[平衡树]牛耳杯程序设计大赛决赛D题——BallIntheBox

题目描述：

BallsInTheBox

Timelimit:1sMemorylimit:32768kb

ProblemDescription

ThereareNboxesinStaginner’shouse,andwemarkthemby1,2,…,N.ThereareNi(1<=Ni<=10^4)ballsintheboxiinitially.NowStaginnerwilldosomeoperations,andhewillaskyousomesimplequestions.

Theoperationscontain:

1.Cij:takealltheballsintheboxiintotheboxjandthrowtheboxiaway,youcansurethatiisdifferentfromj.

2.Ain:addn(1<=n<=10^4)ballsintotheboxi.

3.Bin:taken(n>=0)ballsawayfromboxi,youcansurethatnisnotbiggerthanthenumberofballsintheboxi.

4.Qk:Staginneraskyouforthek-thsmallestnumberofballsinthebox,youcansurethatkisnotbiggerthanthetotalnumberofboxes.

Input:

Thereareseveraltestcases.

Thefirstlineofeachcasecontainstwointegers,N(1<=N<=10^5),M(1<=M<=10^5),indicatesthenumberofboxesandthenumberofoperations.ThenextlinecontainsNintegers,thei-thintegerindicatesthenumberofballsintheboxi.ThentherewillbeMlines,eachlinehasanoperationlikesomeoneoftheoperationsinthedescription.

Output:

Foreachtestcase,youneedonlyprintoneintegerforeachoperation“Qk”,indicatesthek-thsmallestnumberofballsinthebox.

SampleInput:

37

123

A31

C21

Q1

B34

Q1

C31

Q1

SampleOutput:

3

0

3

题目意思很简单，给定N个盒子和M个操作，然后给出初始时每个盒子里球的个数，接下来时M个操作，操作分为四种，

A a b ：把编号为a的盒子里的球的数量增加b

B a b ：把编号为a的盒子里的球的数量减少b

C a b ：把编号为a的盒子里的球全倒进编号为b的盒子里，然后扔掉盒子a

Q k ：查询球数第k小的盒子，对于每一个Q输出一行为该盒子里的球数

思路，如果用数组直接存，A B C 三个操作都容易实现，复杂度为O(1)，然而Q操作每次都会花费O(n)的时间，n达到10^5，而且有10^5个询问，给定时限只有1s，肯定超时，于是想到用平衡二叉树来动态维护这些盒子，这样A B C D的时间复杂度都为O(logn)，应该不会超时。

郁结的过程：比赛当时由于SBT不会写。。AVL写的不熟，然后果断放弃了该题，比赛之后弄到了题目和数据，自己在本地慢慢的搞，起初只用盒子里的球数来作为关键码，这样对于球数相同的情况是不能够处理了（经过旋转会破坏BST的性质），然后想到用盒子的球数和盒子的编号一起做关键码（球数为第一关键码，编号为第二关键码），这样就能够保证所有的关键码都不相同，于是终于AC了。 总算是把AVL写的比较熟练了。虽然代码风格比较挫。

代码：

#include<iostream>
#include<cstdio>
#include<algorithm>
using namespace std;
const int MAXN = 100010;
//关键码结点类 
struct Node
{
   int wgt;
   int num;
   Node(int w=0,int n=0):wgt(w),num(n){}
   bool operator < (const Node &n) const
   {
        if(wgt!=n.wgt)return wgt<n.wgt;
        else return num<n.num;
   }   
   bool operator == (const Node &n) const
   {
        return wgt==n.wgt&&num==n.num;
   }
};
//AVL树 
struct AVLTree
{
   int l[MAXN],r[MAXN],size[MAXN],next[MAXN],h[MAXN],ROOT;
   Node Key[MAXN];
   int newnode(int w,int n)
   {
        int node = next[0];
        Key[node].wgt = w;
        Key[node].num = n;
        next[0] = next[node];
        h[node] = size[node] = 1;
        return node;
   }
   void delnode(int x)
   {
        next[x] = next[0];
        next[0] = x;
        Key[x].wgt = Key[x].num = h[x] = size[x] = l[x] = r[x] = 0;
   }
   void init()
   {
       for(int i=0;i<MAXN;i++)next[i]=i+1;
       memset(l,0,sizeof(l));
       memset(r,0,sizeof(r));
       memset(size,0,sizeof(size));
       memset(h,0,sizeof(h));
       ROOT = 0;
   }    
   void left_rotate(int &p)
   {
       int x = r[p];
       r[p] = l[x];
       l[x] = p;
       size[p] = size[l[p]]+size[r[p]]+1;
       size[x] = size[l[x]]+size[r[x]]+1;
       h[p] = max(h[l[p]],h[r[p]])+1;
       h[x] = max(h[l[x]],h[r[x]])+1;
       p = x;
   }
   void right_rotate(int &p)
   {
       int x = l[p];
       l[p] = r[x];
       r[x] = p;
       size[p] = size[l[p]]+size[r[p]]+1;
       size[x] = size[l[x]]+size[r[x]]+1;
       h[p] = max(h[l[p]],h[r[p]])+1;
       h[x] = max(h[l[x]],h[r[x]])+1;
       p = x;
   }
   void insert(int &p,Node k)
   {
        if(!p)
        {
              
            p = newnode(k.wgt,k.num);
            return;
        }
        if(k<Key[p])insert(l[p],k);
        else insert(r[p],k);
        h[p] = max(h[l[p]],h[r[p]])+1;
        size[p] = size[l[p]]+size[r[p]]+1;
        if(h[l[p]]-h[r[p]]==2)//L
        {
            if(h[l[l[p]]]>h[r[l[p]]])//LL
            {
                right_rotate(p);
            }
            else 
            {
                left_rotate(l[p]);
                right_rotate(p);
            }
        }
        else if(h[l[p]]-h[r[p]]==-2)//R
        {
            if(h[r[r[p]]]>h[l[r[p]]])//RR
            {
                left_rotate(p);
            }
            else 
            {
                right_rotate(r[p]);
                left_rotate(p);
            }
        }
   }
   int findmin(int &p)
   {
       if(l[p])return findmin(l[p]);
       return p;
   }
   void remove(int &p,Node k)
   {
       if(p)
       {
           if(k==Key[p])
           {
               if(!l[p])
               {
                   int x = p;
                   p = r[p];
                   delnode(x);
               }
               else if(!r[p])
               {
                   int x = p;
                   p = l[p];
                   delnode(x);
               }
               else
               {
                   int x = findmin(r[p]);
                   Key[p] = Key[x];
                   remove(r[p],Key[x]);
               }
           }
           else if(k<Key[p])remove(l[p],k);
           else remove(r[p],k);
           if(p)
           {
               h[p] = max(h[l[p]],h[r[p]])+1;
               size[p] = size[l[p]]+size[r[p]]+1;
           }
            if(h[l[p]]-h[r[p]]==2)//L
            {
                if(h[l[l[p]]]>h[r[l[p]]])//LL
                {
                    right_rotate(p);
                }
                else 
                {
                    left_rotate(l[p]);
                    right_rotate(p);
                }
            }
            else if(h[l[p]]-h[r[p]]==-2)//R
            {
                if(h[r[r[p]]]>h[l[r[p]]])//RR
                {
                    left_rotate(p);
                }
                else 
                {
                    right_rotate(r[p]);
                    left_rotate(p);
                }
            }
       } 
   }
   int Select(int p,int k)
   {
       int rank = size[l[p]]+1;
       if(rank==k)return p;
       else if(rank<k) return Select(r[p],k-rank);
       return Select(l[p],k);
   }
}avl;
int a[MAXN];
int main()
{
    int n,m;
    freopen("D.in","r",stdin);
    freopen("DansK2.out","w",stdout);
    while(scanf("%d%d",&n,&m)!=EOF)
    {
        avl.init();
        for(int i=1;i<=n;i++)scanf("%d",&a[i]);
        for(int i=1;i<=n;i++)avl.insert(avl.ROOT,Node(a[i],i));
        for(int i=0;i<m;i++)
        {
            char op[5];int x,y;
            scanf("%s",op);
            if(op[0]=='A')
            {
                scanf("%d%d",&x,&y);
                avl.remove(avl.ROOT,Node(a[x],x));
                a[x]+=y;
                avl.insert(avl.ROOT,Node(a[x],x));
            }
            else if(op[0]=='B')
            {
                scanf("%d%d",&x,&y);
                avl.remove(avl.ROOT,Node(a[x],x));
                a[x]-=y;
                avl.insert(avl.ROOT,Node(a[x],x));
            }
            else if(op[0]=='C')
            {
                scanf("%d%d",&x,&y);
                avl.remove(avl.ROOT,Node(a[x],x));
                avl.remove(avl.ROOT,Node(a[y],y));
                a[y]+=a[x];
                avl.insert(avl.ROOT,Node(a[y],y));
            }
            else
            {
                scanf("%d",&x);
                printf("%d\n",avl.Key[avl.Select(avl.ROOT,x)].wgt);
            }
        }
    }
    return 0;
}