【HDU6514】Monitor（二维树状数组）

题目链接

Monitor Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 163840/163840 K (Java/Others) Total Submission(s): 794    Accepted Submission(s): 251   Problem Description Xiaoteng has a large area of land for growing crops, and the land can be seen as a rectangle of n×m . But recently Xiaoteng found that his crops were often stolen by a group of people, so he decided to install some monitors to find all the people and then negotiate with them. However, Xiao Teng bought bad monitors, each monitor can only monitor the crops inside a rectangle. There are p monitors installed by Xiaoteng, and the rectangle monitored by each monitor is known. Xiao Teng guess that the thieves would also steal q times of crops. he also guessed the range they were going to steal, which was also a rectangle. Xiao Teng wants to know if his monitors can see all the thieves at a time.     Input There are mutiple test cases. Each case starts with a line containing two integers n,m(1≤n,1≤m,n×m≤107) which represent the area of the land. And the secend line contain a integer p(1≤p≤106) which represent the number of the monitor Xiaoteng has installed. This is followed by p lines each describing a rectangle. Each of these lines contains four intergers x1,y1,x2 and y2(1≤x1≤x2≤n,1≤y1≤y2≤m) ,meaning the lower left corner and upper right corner of the rectangle. Next line contain a integer q(1≤q≤106) which represent the number of times that thieves will steal the crops.This is followed by q lines each describing a rectangle. Each of these lines contains four intergers x1,y1,x2 and y2(1≤x1≤x2≤n,1≤y1≤y2≤m) ,meaning the lower left corner and upper right corner of the rectangle.     Output For each case you should print q lines. Each line containing YES or NO mean the all thieves whether can be seen.     Sample Input   6 6 3 2 2 4 4 3 3 5 6 5 1 6 2 2 3 2 5 4 1 5 6 5     Sample Output   YES NO Hint In the picture,the red solid rectangles mean the monitor Xiaoteng installed, and the blue dotted rectangles mean the area will be stolen.     Source 2019中山大学程序设计竞赛（重现赛）

 

【题意】

在一个面积不超过n*m的矩形上，有p个矩形A，问之后的q个矩形B能否被之前的A全部覆盖（每个B的点都在至少一个A中）。

【解题思路】

对于A类矩形(x1,y1,x2,y2)，我们只需要在(x1,y1),(x2+1,y2+1)处+1，在（x1,y2+1）(x2+1,y1)处-1。

之后对整个面积求一个前缀和。则大于0的地方就是被A类矩形覆盖的点。 把值大于0的地方变成1，再一次求一次前缀和，处理好后即可在O(1)的时间算出一个矩形内被覆盖的点的数量。

需要注意的是因为n*m<10^7，所以需要把二维转换成一维来做。

【代码】

#include
using namespace std;
typedef long long LL;
const int maxn=1e7+500;
LL a[maxn],s[maxn];
int n,m,p,q;
int lowbit(int x)
{
    return (-x)&x;
}
void add(int x,int y,int z)
{
    int ty=y;
    while(x<=n)
    {
        y=ty;
        while(y<=m)
        {
            a[x*m+y]+=z;
            y+=lowbit(y);
        }
        x+=lowbit(x);
    }
}
LL query(int x,int y)
{
    LL sum=0;
    int ty=y;
    while(x)
    {
        y=ty;
        while(y)
        {
            sum+=a[x*m+y];
            y-=lowbit(y);
        }
        x-=lowbit(x);
    }
    return sum;
}
int main()
{
    while(~scanf("%d%d",&n,&m))
    {
        memset(a,0,sizeof(a));
        memset(s,0,sizeof(s));
         int x1,y1,x2,y2;
         scanf("%d",&p);
         while(p--)
         {
             scanf("%d%d%d%d",&x1,&y1,&x2,&y2);
             add(x1,y1,1);
             add(x1,y2+1,-1);
             add(x2+1,y1,-1);
             add(x2+1,y2+1,1);
         }
         for(int i=1;i<=n;i++)
         {
             for(int j=1;j<=m;j++)
             {
                 int t=query(i,j);
                 if(t>0)s[i*m+j]=1;
                 else s[i*m+j]=0;
             }
         }
         for(int i=1;i<=n;i++)
         {
             for(int j=1;j<=m;j++)
                s[i*m+j]+=s[(i-1)*m+j]+s[i*m+j-1]-s[(i-1)*m+j-1];
         }
         scanf("%d",&q);
         while(q--)
         {
             scanf("%d%d%d%d",&x1,&y1,&x2,&y2);
             LL t1=(x2-x1+1)*(y2-y1+1);
             LL t2=s[x2*m+y2]+s[(x1-1)*m+y1-1]-s[(x1-1)*m+y2]-s[x2*m+y1-1];
             if(t1==t2)printf("YES\n");
             else printf("NO\n");

         }
    }
}