今天学习了一下AVL,顺便用AVL树 A掉了这道经典的题,以前用树状数组解过这个题,今天记录一下AVL的解法。如有错误之处欢迎指正,各位大牛不要笑话我。
该题需要用平衡树:定义这样的一颗平衡树,根节点大于等于左儿子节点,小于右儿子节点。(也可以规范的定义左儿子小于根,小于右儿子,但是要加入新的数据域)
为了球第K小,除了原本的height域以外加入一个size域,表示以当前节点为跟的子树有多少个节点。
操作1:数据域的维护
首先要在旋转前维护该节点的子树的height和size,然后才能根据更新的数据,判断该树是否平衡,然后旋转
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void
fix(node* &R)
{
R->h = max(R->rchild->h,R->lchild->h) + 1;
R->size = R->rchild->size + R->lchild->size + 1;
}
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操作2:旋转化操作
插入和删除的时候用同样的旋转维护该树
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void
rightsinglerotate(node* &R)
//LL型旋转,单旋一次
{
node * lc = R->lchild;
R->lchild = lc->rchild;
fix(R);
lc->rchild = R;
R = lc;
fix(R);
}
void
leftsinglerotate(node* &R)
//RR型旋转,单选一次
{
node * rc = R->rchild;
R->rchild = rc->lchild;
fix(R);
rc->lchild = R;
R = rc;
fix(R);
}
void
leftdoublerotate(node* &R)
//RL型旋转,双旋
{
rightsinglerotate(R->rchild);
leftsinglerotate(R);
}
void
rightdoublerotate(node* &R)
//LR型旋转,双旋
{
leftsinglerotate(R->lchild);
rightsinglerotate(R);
}
void
maintain(node* &R)
{
if
(R->lchild != MYNULL)
{
if
(R->lchild->lchild->h == R->rchild->h + 1)
rightsinglerotate(R);
else
if
(R->lchild->rchild->h == R->rchild->h + 1)
rightdoublerotate(R);
}
if
(R->rchild != MYNULL)
{
if
(R->rchild->rchild->h == R->lchild->h + 1)
leftsinglerotate(R);
else
if
(R->rchild->lchild->h == R->lchild->h + 1)
leftdoublerotate(R);
}
}
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操作3:求第K小的操作,也就是size域的用途
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int
findK(node* &R,
int
k)
{
if
(k == R->lchild->size + 1)
return
R->value;
else
if
(k <= R->lchild->size)
return
findK(R->lchild,k);
else
if
(k > R->size - R->rchild->size)
return
findK(R->rchild,k + R->rchild->size - R->size);
}
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操作4:erase函数,要完成该题的统计功能,看erase了多少次
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void
erase(node* &R,T value)
{
if
(R == MYNULL)
return
;
if
(R->value == value)
{
if
(R->rchild == MYNULL)
{
node * tmp = R;
R = tmp->lchild;
}
else
{
node *tmp = R->rchild;
while
(tmp->lchild != MYNULL)
tmp = tmp->lchild;
R->value = tmp->value;
erase(R->rchild,tmp->value);
fix(R);
}
return
;
}
else
if
(value < R->value)
erase(R->lchild,value);
else
if
(value < R->value)
erase(R->rchild,value);
fix(R);
maintain(R);
}
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最后在调整树使其不失衡。
有了这些操作就可以轻松的完成题目的要求了,
完整的代码如下
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/*
* =====================================================================================
*
* Filename: AVLtree.cpp
*
* Description: AVLtree with template
*
* Version: 1.0
* Created: 2010年12月27日 09时45分44秒
* Revision: none
* Compiler: gcc
*
* Author: ronaflx
* Company: hit-acm-group
*
* =====================================================================================
*/
#include <iostream>
#include <cstring>
#include <cstdio>
using
namespace
std;
int
delta;
const
int
INF = 10000000;
template
<
typename
T>
class
AVL
{
public
:
AVL()
{
pp = pool;
TMP = node(0,0,NULL,NULL);
MYNULL = &TMP;
roof = MYNULL;
}
void
insert(T k)
{
insert(roof,k);
}
void
erase(T k)
{
erase(roof,k);
}
bool
empty()
{
return
roof == MYNULL;
}
int
findK(
int
k)
{
if
(k <= 0)
return
-INF;
return
findK(roof,k);
}
struct
node
{
node *lchild,*rchild;
T value;
int
h,size;
node(){}
node (
int
h,
int
size,node * lchild,node *rchild)
{
this
->size = size;
this
->h = h;
this
->lchild = lchild;
this
->rchild = rchild;
}
};
node* roof;
private
:
#define max(a,b) ((a) < (b) ? (b) : (a))
static
const
int
N = 1000000;
node* MYNULL,TMP;
//为了快速方便的求高度而设立的虚空节点
node pool[N],*pp;
int
findK(node* &R,
int
k)
{
if
(k == R->lchild->size + 1)
return
R->value;
else
if
(k <= R->lchild->size)
return
findK(R->lchild,k);
else
if
(k > R->size - R->rchild->size)
return
findK(R->rchild,k + R->rchild->size - R->size);
}
void
fix(node* &R)
{
R->h = max(R->rchild->h,R->lchild->h) + 1;
R->size = R->rchild->size + R->lchild->size + 1;
}
void
rightsinglerotate(node* &R)
//LL型旋转,单旋一次
{
node * lc = R->lchild;
R->lchild = lc->rchild;
fix(R);
lc->rchild = R;
R = lc;
fix(R);
}
void
leftsinglerotate(node* &R)
//RR型旋转,单选一次
{
node * rc = R->rchild;
R->rchild = rc->lchild;
fix(R);
rc->lchild = R;
R = rc;
fix(R);
}
void
leftdoublerotate(node* &R)
//RL型旋转,双旋
{
rightsinglerotate(R->rchild);
leftsinglerotate(R);
}
void
rightdoublerotate(node* &R)
//LR型旋转,双旋
{
leftsinglerotate(R->lchild);
rightsinglerotate(R);
}
void
maintain(node* &R)
{
if
(R->lchild != MYNULL)
{
if
(R->lchild->lchild->h == R->rchild->h + 1)
rightsinglerotate(R);
else
if
(R->lchild->rchild->h == R->rchild->h + 1)
rightdoublerotate(R);
}
if
(R->rchild != MYNULL)
{
if
(R->rchild->rchild->h == R->lchild->h + 1)
leftsinglerotate(R);
else
if
(R->rchild->lchild->h == R->lchild->h + 1)
leftdoublerotate(R);
}
}
void
insert(node* &R,T value)
{
if
(R == MYNULL)
{
R = mynew(value);
return
;
}
else
if
(value <= R->value)
insert(R->lchild,value);
else
if
(value > R->value)
insert(R->rchild,value);
fix(R);
maintain(R);
}
void
erase(node* &R,T value)
{
if
(R == MYNULL)
return
;
if
(R->value == value)
{
if
(R->rchild == MYNULL)
{
node * tmp = R;
R = tmp->lchild;
}
else
{
node *tmp = R->rchild;
while
(tmp->lchild != MYNULL)
tmp = tmp->lchild;
R->value = tmp->value;
erase(R->rchild,tmp->value);
fix(R);
}
return
;
}
else
if
(value < R->value)
erase(R->lchild,value);
else
if
(value < R->value)
erase(R->rchild,value);
fix(R);
maintain(R);
}
node* mynew(T value)
{
pp->lchild = pp->rchild = MYNULL;
pp->size = pp->h = 1;
pp->value = value;
return
pp++;
}
#undef max
};
AVL<
int
> avltree;
int
main()
{
int
n,minn;
char
cmd;
int
f,cnt = 0;
while
(
scanf
(
"%d %d"
,&n,&minn) == 2)
{
delta = 0;
cnt = 0;
while
(!avltree.empty())
{
avltree.erase(avltree.roof->value);
}
for
(
int
i = 0;i < n;i++)
{
scanf
(
" %c %d"
,&cmd,&f);
if
(cmd ==
'I'
)
{
if
(f < minn)
continue
;
avltree.insert(f - delta);
}
else
if
(cmd ==
'A'
)
delta += f;
else
if
(cmd ==
'F'
)
{
int
tmp = avltree.findK(avltree.roof->size - f + 1);
if
(tmp == -INF)
printf
(
"-1\n"
);
else
printf
(
"%d\n"
,tmp + delta);
}
else
if
(cmd ==
'S'
)
{
delta -= f;
while
(!avltree.empty())
{
int
tmp = avltree.findK(1);
tmp += delta;
if
(tmp >= minn)
break
;
else
{
cnt++;
avltree.erase(tmp - delta);
}
}
}
}
printf
(
"%d\n"
,cnt);
}
return
0;
}
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