN],i,j;
j=find_components(n,mat,id);
for (i=0;i for (i=0;i for (j=0;j if (id[i]!=id[j]&&mat[i][j])
v[id[j]-1]=0;
for (i=0;i if (v[id[i]-1])
v[id[sets[ret++]=i]-1]=0;
return ret;
}
8、 图论—匹配
8.1 二分图最大匹配(hungary邻接表)
//二分图最大匹配,hungary算法,邻接表形式,复杂度O(m*e)
//返回最大匹配数,传入二分图大小m,n和邻接表 list(只需一边)
//match1,match2返回一个最大匹配,未匹配顶点match值为-1
#include
#define MAXN 310
#define _clr(x) memset(x,0xff,sizeof(int)*MAXN)
struct edge_t{
int from,to;
edge_t* next;
};

int hungary(int m,int n,edge_t* list[],int* match1,int* match2){
int s[MAXN],t[MAXN],p,q,ret=0,i,j,k;edge_t* e;
for (_clr(match1),_clr(match2),i=0;i=0))
for (_clr(t),s[p=q=0]=i;p<=q&&match1[i]<0;p++)
for (e=list[k=s[p]];e&&match1[i]<0;e=e->next)
if (t[j=e->to]<0){
s[++q]=match2[j],t[j]=k;
if (s[q]<0)
for (p=j;p>=0;j=p)
match2[j]=k=t[j],p=match1[k],match1[k]=j;
}
return ret;
} 84
8.2 二分图最大匹配(hungary邻接阵)
//二分图最大匹配,hungary算法,邻接阵形式,复杂度O(m*m*n)
//返回最大匹配数,传入二分图大小m,n和邻接阵 mat,非零元素表示有边
//match1,match2返回一个最大匹配,未匹配顶点match值为-1
#include
#define MAXN 310
#define _clr(x) memset(x,0xff,sizeof(int)*MAXN)

int hungary(int m,int n,int mat[][MAXN],int* match1,int* match2){
int s[MAXN],t[MAXN],p,q,ret=0,i,j,k;
for (_clr(match1),_clr(match2),i=0;i=0))
for (_clr(t),s[p=q=0]=i;p<=q&&match1[i]<0;p++)
for (k=s[p],j=0;j if (mat[k][j]&&t[j]<0){
s[++q]=match2[j],t[j]=k;
if (s[q]<0)
for (p=j;p>=0;j=p)
match2[j]=k=t[j],p=match1[k],match1[k]=j;
}
return ret;
}
8.3 二分图最大匹配(hungary正向表)
//二分图最大匹配,hungary算法,正向表形式,复杂度O(m*e)
//返回最大匹配数,传入二分图大小m,n和正向表 list,buf(只需一边)
//match1,match2返回一个最大匹配,未匹配顶点match值为-1
#include
#define MAXN 310
#define _clr(x) memset(x,0xff,sizeof(int)*MAXN)

int hungary(int m,int n,int* list,int* buf,int* match1,int* match2){
int s[MAXN],t[MAXN],p,q,ret=0,i,j,k,l;
for (_clr(match1),_clr(match2),i=0;i=0))
for (_clr(t),s[p=q=0]=i;p<=q&&match1[i]<0;p++)
for (l=list[k=s[p]];l if (t[j=buf[l]]<0){
s[++q]=match2[j],t[j]=k;
if (s[q]<0)
for (p=j;p>=0;j=p)
match2[j]=k=t[j],p=match1[k],match1[k]=j;
} 85
return ret;
}
8.4二分图最佳匹配(kuhn_munkras 邻接阵)
//二分图最佳匹配,kuhn munkras算法,邻接阵形式,复杂度O(m*m*n)
//返回最佳匹配值,传入二分图大小m,n和邻接阵 mat,表示权值
//match1,match2返回一个最佳匹配,未匹配顶点match值为-1
//一定注意 m<=n,否则循环无法终止
//最小权匹配可将权值取相反数
#include
#define MAXN 310
#define inf 1000000000
#define _clr(x) memset(x,0xff,sizeof(int)*n)

int kuhn_munkras(int m,int n,int mat[][MAXN],int* match1,int* match2){
int s[MAXN],t[MAXN],l1[MAXN],l2[MAXN],p,q,ret=0,i,j,k;
for (i=0;i for (l1[i]=-inf,j=0;j l1[i]=mat[i][j]>l1[i]?mat[i][j]:l1[i];
for (i=0;i for (_clr(match1),_clr(match2),i=0;i for (_clr(t),s[p=q=0]=i;p<=q&&match1[i]<0;p++)
for (k=s[p],j=0;j if (l1[k]+l2[j]==mat[k][j]&&t[j]<0){
s[++q]=match2[j],t[j]=k;
if (s[q]<0)
for (p=j;p>=0;j=p)
match2[j]=k=t[j],p=match1[k],match1[k]=j;
}
if (match1[i]<0){
for (i--,p=inf,k=0;k<=q;k++)
for (j=0;j if (t[j]<0&&l1[s[k]]+l2[j]-mat[s[k]][j] p=l1[s[k]]+l2[j]-mat[s[k]][j];
for (j=0;j for (k=0;k<=q;l1[s[k++]]-=p);
}
}
for (i=0;i ret+=mat[i][match1[i]];
return ret;
} 86
8.5 一般图匹配(邻接表)
//一般图最大匹配,邻接表形式,复杂度O(n*e)
//返回匹配顶点对数,match返回匹配,未匹配顶点 match值为-1
//传入图的顶点数n和邻接表 list
#define MAXN 100
struct edge_t{
int from,to;
edge_t* next;
};

int aug(int n,edge_t* list[],int* match,int* v,int now){
int t,ret=0;edge_t* e;
v[now]=1;
for (e=list[now];e;e=e->next)
if (!v[t=e->to]){
if (match[t]<0)
match[now]=t,match[t]=now,ret=1;
else{
v[t]=1;
if (aug(n,list,match,v,match[t]))
match[now]=t,match[t]=now,ret=1;
v[t]=0;
}
if (ret)
break;
}
v[now]=0;
return ret;
}

int graph_match(int n,edge_t* list[],int* match){
int v[MAXN],i,j;
for (i=0;i v[i]=0,match[i]=-1;
for (i=0,j=n;i=2;)
if (match[i]<0&&aug(n,list,match,v,i))
i=0,j-=2;
else
i++;
for (i=j=0;i j+=(match[i]>=0);
return j/2;
} 87
8.6 一般图匹配(邻接阵)
//一般图最大匹配,邻接阵形式,复杂度O(n^3)
//返回匹配顶点对数,match返回匹配,未匹配顶点 match值为-1
//传入图的顶点数n和邻接阵 mat
#define MAXN 100

int aug(int n,int mat[][MAXN],int* match,int* v,int now){
int i,ret=0;
v[now]=1;
for (i=0;i if (!v[i]&&mat[now][i]){
if (match[i]<0)
match[now]=i,match[i]=now,ret=1;
else{
v[i]=1;
if (aug(n,mat,match,v,match[i]))
match[now]=i,match[i]=now,ret=1;
v[i]=0;
}
if (ret)
break;
}
v[now]=0;
return ret;
}

int graph_match(int n,int mat[][MAXN],int* match){
int v[MAXN],i,j;
for (i=0;i v[i]=0,match[i]=-1;
for (i=0,j=n;i=2;)
if (match[i]<0&&aug(n,mat,match,v,i))
i=0,j-=2;
else
i++;
for (i=j=0;i j+=(match[i]>=0);
return j/2;
}
8.7 一般图匹配(正向表) 88
//一般图最大匹配,正向表形式,复杂度O(n*e)
//返回匹配顶点对数,match返回匹配,未匹配顶点 match值为-1
//传入图的顶点数n和正向表 list,buf
#define MAXN 100

int aug(int n,int* list,int* buf,int* match,int* v,int now){
int i,t,ret=0;
v[now]=1;
for (i=list[now];i if (!v[t=buf[i]]){
if (match[t]<0)
match[now]=t,match[t]=now,ret=1;
else{
v[t]=1;
if (aug(n,list,buf,match,v,match[t]))
match[now]=t,match[t]=now,ret=1;
v[t]=0;
}
if (ret)
break;
}
v[now]=0;
return ret;
}

int graph_match(int n,int* list,int* buf,int* match){
int v[MAXN],i,j;
for (i=0;i v[i]=0,match[i]=-1;
for (i=0,j=n;i=2;)
if (match[i]<0&&aug(n,list,buf,match,v,i))
i=0,j-=2;
else
i++;
for (i=j=0;i j+=(match[i]>=0);
return j/2;
}
9、 图论—网络流
9.1 最大流(邻接阵) 89
//求网络最大流,邻接阵形式
//返回最大流量,flow返回每条边的流量
//传入网络节点数n,容量 mat,源点 source,汇点 sink

#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink,int flow[][MAXN]){
int pre[MAXN],que[MAXN],d[MAXN],p,q,t,i,j;
if (source==sink) return inf;
for (i=0;i for (j=0;j for (;;){
for (i=0;i pre[t=source]=source+1,d[t]=inf;
for (p=q=0;p<=q&&!pre[sink];t=que[p++])
for (i=0;i if (!pre[i]&&j=mat[t][i]-flow[t][i])
pre[que[q++]=i]=t+1,d[i]=d[t] else if (!pre[i]&&j=flow[i][t])
pre[que[q++]=i]=-t-1,d[i]=d[t] if (!pre[sink]) break;
for (i=sink;i!=source;)
if (pre[i]>0)
flow[pre[i]-1][i]+=d[sink],i=pre[i]-1;
else
flow[i][-pre[i]-1]-=d[sink],i=-pre[i]-1;
}
for (j=i=0;i return j;
}
9.2 上下界最大流(邻接阵)
//求上下界网络最大流,邻接阵形式
//返回最大流量,-1 表示无可行流,flow返回每条边的流量
//传入网络节点数n,容量 mat,流量下界bf,源点source,汇点 sink
//MAXN 应比最大结点数多2,无可行流返回-1时mat 未复原!

#define MAXN 100
#define inf 1000000000

int limit_max_flow(int n,int mat[][MAXN],int bf[][MAXN],int source,int sink,int
flow[][MAXN]){ 90
int i,j,sk,ks;
if (source==sink) return inf;
for (mat[n][n+1]=mat[n+1][n]=mat[n][n]=mat[n+1][n+1]=i=0;i for (mat[n][i]=mat[i][n]=mat[n+1][i]=mat[i][n+1]=j=0;j mat[i][j]-=bf[i][j],mat[n][i]+=bf[j][i],mat[i][n+1]+=bf[i][j];
sk=mat[source][sink],ks=mat[sink][source],mat[source][sink]=mat[sink][source]=inf;
for (i=0;i for (j=0;j _max_flow(n+2,mat,n,n+1,flow);
for (i=0;i if (flow[n][i] flow[source][sink]=flow[sink][source]=0,mat[source][sink]=sk,mat[sink][source]=ks;
_max_flow(n,mat,source,sink,flow);
for (i=0;i for (j=0;j mat[i][j]+=bf[i][j],flow[i][j]+=bf[i][j];
for (j=i=0;i return j;
}
9.3 上下界最小流(邻接阵)
//求上下界网络最小流,邻接阵形式
//返回最大流量,-1 表示无可行流,flow返回每条边的流量
//传入网络节点数n,容量 mat,流量下界bf,源点source,汇点 sink
//MAXN 应比最大结点数多2,无可行流返回-1时mat 未复原!

#define MAXN 100
#define inf 1000000000
int limit_min_flow(int n,int mat[][MAXN],int bf[][MAXN],int source,int sink,int
flow[][MAXN]){
int i,j,sk,ks;
if (source==sink) return inf;
for (mat[n][n+1]=mat[n+1][n]=mat[n][n]=mat[n+1][n+1]=i=0;i for (mat[n][i]=mat[i][n]=mat[n+1][i]=mat[i][n+1]=j=0;j mat[i][j]-=bf[i][j],mat[n][i]+=bf[j][i],mat[i][n+1]+=bf[i][j];
sk=mat[source][sink],ks=mat[sink][source],mat[source][sink]=mat[sink][source]=inf;
for (i=0;i for (j=0;j _max_flow(n+2,mat,n,n+1,flow);
for (i=0;i if (flow[n][i] flow[source][sink]=flow[sink][source]=0,mat[source][sink]=sk,mat[sink][source]=ks;
_max_flow(n,mat,sink,source,flow); 91
for (i=0;i for (j=0;j mat[i][j]+=bf[i][j],flow[i][j]+=bf[i][j];
for (j=i=0;i return j;
}
9.4 最大流无流量(邻接阵)
//求网络最大流,邻接阵形式
//返回最大流量
//传入网络节点数n,容量 mat,源点 source,汇点 sink
//注意 mat 矩阵被修改

#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink){
int v[MAXN],c[MAXN],p[MAXN],ret=0,i,j;
for (;;){
for (i=0;i v[i]=c[i]=0;
for (c[source]=inf;;){
for (j=-1,i=0;i if (!v[i]&&c[i]&&(j==-1||c[i]>c[j]))
j=i;
if (j<0) return ret;
if (j==sink) break;
for (v[j]=1,i=0;i if (mat[j][i]>c[i]&&c[j]>c[i])
c[i]=mat[j][i] }
for (ret+=j=c[i=sink];i!=source;i=p[i])
mat[p[i]][i]-=j,mat[i][p[i]]+=j;
}
}
9.5 最小费用最大流(邻接阵)
//求网络最小费用最大流,邻接阵形式
//返回最大流量,flow返回每条边的流量,netcost返回总费用
//传入网络节点数n,容量 mat,单位费用cost,源点 source,汇点 sink
92
#define MAXN 100
#define inf 1000000000

int min_cost_max_flow(int n,int mat[][MAXN],int cost[][MAXN],int source,int sink,int
flow[][MAXN],int& netcost){
int pre[MAXN],min[MAXN],d[MAXN],i,j,t,tag;
if (source==sink) return inf;
for (i=0;i for (j=0;j for (netcost=0;;){
for (i=0;i pre[i]=0,min[i]=inf;
for (pre[source]=source+1,min[source]=0,d[source]=inf,tag=1;tag;)
for (tag=t=0;t if (d[t])
for (i=0;i if (j=mat[t][i]-flow[t][i]&&min[t]+cost[t][i] tag=1,min[i]=min[t]+cost[t][i],pre[i]=t+1,d[i]=d[t] else if (j=flow[i][t]&&min[t] tag=1,min[i]=min[t]-cost[i][t],pre[i]=-t-1,d[i]=d[t] if (!pre[sink]) break;
for (netcost+=min[sink]*d[i=sink];i!=source;)
if (pre[i]>0)
flow[pre[i]-1][i]+=d[sink],i=pre[i]-1;
else
flow[i][-pre[i]-1]-=d[sink],i=-pre[i]-1;
}
for (j=i=0;i return j;
}
10、 图论—应用
10.1 欧拉回路(邻接阵)
//求欧拉回路或欧拉路,邻接阵形式,复杂度O(n^2)
//返回路径长度,path返回路径(有向图时得到的是反向路径)
//传入图的大小 n和邻接阵mat,不相邻点边权 0
//可以有自环与重边,分为无向图和有向图

#define MAXN 100

void find_path_u(int n,int mat[][MAXN],int now,int& step,int* path){ 93
int i;
for (i=n-1;i>=0;i--)
while (mat[now][i]){
mat[now][i]--,mat[i][now]--;
find_path_u(n,mat,i,step,path);
}
path[step++]=now;
}

void find_path_d(int n,int mat[][MAXN],int now,int& step,int* path){
int i;
for (i=n-1;i>=0;i--)
while (mat[now][i]){
mat[now][i]--;
find_path_d(n,mat,i,step,path);
}
path[step++]=now;
}

int euclid_path(int n,int mat[][MAXN],int start,int* path){
int ret=0;
find_path_u(n,mat,start,ret,path);
// find_path_d(n,mat,start,ret,path);
return ret;
}
10.2 树的前序表转化
//将用边表示的树转化为前序表示的树
//传入节点数 n和邻接表 list[],邻接表必须是双向的,会在函数中释放
//pre[]返回前序表,map[]返回前序表中的节点到原来节点的映射
#define MAXN 10000
struct node{
int to;
node* next;
};

void prenode(int n,node* list[],int* pre,int* map,int* v,int now,int last,int& id){
node* t;
int p=id++;
for (v[map[p]=now]=1,pre[p]=last;list[now];){
t=list[now],list[now]=t->next;
if (!v[t->to])
prenode(n,list,pre,map,v,t->to,p,id); 94
}
}

void makepre(int n,node* list[],int* pre,int* map){
int v[MAXN],id=0,i;
for (i=0;i prenode(n,list,pre,map,v,0,-1,id);
}
10.3 树的优化算法
//最大顶点独立集
int max_node_independent(int n,int* pre,int* set){
int c[MAXN],i,ret=0;
for (i=0;i c[i]=set[i]=0;
for (i=n-1;i>=0;i--)
if (!c[i]){
set[i]=1;
if (pre[i]!=-1)
c[pre[i]]=1;
ret++;
}
return ret;
}

//最大边独立集
int max_edge_independent(int n,int* pre,int* set){
int c[MAXN],i,ret=0;
for (i=0;i c[i]=set[i]=0;
for (i=n-1;i>=0;i--)
if (!c[i]&&pre[i]!=-1&&!c[pre[i]]){
set[i]=1;
c[pre[i]]=1;
ret++;
}
return ret;
}

//最小顶点覆盖集
int min_node_cover(int n,int* pre,int* set){
int c[MAXN],i,ret=0;
for (i=0;i c[i]=set[i]=0;
for (i=n-1;i>=0;i--)
if (!c[i]&&pre[i]!=-1&&!c[pre[i]]){
set[i]=1;
c[pre[i]]=1;
ret++;
}
return ret;
}

//最小顶点支配集
int min_node_dominant(int n,int* pre,int* set){
int c[MAXN],i,ret=0;
for (i=0;i c[i]=set[i]=0;
for (i=n-1;i>=0;i--)
if (!c[i]&&(pre[i]==-1||!set[pre[i]])){
if (pre[i]!=-1){
set[pre[i]]=1;
c[pre[i]]=1;
if (pre[pre[i]]!=-1)
c[pre[pre[i]]]=1;
}
else
set[i]=1;
ret++;
}
return ret;
}
10.4 拓扑排序(邻接阵)
//拓扑排序,邻接阵形式,复杂度O(n^2)
//如果无法完成排序,返回0,否则返回1,ret返回有序点列
//传入图的大小 n和邻接阵mat,不相邻点边权 0
#define MAXN 100

int toposort(int n,int mat[][MAXN],int* ret){
int d[MAXN],i,j,k;
for (i=0;i for (d[i]=j=0;j for (k=0;k for (i=0;d[i]&&i if (i==n) 96
return 0;
for (d[i]=-1,j=0;j d[j]-=mat[i][j];
}
return 1;
}
10.5 最佳边割集
//最佳边割集
#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink){
int v[MAXN],c[MAXN],p[MAXN],ret=0,i,j;
for (;;){
for (i=0;i v[i]=c[i]=0;
for (c[source]=inf;;){
for (j=-1,i=0;i if (!v[i]&&c[i]&&(j==-1||c[i]>c[j]))
j=i;
if (j<0) return ret;
if (j==sink) break;
for (v[j]=1,i=0;i if (mat[j][i]>c[i]&&c[j]>c[i])
c[i]=mat[j][i] }
for (ret+=j=c[i=sink];i!=source;i=p[i])
mat[p[i]][i]-=j,mat[i][p[i]]+=j;
}
}

int best_edge_cut(int n,int mat[][MAXN],int source,int sink,int set[][2],int& mincost){
int m0[MAXN][MAXN],m[MAXN][MAXN],i,j,k,l,ret=0,last;
if (source==sink)
return -1;
for (i=0;i for (j=0;j m0[i][j]=mat[i][j];
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
mincost=last=max_flow(n,m,source,sink); 97
for (k=0;k for (l=0;l if (m0[k][l]){
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
m[k][l]=0;
if (max_flow(n,m,source,sink)==last-mat[k][l]){
set[ret][0]=k;
set[ret++][1]=l;
m0[k][l]=0;
last-=mat[k][l];
}
}
return ret;
}
10.6 最佳点割集
//最佳顶点割集
#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink){
int v[MAXN],c[MAXN],p[MAXN],ret=0,i,j;
for (;;){
for (i=0;i v[i]=c[i]=0;
for (c[source]=inf;;){
for (j=-1,i=0;i if (!v[i]&&c[i]&&(j==-1||c[i]>c[j]))
j=i;
if (j<0) return ret;
if (j==sink) break;
for (v[j]=1,i=0;i if (mat[j][i]>c[i]&&c[j]>c[i])
c[i]=mat[j][i] }
for (ret+=j=c[i=sink];i!=source;i=p[i])
mat[p[i]][i]-=j,mat[i][p[i]]+=j;
}
}

int best_vertex_cut(int n,int mat[][MAXN],int* cost,int source,int sink,int* set,int& mincost){ 98
int m0[MAXN][MAXN],m[MAXN][MAXN],i,j,k,ret=0,last;
if (source==sink||mat[source][sink])
return -1;
for (i=0;i for (j=0;j m0[i][j]=0;
for (i=0;i for (j=0;j if (mat[i][j])
m0[i][n+j]=inf;
for (i=0;i m0[n+i][i]=cost[i];
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
mincost=last=max_flow(n+n,m,source,n+sink);
for (k=0;k if (k!=source&&k!=sink){
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
m[n+k][k]=0;
if (max_flow(n+n,m,source,n+sink)==last-cost[k]){
set[ret++]=k;
m0[n+k][k]=0;
last-=cost[k];
}
}
return ret;
}
10.7 最小边割集
//最小边割集
#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink){
int v[MAXN],c[MAXN],p[MAXN],ret=0,i,j;
for (;;){
for (i=0;i v[i]=c[i]=0;
for (c[source]=inf;;){
for (j=-1,i=0;i if (!v[i]&&c[i]&&(j==-1||c[i]>c[j]))
j=i;
if (j<0) return ret;
if (j==sink) break;
for (v[j]=1,i=0;i if (mat[j][i]>c[i]&&c[j]>c[i])
c[i]=mat[j][i] }
for (ret+=j=c[i=sink];i!=source;i=p[i])
mat[p[i]][i]-=j,mat[i][p[i]]+=j;
}
}

int min_edge_cut(int n,int mat[][MAXN],int source,int sink,int set[][2]){
int m0[MAXN][MAXN],m[MAXN][MAXN],i,j,k,l,ret=0,last;
if (source==sink)
return -1;
for (i=0;i for (j=0;j m0[i][j]=(mat[i][j]!=0);
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
last=max_flow(n,m,source,sink);
for (k=0;k for (l=0;l if (m0[k][l]){
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
m[k][l]=0;
if (max_flow(n,m,source,sink) set[ret][0]=k;
set[ret++][1]=l;
m0[k][l]=0;
last--;
}
}
return ret;
}
10.8 最小点割集
//最小顶点割集 100
#define MAXN 100
#define inf 1000000000

int max_flow(int n,int mat[][MAXN],int source,int sink){
int v[MAXN],c[MAXN],p[MAXN],ret=0,i,j;
for (;;){
for (i=0;i v[i]=c[i]=0;
for (c[source]=inf;;){
for (j=-1,i=0;i if (!v[i]&&c[i]&&(j==-1||c[i]>c[j]))
j=i;
if (j<0) return ret;
if (j==sink) break;
for (v[j]=1,i=0;i if (mat[j][i]>c[i]&&c[j]>c[i])
c[i]=mat[j][i] }
for (ret+=j=c[i=sink];i!=source;i=p[i])
mat[p[i]][i]-=j,mat[i][p[i]]+=j;
}
}

int min_vertex_cut(int n,int mat[][MAXN],int source,int sink,int* set){
int m0[MAXN][MAXN],m[MAXN][MAXN],i,j,k,ret=0,last;
if (source==sink||mat[source][sink])
return -1;
for (i=0;i for (j=0;j m0[i][j]=0;
for (i=0;i for (j=0;j if (mat[i][j])
m0[i][n+j]=inf;
for (i=0;i m0[n+i][i]=1;
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
last=max_flow(n+n,m,source,n+sink);
for (k=0;k if (k!=source&&k!=sink){
for (i=0;i for (j=0;j m[i][j]=m0[i][j];
m[n+k][k]=0;
if (max_flow(n+n,m,source,n+sink) set[ret++]=k;
m0[n+k][k]=0;
last--;
}
}
return ret;
}
10.9 最小路径覆盖
//最小路径覆盖,O(n^3)
//求解最小的路径覆盖图中所有点,有向图无向图均适用
//注意此问题等价二分图最大匹配,可以用邻接表或正向表减小复杂度
//返回最小路径条数,pre返回前指针(起点-1),next返回后指针(终点-1)
#include
#define MAXN 310
#define _clr(x) memset(x,0xff,sizeof(int)*n)

int hungary(int n,int mat[][MAXN],int* match1,int* match2){
int s[MAXN],t[MAXN],p,q,ret=0,i,j,k;
for (_clr(match1),_clr(match2),i=0;i=0))
for (_clr(t),s[p=q=0]=i;p<=q&&match1[i]<0;p++)
for (k=s[p],j=0;j if (mat[k][j]&&t[j]<0){
s[++q]=match2[j],t[j]=k;
if (s[q]<0)
for (p=j;p>=0;j=p)
match2[j]=k=t[j],p=match1[k],match1[k]=j;
}
return ret;
}

inline int path_cover(int n,int mat[][MAXN],int* pre,int* next){
return n-hungary(n,mat,next,pre);
}
11、 图论—支撑树
11.1 最小生成树(kruskal 邻接表) 102
//无向图最小生成树,kruskal 算法,邻接表形式,复杂度O(mlogm)
//返回最小生成树的长度,传入图的大小n和邻接表 list
//可更改边权的类型,edge[][2]返回树的构造,用边集表示
//如果图不连通,则对各连通分支构造最小生成树,返回总长度
#include
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int from,to;
elem_t len;
edge_t* next;
};

#define _ufind_run(x) for(;p[t=x];x=p[x],p[t]=(p[x]?p[x]:x))
#define _run_both _ufind_run(i);_ufind_run(j)
struct ufind{
int p[MAXN],t;
void init(){memset(p,0,sizeof(p));}
void set_friend(int i,int j){_run_both;p[i]=(i==j?0:j);}
int is_friend(int i,int j){_run_both;return i==j&&i;}
};

#define _cp(a,b) ((a).len<(b).len)
struct heap_t{int a,b;elem_t len;};
struct minheap{
heap_t h[MAXN*MAXN];
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

elem_t kruskal(int n,edge_t* list[],int edge[][2]){
ufind u;minheap h;
edge_t* t;heap_t e; 103
elem_t ret=0;int i,m=0;
u.init(),h.init();
for (i=0;i for (t=list[i];t;t=t->next)
if (ito)
e.a=i,e.b=t->to,e.len=t->len,h.ins(e);
while (m if (!u.is_friend(e.a+1,e.b+1))
edge[m][0]=e.a,edge[m][1]=e.b,ret+=e.len,u.set_friend(e.a+1,e.b+1);
return ret;
}
11.2 最小生成树(kruskal 正向表)
//无向图最小生成树,kruskal 算法,正向表形式,复杂度O(mlogm)
//返回最小生成树的长度,传入图的大小n和正向表 list,buf
//可更改边权的类型,edge[][2]返回树的构造,用边集表示
//如果图不连通,则对各连通分支构造最小生成树,返回总长度
#include
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int to;
elem_t len;
};

#define _ufind_run(x) for(;p[t=x];x=p[x],p[t]=(p[x]?p[x]:x))
#define _run_both _ufind_run(i);_ufind_run(j)
struct ufind{
int p[MAXN],t;
void init(){memset(p,0,sizeof(p));}
void set_friend(int i,int j){_run_both;p[i]=(i==j?0:j);}
int is_friend(int i,int j){_run_both;return i==j&&i;}
};

#define _cp(a,b) ((a).len<(b).len)
struct heap_t{int a,b;elem_t len;};
struct minheap{
heap_t h[MAXN*MAXN];
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1); 104
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

elem_t kruskal(int n,int* list,edge_t* buf,int edge[][2]){
ufind u;minheap h;
heap_t e;elem_t ret=0;
int i,j,m=0;
u.init(),h.init();
for (i=0;i for (j=list[i];j if (i e.a=i,e.b=buf[j].to,e.len=buf[j].len,h.ins(e);
while (m if (!u.is_friend(e.a+1,e.b+1))
edge[m][0]=e.a,edge[m][1]=e.b,ret+=e.len,u.set_friend(e.a+1,e.b+1);
return ret;
}
11.3 最小生成树(prim+binary_heap邻接表)
//无向图最小生成树,prim算法+二分堆,邻接表形式,复杂度O(mlogm)
//返回最小生成树的长度,传入图的大小n和邻接表 list
//可更改边权的类型,pre[]返回树的构造,用父结点表示,根节点(第一个)pre值为-1
//必须保证图的连通的!
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int from,to;
elem_t len;
edge_t* next;
};

#define _cp(a,b) ((a).d<(b).d)
struct heap_t{elem_t d;int v;};
struct heap{
heap_t h[MAXN*MAXN]; 105
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

elem_t prim(int n,edge_t* list[],int* pre){
heap h;
elem_t min[MAXN],ret=0;
edge_t* t;heap_t e;
int v[MAXN],i;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
h.init();e.v=0,e.d=0,h.ins(e);
while (h.del(e))
if (!v[e.v])
for (v[e.v]=1,ret+=e.d,t=list[e.v];t;t=t->next)
if (!v[t->to]&&t->lento])
pre[t->to]=t->from,min[e.v=t->to]=e.d=t->len,h.ins(e);
return ret;
}
11.4 最小生成树(prim+binary_heap正向表)
//无向图最小生成树,prim算法+二分堆,正向表形式,复杂度O(mlogm)
//返回最小生成树的长度,传入图的大小n和正向表 list,buf
//可更改边权的类型,pre[]返回树的构造,用父结点表示,根节点(第一个)pre值为-1
//必须保证图的连通的!
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int to;
elem_t len;
};
106
#define _cp(a,b) ((a).d<(b).d)
struct heap_t{elem_t d;int v;};
struct heap{
heap_t h[MAXN*MAXN];
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

elem_t prim(int n,int* list,edge_t* buf,int* pre){
heap h;heap_t e;
elem_t min[MAXN],ret=0;
int v[MAXN],i,j;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
h.init();e.v=0,e.d=0,h.ins(e);
while (h.del(e))
if (!v[i=e.v])
for (v[i]=1,ret+=e.d,j=list[i];j if (!v[buf[j].to]&&buf[j].len pre[buf[j].to]=i,min[e.v=buf[j].to]=e.d=buf[j].len,h.ins(e);
return ret;
}
11.5 最小生成树(prim+mapped_heap邻接表)
//无向图最小生成树,prim算法+映射二分堆,邻接表形式,复杂度O(mlogn)
//返回最小生成树的长度,传入图的大小n和邻接表list
//可更改边权的类型,pre[]返回树的构造,用父结点表示,根节点(第一个)pre值为-1
//必须保证图的连通的!
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int from,to; 107
elem_t len;
edge_t* next;
};

#define _cp(a,b) ((a)<(b))
struct heap{
elem_t h[MAXN+1];
int ind[MAXN+1],map[MAXN+1],n,p,c;
void init(){n=0;}
void ins(int i,elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
h[map[ind[p]=i]=p]=e;
}
int del(int i,elem_t& e){
i=map[i];if (i<1||i>n) return 0;
for (e=h[p=i];p>1;h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
for
(c=2;c =1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
int delmin(int& i,elem_t& e){
if (n<1) return 0;i=ind[1];
for
(e=h[p=1],c=2;c ],p=c,c<<=1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
};

elem_t prim(int n,edge_t* list[],int* pre){
heap h;
elem_t min[MAXN],ret=0,e;
edge_t* t;
int v[MAXN],i;
for (h.init(),i=0;i min[i]=(i?inf:0),v[i]=0,pre[i]=-1,h.ins(i,min[i]);
while (h.delmin(i,e))
for (v[i]=1,ret+=e,t=list[i];t;t=t->next)
if (!v[t->to]&&t->lento])
pre[t->to]=t->from,h.del(t->to,e),h.ins(t->to,min[t->to]=t->len);
return ret;
} 108
11.6 最小生成树(prim+mapped_heap正向表)
//无向图最小生成树,prim算法+映射二分堆,正向表形式,复杂度O(mlogn)
//返回最小生成树的长度,传入图的大小n和正向表list,buf
//可更改边权的类型,pre[]返回树的构造,用父结点表示,根节点(第一个)pre值为-1
//必须保证图的连通的!
#define MAXN 200
#define inf 1000000000
typedef double elem_t;
struct edge_t{
int to;
elem_t len;
};

#define _cp(a,b) ((a)<(b))
struct heap{
elem_t h[MAXN+1];
int ind[MAXN+1],map[MAXN+1],n,p,c;
void init(){n=0;}
void ins(int i,elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
h[map[ind[p]=i]=p]=e;
}
int del(int i,elem_t& e){
i=map[i];if (i<1||i>n) return 0;
for (e=h[p=i];p>1;h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
for
(c=2;c =1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
int delmin(int& i,elem_t& e){
if (n<1) return 0;i=ind[1];
for
(e=h[p=1],c=2;c ],p=c,c<<=1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
};

elem_t prim(int n,int* list,edge_t* buf,int* pre){
heap h;
elem_t min[MAXN],ret=0,e;
int v[MAXN],i,j; 109
for (h.init(),i=0;i min[i]=(i?inf:0),v[i]=0,pre[i]=-1,h.ins(i,min[i]);
while (h.delmin(i,e))
for (v[i]=1,ret+=e,j=list[i];j if (!v[buf[j].to]&&buf[j].len pre[buf[j].to]=i,h.del(buf[j].to,e),h.ins(buf[j].to,min[buf[j].to]=buf[j].len);
return ret;
}
11.7 最小生成树(prim邻接阵)
//无向图最小生成树,prim算法,邻接阵形式,复杂度O(n^2)
//返回最小生成树的长度,传入图的大小n和邻接阵 mat,不相邻点边权inf
//可更改边权的类型,pre[]返回树的构造,用父结点表示,根节点(第一个)pre值为-1
//必须保证图的连通的!
#define MAXN 200
#define inf 1000000000
typedef double elem_t;

elem_t prim(int n,elem_t mat[][MAXN],int* pre){
elem_t min[MAXN],ret=0;
int v[MAXN],i,j,k;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
for (min[j=0]=0;j for (k=-1,i=0;i if (!v[i]&&(k==-1||min[i] k=i;
for (v[k]=1,ret+=min[k],i=0;i if (!v[i]&&mat[k][i] min[i]=mat[pre[i]=k][i];
}
return ret;
}
11.8 最小树形图(邻接阵)
//多源最小树形图,edmonds算法,邻接阵形式,复杂度O(n^3)
//返回最小生成树的长度,构造失败返回负值
//传入图的大小 n和邻接阵mat,不相邻点边权 inf
//可更改边权的类型,pre[]返回树的构造,用父结点表示
//传入时 pre[]数组清零,用-1标出源点
#include 110
#define MAXN 120
#define inf 1000000000
typedef int elem_t;

elem_t edmonds(int n,elem_t mat[][MAXN*2],int* pre){
elem_t ret=0;
int c[MAXN*2][MAXN*2],l[MAXN*2],p[MAXN*2],m=n,t,i,j,k;
for (i=0;i do{
memset(c,0,sizeof(c)),memset(p,0xff,sizeof(p));
for (t=m,i=0;i for (i=0;i if (l[i]==i&&pre[i]!=-1){
for (j=0;j if (l[j]==j&&i!=j&&mat[j][i] p[i]=j;
if ((pre[i]=p[i])==-1)
return -1;
if (c[i][p[i]]){
for (j=0;j<=m;mat[j][m]=mat[m][j]=inf,j++);
for (k=i;l[k]!=m;l[k]=m,k=p[k])
for (j=0;j if (l[j]==j){
if (mat[j][k]-mat[p[k]][k] mat[j][m]=mat[j][k]-mat[p[k]][k];
if (mat[k][j] mat[m][j]=mat[k][j];
}
c[m][m]=1,l[m]=m,m++;
}
for (j=0;j if (c[i][j])
for (k=p[i];k!=-1&&l[k]==k;c[k][j]=1,k=p[k]);
}
}
while (t for (;m-->n;pre[k]=pre[m])
for (i=0;i if (l[i]==m){
for (j=0;j if (pre[j]==m&&mat[i][j]==mat[m][j])
pre[j]=i;
if (mat[pre[m]][m]==mat[pre[m]][i]-mat[pre[i]][i])
k=i; 111
}
for (i=0;i if (pre[i]!=-1)
ret+=mat[pre[i]][i];
return ret;
}
12、 图论—最短路径
12.1 最短路径(单源 bellman_ford邻接阵)
//单源最短路径,bellman_ford算法,邻接阵形式,复杂度O(n^3)
//求出源 s到所有点的最短路经,传入图的大小 n和邻接阵 mat
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,路权可为负,若图包含负环则求解失败,返回 0
//优化:先删去负边使用dijkstra 求出上界,加速迭代过程
#define MAXN 200
#define inf 1000000000
typedef int elem_t;

int bellman_ford(int n,elem_t mat[][MAXN],int s,elem_t* min,int* pre){
int v[MAXN],i,j,k,tag;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
for (min[s]=0,j=0;j for (k=-1,i=0;i if (!v[i]&&(k==-1||min[i] k=i;
for (v[k]=1,i=0;i if (!v[i]&&mat[k][i]>=0&&min[k]+mat[k][i] min[i]=min[k]+mat[pre[i]=k][i];
}
for (tag=1,j=0;tag&&j<=n;j++)
for (tag=i=0;i for (k=0;k if (min[k]+mat[k][i] min[i]=min[k]+mat[pre[i]=k][i],tag=1;
return j<=n;
}
12.2 最短路径(单源 dijkstra+bfs邻接表) 112
//单源最短路径,用于路权相等的情况,dijkstra 优化为 bfs,邻接表形式,复杂度O(m)
//求出源 s到所有点的最短路经,传入图的大小 n和邻接表 list,边权值 len
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负且相等!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;
struct edge_t{
int from,to;
edge_t* next;
};

void dijkstra(int n,edge_t* list[],elem_t len,int s,elem_t* min,int* pre){
edge_t* t;
int i,que[MAXN],f=0,r=0,p=1,l=1;
for (i=0;i min[i]=inf;
min[que[0]=s]=0,pre[s]=-1;
for (;r<=f;l++,r=f+1,f=p-1)
for (i=r;i<=f;i++)
for (t=list[que[i]];t;t=t->next)
if (min[t->to]==inf)
min[que[p++]=t->to]=len*l,pre[t->to]=que[i];
}
12.3 最短路径(单源 dijkstra+bfs正向表)
//单源最短路径,用于路权相等的情况,dijkstra 优化为 bfs,正向表形式,复杂度O(m)
//求出源 s到所有点的最短路经,传入图的大小 n和正向表 list,buf,边权值 len
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负且相等!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;

void dijkstra(int n,int* list,int* buf,elem_t len,int s,elem_t* min,int* pre){
int i,que[MAXN],f=0,r=0,p=1,l=1,t;
for (i=0;i min[i]=inf;
min[que[0]=s]=0,pre[s]=-1;
for (;r<=f;l++,r=f+1,f=p-1)
for (i=r;i<=f;i++)
for (t=list[que[i]];t if (min[buf[t]]==inf) 113
min[que[p++]=buf[t]]=len*l,pre[buf[t]]=que[i];
}
12.4 最短路径(单源 dijkstra+binary_heap邻接表)
//单源最短路径,dijkstra 算法+二分堆,邻接表形式,复杂度 O(mlogm)
//求出源 s到所有点的最短路经,传入图的大小 n和邻接表 list
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;
struct edge_t{
int from,to;
elem_t len;
edge_t* next;
};

#define _cp(a,b) ((a).d<(b).d)
struct heap_t{elem_t d;int v;};
struct heap{
heap_t h[MAXN*MAXN];
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

void dijkstra(int n,edge_t* list[],int s,elem_t* min,int* pre){
heap h;
edge_t* t;heap_t e;
int v[MAXN],i;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
h.init();min[e.v=s]=e.d=0,h.ins(e);
while (h.del(e)) 114
if (!v[e.v])
for (v[e.v]=1,t=list[e.v];t;t=t->next)
if (!v[t->to]&&min[t->from]+t->lento])
pre[t->to]=t->from,min[e.v=t->to]=e.d=min[t->from]+t->len,h.ins(e);
}
12.5 最短路径(单源 dijkstra+binary_heap正向表)
//单源最短路径,dijkstra 算法+二分堆,正向表形式,复杂度 O(mlogm)
//求出源 s到所有点的最短路经,传入图的大小 n和正向表 list,buf
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;
struct edge_t{
int to;
elem_t len;
};

#define _cp(a,b) ((a).d<(b).d)
struct heap_t{elem_t d;int v;};
struct heap{
heap_t h[MAXN*MAXN];
int n,p,c;
void init(){n=0;}
void ins(heap_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[p]=h[p>>1],p>>=1);
h[p]=e;
}
int del(heap_t& e){
if (!n) return 0;
for
(e=h[p=1],c=2;c h[p]=h[n--];return 1;
}
};

void dijkstra(int n,int* list,edge_t* buf,int s,elem_t* min,int* pre){
heap h;heap_t e;
int v[MAXN],i,t,f;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
h.init();min[e.v=s]=e.d=0,h.ins(e); 115
while (h.del(e))
if (!v[e.v])
for (v[f=e.v]=1,t=list[f];t if (!v[buf[t].to]&&min[f]+buf[t].len pre[buf[t].to]=f,min[e.v=buf[t].to]=e.d=min[f]+buf[t].len,h.ins(e);
}
12.6 最短路径(单源 dijkstra+mapped_heap邻接表)
//单源最短路径,dijkstra 算法+映射二分堆,邻接表形式,复杂度 O(mlogn)
//求出源s到所有点的最短路经,传入图的大小 n和邻接表 list
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;
struct edge_t{
int from,to;
elem_t len;
edge_t* next;
};

#define _cp(a,b) ((a)<(b))
struct heap{
elem_t h[MAXN+1];
int ind[MAXN+1],map[MAXN+1],n,p,c;
void init(){n=0;}
void ins(int i,elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
h[map[ind[p]=i]=p]=e;
}
int del(int i,elem_t& e){
i=map[i];if (i<1||i>n) return 0;
for (e=h[p=i];p>1;h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
for
(c=2;c =1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
int delmin(int& i,elem_t& e){
if (n<1) return 0;i=ind[1];
for
(e=h[p=1],c=2;c ],p=c,c<<=1); 116
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
};

void dijkstra(int n,edge_t* list[],int s,elem_t* min,int* pre){
heap h;
edge_t* t;elem_t e;
int v[MAXN],i;
for (h.init(),i=0;i min[i]=((i==s)?0:inf),v[i]=0,pre[i]=-1,h.ins(i,min[i]);
while (h.delmin(i,e))
for (v[i]=1,t=list[i];t;t=t->next)
if (!v[t->to]&&min[i]+t->lento])
pre[t->to]=i,h.del(t->to,e),min[t->to]=e=min[i]+t->len,h.ins(t->to,e);
}
12.7 最短路径(单源 dijkstra+mapped_heap正向表)
//单源最短路径,dijkstra 算法+映射二分堆,正向表形式,复杂度 O(mlogn)
//求出源s到所有点的最短路经,传入图的大小 n和正向表 list,buf
//返回到各点最短距离min[]和路径 pre[],pre[i]记录 s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;
struct edge_t{
int to;
elem_t len;
};

#define _cp(a,b) ((a)<(b))
struct heap{
elem_t h[MAXN+1];
int ind[MAXN+1],map[MAXN+1],n,p,c;
void init(){n=0;}
void ins(int i,elem_t e){
for (p=++n;p>1&&_cp(e,h[p>>1]);h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
h[map[ind[p]=i]=p]=e;
}
int del(int i,elem_t& e){
i=map[i];if (i<1||i>n) return 0;
for (e=h[p=i];p>1;h[map[ind[p]=ind[p>>1]]=p]=h[p>>1],p>>=1);
for
(c=2;c =1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
int delmin(int& i,elem_t& e){
if (n<1) return 0;i=ind[1];
for
(e=h[p=1],c=2;c ],p=c,c<<=1);
h[map[ind[p]=ind[n]]=p]=h[n];n--;return 1;
}
};

void dijkstra(int n,int* list,edge_t* buf,int s,elem_t* min,int* pre){
heap h;elem_t e;
int v[MAXN],i,t;
for (h.init(),i=0;i min[i]=((i==s)?0:inf),v[i]=0,pre[i]=-1,h.ins(i,min[i]);
while (h.delmin(i,e))
for (v[i]=1,t=list[i];t if (!v[buf[t].to]&&min[i]+buf[t].len
pre[buf[t].to]=i,h.del(buf[t].to,e),min[buf[t].to]=e=min[i]+buf[t].len,h.ins(buf[t].to,e);
}
12.8 最短路径(单源 dijkstra邻接阵)
//单源最短路径,dijkstra 算法,邻接阵形式,复杂度O(n^2)
//求出源 s到所有点的最短路经,传入图的顶点数 n,(有向)邻接矩阵mat
//返回到各点最短距离min[]和路径 pre[],pre[i]记录s到 i路径上 i的父结点,pre[s]=-1
//可更改路权类型,但必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;

void dijkstra(int n,elem_t mat[][MAXN],int s,elem_t* min,int* pre){
int v[MAXN],i,j,k;
for (i=0;i min[i]=inf,v[i]=0,pre[i]=-1;
for (min[s]=0,j=0;j for (k=-1,i=0;i if (!v[i]&&(k==-1||min[i] k=i;
for (v[k]=1,i=0;i if (!v[i]&&min[k]+mat[k][i] min[i]=min[k]+mat[pre[i]=k][i];
}
}
12.9 最短路径(多源 floyd_warshall邻接阵)
//多源最短路径,floyd_warshall算法,复杂度 O(n^3)
//求出所有点对之间的最短路经,传入图的大小和邻接阵
//返回各点间最短距离min[]和路径 pre[],pre[i][j]记录i到 j最短路径上 j的父结点
//可更改路权类型,路权必须非负!
#define MAXN 200
#define inf 1000000000
typedef int elem_t;

void floyd_warshall(int n,elem_t mat[][MAXN],elem_t min[][MAXN],int pre[][MAXN]){
int i,j,k;
for (i=0;i for (j=0;j min[i][j]=mat[i][j],pre[i][j]=(i==j)?-1:i;
for (k=0;k for (i=0;i for (j=0;j if (min[i][k]+min[k][j] min[i][j]=min[i][k]+min[k][j],pre[i][j]=pre[k][j];
}
13、 应用
13.1 Joseph问题
// Joseph's Problem
// input: n,m -- the number of persons, the inteval between persons
// output: -- return the reference of last person

int josephus0(int n, int m)
{
if (n == 2) return (m%2) ? 2 : 1;
int v = (m+josephus0(n-1,m)) % n;
if (v == 0) v = n;
return v;
}
int josephus(int n, int m) 119
{
if (m == 1) return n;
if (n == 1) return 1;
if (m >=n) return josephus0(n,m);
int l = (n/m)*m;
int j = josephus(n - (n/m), m);
if (j <= n-l) return l+j;
j -= n-l;
int t = (j/(m-1))*m;
if ((j % (m-1)) == 0) return t-1;
return t + (j % (m-1));
}
13.2 N皇后构造解
//N皇后构造解,n>=4

void even1(int n,int *p){
int i;
for (i=1;i<=n/2;i++)
p[i-1]=2*i;
for (i=n/2+1;i<=n;i++)
p[i-1]=2*i-n-1;
}

void even2(int n,int *p){
int i;
for (i=1;i<=n/2;i++)
p[i-1]=(2*i+n/2-3)%n+1;
for (i=n/2+1;i<=n;i++)
p[i-1]=n-(2*(n-i+1)+n/2-3)%n;
}

void generate(int,int*);
void odd(int n,int *p){
generate(n-1,p),p[n-1]=n;
}

void generate(int n,int *p){
if (n&1)
odd(n,p);
else if (n%6!=2)
even1(n,p);
else 120
even2(n,p);
}
13.3 布尔母函数
//布尔母函数
//判 m[]个价值为 w[]的货币能否构成 value
//适合 m[]较大w[]较小的情况
//返回布尔量
//传入货币种数 n,个数 m[],价值 w[]和目标值value
#define MAXV 100000

int genfunc(int n,int* m,int* w,int value){
int i,j,k,c;
char r[MAXV];
for (r[0]=i=1;i<=value;r[i++]=0);
for (i=0;i for (j=0;j c=m[i]*r[k=j];
while ((k+=w[i])<=value)
if (r[k])
c=m[i];
else if (c)
r[k]=1,c--;
if (r[value])
return 1;
}
}
return 0;
}
13.4 第 k元素
//取第 k个元素,k=0..n-1
//平均复杂度O(n)
//注意 a[]中的顺序被改变

#define _cp(a,b) ((a)<(b))
typedef int elem_t;
elem_t kth_element(int n,elem_t* a,int k){
elem_t t,key;
int l=0,r=n-1,i,j;
while (l for (key=a[((i=l-1)+(j=r+1))>>1];i for (j--;_cp(key,a[j]);j--);
for (i++;_cp(a[i],key);i++);
if (i }
if (k>j) l=j+1;
else r=j;
}
return a[k];
}
13.5 幻方构造
//幻方构造(l!=2)
#define MAXN 100

void dllb(int l,int si,int sj,int sn,int d[][MAXN]){
int n,i=0,j=l/2;
for (n=1;n<=l*l;n++){
d[i+si][j+sj]=n+sn;
if (n%l){
i=(i)?(i-1):(l-1);
j=(j==l-1)?0:(j+1);
}
else
i=(i==l-1)?0:(i+1);
}
}

void magic_odd(int l,int d[][MAXN]){
dllb(l,0,0,0,d);
}

void magic_4k(int l,int d[][MAXN]){
int i,j;
for (i=0;i for (j=0;j
d[i][j]=((i%4==0||i%4==3)&&(j%4==0||j%4==3)||(i%4==1||i%4==2)&&(j%4==1||j%4==2))
?(l*l-(i*l+j)):(i*l+j+1);
}

void magic_other(int l,int d[][MAXN]){
int i,j,t; 122
dllb(l/2,0,0,0,d);
dllb(l/2,l/2,l/2,l*l/4,d);
dllb(l/2,0,l/2,l*l/2,d);
dllb(l/2,l/2,0,l*l/4*3,d);
for (i=0;i for (j=0;j if (i!=l/4||j)
t=d[i][j],d[i][j]=d[i+l/2][j],d[i+l/2][j]=t;
t=d[l/4][l/4],d[l/4][l/4]=d[l/4+l/2][l/4],d[l/4+l/2][l/4]=t;
for (i=0;i for (j=l-l/4+1;j t=d[i][j],d[i][j]=d[i+l/2][j],d[i+l/2][j]=t;
}

void generate(int l,int d[][MAXN]){
if (l%2)
magic_odd(l,d);
else if (l%4==0)
magic_4k(l,d);
else
magic_other(l,d);
}
13.6 模式匹配(kmp)
//模式匹配,kmp算法,复杂度 O(m+n)
//返回匹配位置,-1 表示匹配失败,传入匹配串和模式串和长度
//可更改元素类型,更换匹配函数
#define MAXN 10000
#define _match(a,b) ((a)==(b))
typedef char elem_t;

int pat_match(int ls,elem_t* str,int lp,elem_t* pat){
int fail[MAXN]={-1},i=0,j;
for (j=1;j for (i=fail[j-1];i>=0&&!_match(pat[i+1],pat[j]);i=fail[i]);
fail[j]=(_match(pat[i+1],pat[j])?i+1:-1);
}
for (i=j=0;i if (_match(str[i],pat[j]))
j++;
else if (j)
j=fail[j-1]+1,i--;
return j==lp?(i-lp):-1; 123
}
13.7 逆序对数
//序列逆序对数,复杂度O(nlogn)
//传入序列长度和内容,返回逆序对数
//可更改元素类型和比较函数
#include
#define MAXN 1000000
#define _cp(a,b) ((a)<=(b))
typedef int elem_t;
elem_t _tmp[MAXN];

int inv(int n,elem_t* a){
int l=n>>1,r=n-l,i,j;
int ret=(r>1?(inv(l,a)+inv(r,a+l)):0);
for (i=j=0;i<=l;_tmp[i+j]=a[i],i++)
for (ret+=j;j memcpy(a,_tmp,sizeof(elem_t)*n);
return ret;
}
13.8 字符串最小表示
/*
求字符串的最小表示
输入：字符串
返回：字符串最小表示的首字母位置(0...size-1)
*/
template
int MinString(vector &str)
{
int i, j, k;
vector ss(str.size() << 1);
for (i = 0; i < str.size(); i ++) ss[i] = ss[i + str.size()] = str[i];
for (i = k = 0, j = 1; k < str.size() && i < str.size() && j < str.size(); ) {
for (k = 0; k < str.size() && ss[i + k] == ss[j + k]; k ++);
if (k < str.size()) {
if (ss[i + k] > ss[j + k])
i += k + 1;
else j += k + 1;
if (i == j) j ++;
} 124
}
return i < j ? i : j;
}
13.9 最长公共单调子序列
// 最长公共递增子序列， 时间复杂度O(n^2 * logn)，空间 O(n^2)

/**
* n为 a 的大小, m为b 的大小
* 结果在 ans中
* "define _cp(a,b) ((a)<(b))"求解最长严格递增序列
*/
#define MAXN 1000
#define _cp(a,b) ((a)<(b))
typedef int elem_t;

elem_t DP[MAXN][MAXN];
int num[MAXN], p[1<<20];

int LIS(int n, elem_t *a, int m, elem_t *b, elem_t *ans){
int i, j, l, r, k;

DP[0][0] = 0;
num[0] = (b[0] == a[0]);
for(i = 1; i < m; i++) {
num[i] = (b[i] == a[0]) || num[i-1];
DP[i][0] = 0;
}
for(i = 1; i < n; i++){
if(b[0] == a[i] && !num[0]) {
num[0] = 1;
DP[0][0] = i<<10;
}

for(j = 1; j < m; j++){
for(k=((l=0)+(r=num[j-1]-1))>>1; l<=r; k=(l+r)>>1)
if(_cp(a[DP[j-1][k]>>10], a[i]))
l=k+1;
else
r=k-1;

if(l < num[j-1] && i == (DP[j-1][l]>>10) ){
if(l >= num[j]) DP[j][num[j]++] = DP[j-1][l]; 125
else DP[j][l] = _cp(a[DP[j][l]>>10],a[i]) ? DP[j][l] : DP[j-1][l];
}
if(b[j] == a[i]){
for(k=((l=0)+(r=num[j]-1))>>1; l<=r; k=(l+r)>>1)
if(_cp(a[DP[j][k]>>10], a[i]))
l=k+1;
else
r=k-1;

DP[j][l] = (i<<10) + j;
num[j] += (l>=num[j]);
p[DP[j][l]] = l ? DP[j][l-1] : -1;
}
}
}

for (k=DP[m-1][i=num[m-1]-1];i>=0;ans[i--]=a[k>>10],k=p[k]);
return num[m-1];
}
13.10 最长子序列
//最长单调子序列,复杂度O(nlogn)
//注意最小序列覆盖和最长序列的对应关系,例如
//"define _cp(a,b) ((a)>(b))"求解最长严格递减序列,则
//"define _cp(a,b) (!((a)>(b)))"求解最小严格递减序列覆盖
//可更改元素类型和比较函数
#define MAXN 10000
#define _cp(a,b) ((a)>(b))
typedef int elem_t;

int subseq(int n,elem_t* a){
int b[MAXN],i,l,r,m,ret=0;
for (i=0;iret))
for (m=((l=1)+(r=ret))>>1;l<=r;m=(l+r)>>1)
if (_cp(a[b[m]],a[i]))
l=m+1;
else
r=m-1;
return ret;
}

int subseq(int n,elem_t* a,elem_t* ans){
int b[MAXN],p[MAXN],i,l,r,m,ret=0; 126
for (i=0;iret))
for (m=((l=1)+(r=ret))>>1;l<=r;m=(l+r)>>1)
if (_cp(a[b[m]],a[i]))
l=m+1;
else
r=m-1;
for (m=b[i=ret];i;ans[--i]=a[m],m=p[m]);
return ret;
}
13.11 最大子串匹配
//最大子串匹配,复杂度O(mn)
//返回最大匹配值,传入两个串和串的长度,重载返回一个最大匹配
//注意做字符串匹配是串末的'\0'没有置!
//可更改元素类型,更换匹配函数和匹配价值函数
#include
#define MAXN 100
#define max(a,b) ((a)>(b)?(a):(b))
#define _match(a,b) ((a)==(b))
#define _value(a,b) 1
typedef char elem_t;

int str_match(int m,elem_t* a,int n,elem_t* b){
int match[MAXN+1][MAXN+1],i,j;
memset(match,0,sizeof(match));
for (i=0;i for (j=0;j match[i+1][j+1]=max(max(match[i][j+1],match[i+1][j]),
(match[i][j]+_value(a[i],b[i]))*_match(a[i],b[j]));
return match[m][n];
}

int str_match(int m,elem_t* a,int n,elem_t* b,elem_t* ret){
int match[MAXN+1][MAXN+1],last[MAXN+1][MAXN+1],i,j,t;
memset(match,0,sizeof(match));
for (i=0;i for (j=0;j match[i+1][j+1]=(match[i][j+1]>match[i+1][j]?match[i][j+1]:match[i+1][j]);
last[i+1][j+1]=(match[i][j+1]>match[i+1][j]?3:1);
if ((t=(match[i][j]+_value(a[i],b[i]))*_match(a[i],b[j]))>match[i+1][j+1])
match[i+1][j+1]=t,last[i+1][j+1]=2;
}
for (;match[i][j];i-=(last[t=i][j]>1),j-=(last[t][j]<3)) 127
ret[match[i][j]-1]=(last[i][j]<3?a[i-1]:b[j-1]);
return match[m][n];
}
13.12 最大子段和
//求最大子段和,复杂度O(n)
//传入串长n和内容list[]
//返回最大子段和,重载返回子段位置(maxsum=list[start]+...+list[end])
//可更改元素类型
typedef int elem_t;

elem_t maxsum(int n,elem_t* list){
elem_t ret,sum=0;
int i;
for (ret=list[i=0];i sum=(sum>0?sum:0)+list[i],ret=(sum>ret?sum:ret);
return ret;
}

elem_t maxsum(int n,elem_t* list,int& start,int& end){
elem_t ret,sum=0;
int s,i;
for (ret=list[start=end=s=i=0];i0?s:i))
if ((sum=(sum>0?sum:0)+list[i])>ret)
ret=sum,start=s,end=i;
return ret;
}
13.13 最大子阵和
//求最大子阵和,复杂度O(n^3)
//传入阵的大小 m,n和内容 mat[][]
//返回最大子阵和,重载返回子阵位置(maxsum=list[s1][s2]+...+list[e1][e2])
//可更改元素类型
#define MAXN 100
typedef int elem_t;

elem_t maxsum(int m,int n,elem_t mat[][MAXN]){
elem_t matsum[MAXN][MAXN+1],ret,sum;
int i,j,k;
for (i=0;i for (matsum[i][j=0]=0;j matsum[i][j+1]=matsum[i][j]+mat[i][j];
for (ret=mat[0][j=0];j for (k=j;k for (sum=0,i=0;i sum=(sum>0?sum:0)+matsum[i][k+1]-matsum[i][j],ret=(sum>ret?sum:ret);
return ret;
}

elem_t maxsum(int m,int n,elem_t mat[][MAXN],int& s1,int& s2,int& e1,int& e2){
elem_t matsum[MAXN][MAXN+1],ret,sum;
int i,j,k,s;
for (i=0;i for (matsum[i][j=0]=0;j matsum[i][j+1]=matsum[i][j]+mat[i][j];
for (ret=mat[s1=e1=0][s2=e2=j=0];j for (k=j;k for (sum=0,s=i=0;i0?s:i))
if ((sum=(sum>0?sum:0)+matsum[i][k+1]-matsum[i][j])>ret)
ret=sum,s1=s,s2=i,e1=j,e2=k;
return ret;
}
14、 其它
14.1 大数(只能处理正数)
#include
#include

#define DIGIT 4
#define DEPTH 10000
#define MAX 100
typedef int bignum_t[MAX+1];

int read(bignum_t a,istream& is=cin){
char buf[MAX*DIGIT+1],ch;
int i,j;
memset((void*)a,0,sizeof(bignum_t));
if (!(is>>buf)) return 0;
for (a[0]=strlen(buf),i=a[0]/2-1;i>=0;i--)
ch=buf[i],buf[i]=buf[a[0]-1-i],buf[a[0]-1-i]=ch;
for (a[0]=(a[0]+DIGIT-1)/DIGIT,j=strlen(buf);j for (i=1;i<=a[0];i++) 129
for (a[i]=0,j=0;j a[i]=a[i]*10+buf[i*DIGIT-1-j]-'0';
for (;!a[a[0]]&&a[0]>1;a[0]--);
return 1;
}

void write(const bignum_t a,ostream& os=cout){
int i,j;
for (os< for (j=DEPTH/10;j;j/=10)
os< }

int comp(const bignum_t a,const bignum_t b){
int i;
if (a[0]!=b[0])
return a[0]-b[0];
for (i=a[0];i;i--)
if (a[i]!=b[i])
return a[i]-b[i];
return 0;
}

int comp(const bignum_t a,const int b){
int c[12]={1};
for (c[1]=b;c[c[0]]>=DEPTH;c[c[0]+1]=c[c[0]]/DEPTH,c[c[0]]%=DEPTH,c[0]++);
return comp(a,c);
}

int comp(const bignum_t a,const int c,const int d,const bignum_t b){
int i,t=0,O=-DEPTH*2;
if (b[0]-a[0] return 1;
for (i=b[0];i>d;i--){
t=t*DEPTH+a[i-d]*c-b[i];
if (t>0) return 1;
if (t }
for (i=d;i;i--){
t=t*DEPTH-b[i];
if (t>0) return 1;
if (t }
return t>0; 130
}

void add(bignum_t a,const bignum_t b){
int i;
for (i=1;i<=b[0];i++)
if ((a[i]+=b[i])>=DEPTH)
a[i]-=DEPTH,a[i+1]++;
if (b[0]>=a[0])
a[0]=b[0];
else
for (;a[i]>=DEPTH&&i a[0]+=(a[a[0]+1]>0);
}

void add(bignum_t a,const int b){
int i=1;
for (a[1]+=b;a[i]>=DEPTH&&i for (;a[a[0]]>=DEPTH;a[a[0]+1]=a[a[0]]/DEPTH,a[a[0]]%=DEPTH,a[0]++);
}

void sub(bignum_t a,const bignum_t b){
int i;
for (i=1;i<=b[0];i++)
if ((a[i]-=b[i])<0)
a[i+1]--,a[i]+=DEPTH;
for (;a[i]<0;a[i]+=DEPTH,i++,a[i]--);
for (;!a[a[0]]&&a[0]>1;a[0]--);
}

void sub(bignum_t a,const int b){
int i=1;
for
(a[1]-=b;a[i]<0;a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH,i++);
for (;!a[a[0]]&&a[0]>1;a[0]--);
}

void sub(bignum_t a,const bignum_t b,const int c,const int d){
int i,O=b[0]+d;
for (i=1+d;i<=O;i++)
if ((a[i]-=b[i-d]*c)<0)
a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH;
for (;a[i]<0;a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH,i++);
for (;!a[a[0]]&&a[0]>1;a[0]--);
} 131

void mul(bignum_t c,const bignum_t a,const bignum_t b){
int i,j;
memset((void*)c,0,sizeof(bignum_t));
for (c[0]=a[0]+b[0]-1,i=1;i<=a[0];i++)
for (j=1;j<=b[0];j++)
if ((c[i+j-1]+=a[i]*b[j])>=DEPTH)
c[i+j]+=c[i+j-1]/DEPTH,c[i+j-1]%=DEPTH;
for (c[0]+=(c[c[0]+1]>0);!c[c[0]]&&c[0]>1;c[0]--);
}

void mul(bignum_t a,const int b){
int i;
for (a[1]*=b,i=2;i<=a[0];i++){
a[i]*=b;
if (a[i-1]>=DEPTH)
a[i]+=a[i-1]/DEPTH,a[i-1]%=DEPTH;
}
for (;a[a[0]]>=DEPTH;a[a[0]+1]=a[a[0]]/DEPTH,a[a[0]]%=DEPTH,a[0]++);
for (;!a[a[0]]&&a[0]>1;a[0]--);
}

void mul(bignum_t b,const bignum_t a,const int c,const int d){
int i;
memset((void*)b,0,sizeof(bignum_t));
for (b[0]=a[0]+d,i=d+1;i<=b[0];i++)
if ((b[i]+=a[i-d]*c)>=DEPTH)
b[i+1]+=b[i]/DEPTH,b[i]%=DEPTH;
for (;b[b[0]+1];b[0]++,b[b[0]+1]=b[b[0]]/DEPTH,b[b[0]]%=DEPTH);
for (;!b[b[0]]&&b[0]>1;b[0]--);
}

void div(bignum_t c,bignum_t a,const bignum_t b){
int h,l,m,i;
memset((void*)c,0,sizeof(bignum_t));
c[0]=(b[0] for (i=c[0];i;sub(a,b,c[i]=m,i-1),i--)
for (h=DEPTH-1,l=0,m=(h+l+1)>>1;h>l;m=(h+l+1)>>1)
if (comp(b,m,i-1,a)) h=m-1;
else l=m;
for (;!c[c[0]]&&c[0]>1;c[0]--);
c[0]=c[0]>1?c[0]:1;
}
132
void div(bignum_t a,const int b,int& c){
int i;
for (c=0,i=a[0];i;c=c*DEPTH+a[i],a[i]=c/b,c%=b,i--);
for (;!a[a[0]]&&a[0]>1;a[0]--);
}

void sqrt(bignum_t b,bignum_t a){
int h,l,m,i;
memset((void*)b,0,sizeof(bignum_t));
for (i=b[0]=(a[0]+1)>>1;i;sub(a,b,m,i-1),b[i]+=m,i--)
for (h=DEPTH-1,l=0,b[i]=m=(h+l+1)>>1;h>l;b[i]=m=(h+l+1)>>1)
if (comp(b,m,i-1,a)) h=m-1;
else l=m;
for (;!b[b[0]]&&b[0]>1;b[0]--);
for (i=1;i<=b[0];b[i++]>>=1);
}

int length(const bignum_t a){
int t,ret;
for (ret=(a[0]-1)*DIGIT,t=a[a[0]];t;t/=10,ret++);
return ret>0?ret:1;
}

int digit(const bignum_t a,const int b){
int i,ret;
for (ret=a[(b-1)/DIGIT+1],i=(b-1)%DIGIT;i;ret/=10,i--);
return ret%10;
}

int zeronum(const bignum_t a){
int ret,t;
for (ret=0;!a[ret+1];ret++);
for (t=a[ret+1],ret*=DIGIT;!(t%10);t/=10,ret++);
return ret;
}

void comp(int* a,const int l,const int h,const int d){
int i,j,t;
for (i=l;i<=h;i++)
for (t=i,j=2;t>1;j++)
while (!(t%j))
a[j]+=d,t/=j;
}
133
void convert(int* a,const int h,bignum_t b){
int i,j,t=1;
memset(b,0,sizeof(bignum_t));
for (b[0]=b[1]=1,i=2;i<=h;i++)
if (a[i])
for (j=a[i];j;t*=i,j--)
if (t*i>DEPTH)
mul(b,t),t=1;
mul(b,t);
}

void combination(bignum_t a,int m,int n){
int* t=new int[m+1];
memset((void*)t,0,sizeof(int)*(m+1));
comp(t,n+1,m,1);
comp(t,2,m-n,-1);
convert(t,m,a);
delete []t;
}

void permutation(bignum_t a,int m,int n){
int i,t=1;
memset(a,0,sizeof(bignum_t));
a[0]=a[1]=1;
for (i=m-n+1;i<=m;t*=i++)
if (t*i>DEPTH)
mul(a,t),t=1;
mul(a,t);
}

#define SGN(x) ((x)>0?1:((x)<0?-1:0))
#define ABS(x) ((x)>0?(x):-(x))

int read(bignum_t a,int &sgn,istream& is=cin){
char str[MAX*DIGIT+2],ch,*buf;
int i,j;
memset((void*)a,0,sizeof(bignum_t));
if (!(is>>str)) return 0;
buf=str,sgn=1;
if (*buf=='-') sgn=-1,buf++;
for (a[0]=strlen(buf),i=a[0]/2-1;i>=0;i--)
ch=buf[i],buf[i]=buf[a[0]-1-i],buf[a[0]-1-i]=ch;
for (a[0]=(a[0]+DIGIT-1)/DIGIT,j=strlen(buf);j for (i=1;i<=a[0];i++) 134
for (a[i]=0,j=0;j a[i]=a[i]*10+buf[i*DIGIT-1-j]-'0';
for (;!a[a[0]]&&a[0]>1;a[0]--);
if (a[0]==1&&!a[1]) sgn=0;
return 1;
}
14.2 分数
struct frac{
int num,den;
};

double fabs(double x){
return x>0?x:-x;
}

int gcd(int a,int b){
int t;
if (a<0)
a=-a;
if (b<0)
b=-b;
if (!b)
return a;
while (t=a%b)
a=b,b=t;
return b;
}

void simplify(frac& f){
int t;
if (t=gcd(f.num,f.den))
f.num/=t,f.den/=t;
else
f.den=1;
}

frac f(int n,int d,int s=1){
frac ret;
if (d<0)
ret.num=-n,ret.den=-d;
else
ret.num=n,ret.den=d; 135
if (s)
simplify(ret);
return ret;
}

frac convert(double x){
frac ret;
for (ret.den=1;fabs(x-int(x))>1e-10;ret.den*=10,x*=10);
ret.num=(int)x;
simplify(ret);
return ret;
}

int fraqcmp(frac a,frac b){
int g1=gcd(a.den,b.den),g2=gcd(a.num,b.num);
if (!g1||!g2)
return 0;
return b.den/g1*(a.num/g2)-a.den/g1*(b.num/g2);
}

frac add(frac a,frac b){
int g1=gcd(a.den,b.den),g2,t;
if (!g1)
return f(1,0,0);
t=b.den/g1*a.num+a.den/g1*b.num;
g2=gcd(g1,t);
return f(t/g2,a.den/g1*(b.den/g2),0);
}

frac sub(frac a,frac b){
return add(a,f(-b.num,b.den,0));
}

frac mul(frac a,frac b){
int t1=gcd(a.den,b.num),t2=gcd(a.num,b.den);
if (!t1||!t2)
return f(1,1,0);
return f(a.num/t2*(b.num/t1),a.den/t1*(b.den/t2),0);
}

frac div(frac a,frac b){
return mul(a,f(b.den,b.num,0));
} 136
14.3 矩阵
define MAXN 100

#define fabs(x) ((x)>0?(x):-(x))
#define zero(x) (fabs(x)<1e-10)

struct mat{
int n,m;
double data[MAXN][MAXN];
};

int mul(mat& c,const mat& a,const mat& b){
int i,j,k;
if (a.m!=b.n)
return 0;
c.n=a.n,c.m=b.m;
for (i=0;i for (j=0;j for (c.data[i][j]=k=0;k c.data[i][j]+=a.data[i][k]*b.data[k][j];
return 1;
}

int inv(mat& a){
int i,j,k,is[MAXN],js[MAXN];
double t;
if (a.n!=a.m)
return 0;
for (k=0;k for (t=0,i=k;i for (j=k;j if (fabs(a.data[i][j])>t)
t=fabs(a.data[is[k]=i][js[k]=j]);
if (zero(t))
return 0;
if (is[k]!=k)
for (j=0;j t=a.data[k][j],a.data[k][j]=a.data[is[k]][j],a.data[is[k]][j]=t;
if (js[k]!=k)
for (i=0;i t=a.data[i][k],a.data[i][k]=a.data[i][js[k]],a.data[i][js[k]]=t;
a.data[k][k]=1/a.data[k][k];
for (j=0;j if (j!=k)
a.data[k][j]*=a.data[k][k];
for (i=0;i if (i!=k)
for (j=0;j if (j!=k)
a.data[i][j]-=a.data[i][k]*a.data[k][j];
for (i=0;i if (i!=k)
a.data[i][k]*=-a.data[k][k];
}
for (k=a.n-1;k>=0;k--){
for (j=0;j if (js[k]!=k)
t=a.data[k][j],a.data[k][j]=a.data[js[k]][j],a.data[js[k]][j]=t;
for (i=0;i if (is[k]!=k)
t=a.data[i][k],a.data[i][k]=a.data[i][is[k]],a.data[i][is[k]]=t;
}
return 1;
}

double det(const mat& a){
int i,j,k,sign=0;
double b[MAXN][MAXN],ret=1,t;
if (a.n!=a.m)
return 0;
for (i=0;i for (j=0;j b[i][j]=a.data[i][j];
for (i=0;i if (zero(b[i][i])){
for (j=i+1;j if (!zero(b[j][i]))
break;
if (j==a.n)
return 0;
for (k=i;k t=b[i][k],b[i][k]=b[j][k],b[j][k]=t;
sign++;
}
ret*=b[i][i];
for (k=i+1;k b[i][k]/=b[i][i]; 138
for (j=i+1;j for (k=i+1;k b[j][k]-=b[j][i]*b[i][k];
}
if (sign&1)
ret=-ret;
return ret;
}
14.4 线性方程组
#define MAXN 100
#define fabs(x) ((x)>0?(x):-(x))
#define eps 1e-10

//列主元gauss消去求解 a[][]x[]=b[]
//返回是否有唯一解,若有解在b[]中
int gauss_cpivot(int n,double a[][MAXN],double b[]){
int i,j,k,row;
double maxp,t;
for (k=0;k for (maxp=0,i=k;i if (fabs(a[i][k])>fabs(maxp))
maxp=a[row=i][k];
if (fabs(maxp) return 0;
if (row!=k){
for (j=k;j t=a[k][j],a[k][j]=a[row][j],a[row][j]=t;
t=b[k],b[k]=b[row],b[row]=t;
}
for (j=k+1;j a[k][j]/=maxp;
for (i=k+1;i a[i][j]-=a[i][k]*a[k][j];
}
b[k]/=maxp;
for (i=k+1;i b[i]-=b[k]*a[i][k];
}
for (i=n-1;i>=0;i--)
for (j=i+1;j b[i]-=a[i][j]*b[j];
return 1; 139
}

//全主元gauss消去解a[][]x[]=b[]
//返回是否有唯一解,若有解在b[]中
int gauss_tpivot(int n,double a[][MAXN],double b[]){
int i,j,k,row,col,index[MAXN];
double maxp,t;
for (i=0;i index[i]=i;
for (k=0;k for (maxp=0,i=k;i for (j=k;j if (fabs(a[i][j])>fabs(maxp))
maxp=a[row=i][col=j];
if (fabs(maxp) return 0;
if (col!=k){
for (i=0;i t=a[i][col],a[i][col]=a[i][k],a[i][k]=t;
j=index[col],index[col]=index[k],index[k]=j;
}
if (row!=k){
for (j=k;j t=a[k][j],a[k][j]=a[row][j],a[row][j]=t;
t=b[k],b[k]=b[row],b[row]=t;
}
for (j=k+1;j a[k][j]/=maxp;
for (i=k+1;i a[i][j]-=a[i][k]*a[k][j];
}
b[k]/=maxp;
for (i=k+1;i b[i]-=b[k]*a[i][k];
}
for (i=n-1;i>=0;i--)
for (j=i+1;j b[i]-=a[i][j]*b[j];
for (k=0;k a[0][index[k]]=b[k];
for (k=0;k b[k]=a[0][k];
return 1;
} 140
14.5 线性相关
//判线性相关(正交化)
//传入 m个 n维向量
#include
#define MAXN 100
#define eps 1e-10

int linear_dependent(int m,int n,double vec[][MAXN]){
double ort[MAXN][MAXN],e;
int i,j,k;
if (m>n)
return 1;
for (i=0;i for (j=0;j ort[i][j]=vec[i][j];
for (k=0;k for (e=j=0;j e+=ort[i][j]*ort[k][j];
for (j=0;j ort[i][j]-=e*ort[k][j];
for (e=j=0;j e+=ort[i][j]*ort[i][j];
if (fabs(e=sqrt(e)) return 1;
for (j=0;j ort[i][j]/=e;
}
}
return 0;
}
14.6 日期
//日期函数

int days[12]={31,28,31,30,31,30,31,31,30,31,30,31};
struct date{
int year,month,day;
};

//判闰年
inline int leap(int year){ 141
return (year%4==0&&year%100!=0)||year%400==0;
}

//判合法性
inline int legal(date a){
if (a.month<0||a.month>12)
return 0;
if (a.month==2)
return a.day>0&&a.day<=28+leap(a.year);
return a.day>0&&a.day<=days[a.month-1];
}

//比较日期大小
inline int datecmp(date a,date b){
if (a.year!=b.year)
return a.year-b.year;
if (a.month!=b.month)
return a.month-b.month;
return a.day-b.day;
}

//返回指定日期是星期几
int weekday(date a){
int tm=a.month>=3?(a.month-2):(a.month+10);
int ty=a.month>=3?a.year:(a.year-1);
return (ty+ty/4-ty/100+ty/400+(int)(2.6*tm-0.2)+a.day)%7;
}

//日期转天数偏移
int date2int(date a){
int ret=a.year*365+(a.year-1)/4-(a.year-1)/100+(a.year-1)/400,i;
days[1]+=leap(a.year);
for (i=0;i days[1]=28;
return ret+a.day;
}

//天数偏移转日期
date int2date(int a){
date ret;
ret.year=a/146097*400;
for (a%=146097;a>=365+leap(ret.year);a-=365+leap(ret.year),ret.year++);
days[1]+=leap(ret.year);
for (ret.month=1;a>=days[ret.month-1];a-=days[ret.month-1],ret.month++); 142
days[1]=28;
ret.day=a+1;
return ret;
}