这种树是一种无需旋转操作的Treap树,由FHQ(范浩强)大佬发明,堪称是神级的数据结构!他短小精悍,易于学习,而其思想之优雅令无数OIer/ACMer以及程序员们为之着迷!
而这一切都来源于这一份ppt:范浩强谈数据结构
在了解这个数据结构之前最好先了解一下treap树以及笛卡尔树,这两种树本身具有一定的相似性,而为FHQ_Treap提供了思想延申的起点。
FHQ_Treap代替旋转操作的核心就是树的分离(split)与合并(merge)两种操作,使得我们可以将原树按照值/位置(下标)分解为两棵树,以及将两棵树合并为一棵平衡的树,而剩下的一切操作均基于这两种操作!
按值的split:
void split(int x ,int k, int &l, int &r)
{
if(!x) {l=r=0;return;}//到空节点返回0
pushdown(x);
if(val[x]<=k){l=x;split(rs[l],rs[l],r,k); pushup(l);}//x分给左树,接着分x的右儿子
else{r=x;split(ls[r],l,ls[r],k);pushup(r);}//x分给右树,接着分x的左儿子
}
按权值分裂应用于维护有序序列时(类比于STL的set,或堆),可以查询kth大等
按下标的split:
void split(int x, int k, int &l, int &r)
{
if(!x) {l=r=0;return;}
pushdown(x);
if(sz[ls[x]]+1<=k){l=x;split(rs[l],rs[l],r,k-sz[ls[x]]-1);pushup(l);}//注意修改k
else{r=x;split(ls[r],l,ls[r],k);pushup(r);}
}
按下标分裂用于维护普通序列,可进行插入节点、区间和查询、RMQ、区间修改等
合并(merge):
int merge(int x, int y)// 合并
{
if(!x || !y) return x+y;// x和y中必定有一个是0
if(pri[x] < pri[y])// 把x加到左边的树上
{
pushdown(x);
rs[x] = merge(rs[x],y);
pushup(x);
return x;
}
else
{
pushdown(y);
ls[y] = merge(x, ls[y]);
pushup(y);
return y;
}
}
split与merge的pushup与pushdown按具体情况自行增删修改即可。
鉴于本帖为模板向,不详细解释其背后原理(其实是自己不太会啦。。)
直接上题!!
#include
using namespace std;
typedef long long ll;
typedef unsigned long long usi;
const ll MOD = 998244353;
const int MAXN = 1e5 + 7;
const int INF = 0x3f3f3f3f;
const double PI = acos(-1.0);
const double eps = 1e-12;
int ch[MAXN][3], val[MAXN], pri[MAXN], sz[MAXN];
class FHQ_Treap
{
public:
int size = 0;
bool empty() {return size == 0;}
void insert(int a)
{
split(root, a, x, y);
root = merge(merge(x, newnode(a)), y);
}
void erase(int a)
{
split(root, a, x, z);
split(x, a-1, x, y);
y = merge(ch[y][0], ch[y][1]);
root = merge(merge(x, y), z);
}
int numrank(int a)
{
split(root, a-1, x, y);
int res = sz[x] + 1;
root = merge(x, y);
return res;
}
int kth(int k)
{
return val[myrank(root, k)];
}
int pre(int a)
{
split(root, a-1, x, y);
int res = val[myrank(x, sz[x])];
root = merge(x, y);
return res;
}
int next(int a)
{
split(root, a, x, y);
int res = val[myrank(y, 1)];
root = merge(x, y);
return res;
}
private:
int root = 0, x, y, z;
void update(int x)
{
sz[x] = sz[ch[x][0]] + sz[ch[x][1]] + 1;
}
int newnode(int v)
{
sz[++size] = 1;
val[size] = v;
pri[size] = rand();
return size;
}
int merge(int x, int y)
{
if(!x || !y) return x + y;
if(pri[x] < pri[y])
{
ch[x][1] = merge(ch[x][1],y);
update(x);
return x;
}
else
{
ch[y][0] = merge(x, ch[y][0]);
update(y);
return y;
}
}
void split(int now, int k, int &x, int &y)
{
if(!now) x = y = 0;
else
{
if(val[now] <= k) x = now, split(ch[now][1], k, ch[now][1], y);
else y = now, split(ch[now][0], k, x, ch[now][0]);
update(now);
}
}
int myrank(int now, int a)
{
while(true)
{
if(a < sz[ch[now][0]]) now = ch[now][0];
else if(a == sz[ch[now][0]] + 1) return now;
else a -= sz[ch[now][0]] + 1, now = ch[now][1];
}
}
};
FHQ_Treap mytree;
int main()
{
srand((unsigned)time(NULL));
int n, opt, a;
cin >> n;
while(n--)
{
scanf("%d%d", &opt, &a);
if(opt == 1) mytree.insert(a);
else if(opt == 2) mytree.erase(a);
else if(opt == 3) printf("%d\n", mytree.numrank(a));
else if(opt == 4) printf("%d\n", mytree.kth(a));
else if(opt == 5) printf("%d\n", mytree.pre(a));
else if(opt == 6) printf("%d\n", mytree.next(a));
}
return 0;
}
本题使用了建笛卡尔树的方法优化建树到O(n)
#include
using namespace std;
typedef long long ll;
const int MAXN = 2e5 + 5;
struct FHQ_Treap
{
int size = 0, root = 0;
struct node
{
int ls, rs, rnd, siz;
ll val, sum, lazy;
node() {}
node(int LS, int RS, int RND, int SIZ, ll VAL, ll SUM, ll LAZY):
ls(LS), rs(RS), rnd(RND), siz(SIZ), val(VAL), sum(SUM), lazy(LAZY) {}
}tree[MAXN];
inline void pushUp(int p)
{
tree[p].siz = tree[tree[p].ls].siz + tree[tree[p].rs].siz + 1;
tree[p].sum = tree[tree[p].ls].sum + tree[tree[p].rs].sum + tree[p].val;
}
inline void pushDown(int p)
{
if(tree[p].lazy)
{
tree[tree[p].ls].sum += 1ll * tree[tree[p].ls].siz * tree[p].lazy;
tree[tree[p].ls].lazy += tree[p].lazy;
tree[tree[p].ls].val += tree[p].lazy;
tree[tree[p].rs].sum += 1ll * tree[tree[p].rs].siz * tree[p].lazy;
tree[tree[p].rs].lazy += tree[p].lazy;
tree[tree[p].rs].val += tree[p].lazy;
tree[p].lazy = 0;
}
}
void split(int p, int rk, int &x, int &y)
{
if(!p) {x = y = 0; return;}
pushDown(p);
if(rk <= tree[tree[p].ls].siz)
{
y = p;
split(tree[p].ls, rk, x, tree[p].ls);
pushUp(y);
}
else
{
x = p;
split(tree[p].rs, rk-tree[tree[p].ls].siz-1, tree[p].rs, y);
pushUp(x);
}
}
int merge(int x, int y)
{
if(!x || !y) return x | y;
if(tree[x].rnd < tree[y].rnd)
{
pushDown(x);
tree[x].rs = merge(tree[x].rs, y);
pushUp(x);
return x;
}
else
{
pushDown(y);
tree[y].ls = merge(x, tree[y].ls);
pushUp(y);
return y;
}
}
int newnode(ll v)
{
int p = ++size;
tree[p] = node(0, 0, rand(), 1, v, v, 0);
return p;
}
void insert(ll val, int p)
{
int x, y;
split(root, p - 1, x, y);
root = merge(merge(x, newnode(val)), y);
}
void add(int l, int r, ll v)
{
int x, y, z;
split(root, l - 1, x, y);
split(y, r + 1 - l, y, z);
tree[y].sum += 1ll * tree[y].siz * v;
tree[y].lazy += v;
tree[y].val += v;
root = merge(merge(x, y), z);
}
void query(int l, int r)
{
int x, y, z;
split(root, l - 1, x, y);
split(y, r + 1 - l, y, z);
printf("%lld\n", tree[y].sum);
root = merge(merge(x, y), z);
}
int st[MAXN], top;
inline void build(int n)
//用建笛卡尔树的方法O(n)建树,st为栈,返回新树的根
{
top = 0;
for(int i = 0; i < n; ++i)
{
ll val;
scanf("%lld", &val);
int tmp = newnode(val), last = 0;
while(top && tree[st[top]].rnd > tree[tmp].rnd)
{
last = st[top];
pushUp(last);
st[top--] = 0;
}
if(top) tree[st[top]].rs = tmp;
tree[tmp].ls = last;
st[++top] = tmp;
}
while(top) pushUp(st[top--]);
root = st[1];
}
} mytree;
int main()
{
int n, m, op, pos, l, r;
ll v;
scanf("%d", &n);
mytree.build(n);
scanf("%d", &m);
while(m--)
{
scanf("%d", &op);
if(op == 1) scanf("%d", &pos), mytree.insert(0, pos);
else if(op == 2) scanf("%d%d%lld", &l, &r, &v), mytree.add(l, r, v);
else scanf("%d%d", &l, &r), mytree.query(l, r);
}
return 0;
}
#include
#include
#include
#define N 100005
int Root;
int lazy[N];
int n,m,cnt;
int val[N],sze[N];
int ch[N][2],prio[N];
void pushup(int o){
sze[o]=sze[ch[o][0]]+sze[ch[o][1]]+1;
}
void pushdown(int o){
if(!lazy[o] or !o) return;
ch[o][0]^=ch[o][1]^=ch[o][0]^=ch[o][1];
lazy[ch[o][0]]^=1;
lazy[ch[o][1]]^=1;
lazy[o]=0;
}
void split(int o,int k,int &x,int &y){
if(!o) x=y=0;
else{
pushdown(o);
if(k>sze[ch[o][0]]) x=o,split(ch[o][1],k-sze[ch[o][0]]-1,ch[o][1],y);
else y=o,split(ch[o][0],k,x,ch[o][0]);
pushup(o);
}
}
int merge(int x,int y){
if(!x or !y) return x+y;
pushdown(x); pushdown(y);
if(prio[x]<prio[y]){
ch[x][1]=merge(ch[x][1],y);
pushup(x);
return x;
}
else{
ch[y][0]=merge(x,ch[y][0]);
pushup(y);
return y;
}
}
int newnode(int v){
val[++cnt]=v;
sze[cnt]=1;
prio[cnt]=rand();
return cnt;
}
void res(int l,int r){
int a,b,c,d;
split(Root,r,a,b);
split(a,l-1,c,d);
lazy[d]^=1;
Root=merge(merge(c,d),b);
}
void dfs(int now){
if(!now) return;
pushdown(now);
dfs(ch[now][0]);
printf("%d ",val[now]);
dfs(ch[now][1]);
}
signed main(){
srand(time(0));
scanf("%d%d",&n,&m);
for(int i=1;i<=n;i++)
Root=merge(Root,newnode(i));
//printf("Root=%d\n",Root);
for(int x,y,i=1;i<=m;i++){
scanf("%d%d",&x,&y);
res(x,y);
//printf("i=%d\n",i);
//dfs(Root);
}
//printf("Root=%d\n",Root);
dfs(Root);
return 0;
}