POJ 1054 The Troublesome Frog (hash散列）

The Troublesome Frog

Time Limit: 5000MS		Memory Limit: 100000K
Total Submissions: 9126		Accepted: 2749
Case Time Limit: 500MS

Description

In Korea, the naughtiness of the cheonggaeguri, a small frog, is legendary. This is a well-deserved reputation, because the frogs jump through your rice paddy at night, flattening rice plants. In the morning, after noting which plants have been flattened, you want to identify the path of the frog which did the most damage. A frog always jumps through the paddy in a straight line, with every hop the same length: 
 
Your rice paddy has plants arranged on the intersection points of a grid as shown in Figure-1, and the troublesome frogs hop completely through your paddy, starting outside the paddy on one side and ending outside the paddy on the other side as shown in Figure-2: 
 
Many frogs can jump through the paddy, hopping from rice plant to rice plant. Every hop lands on a plant and flattens it, as in Figure-3. Note that some plants may be landed on by more than one frog during the night. Of course, you can not see the lines showing the paths of the frogs or any of their hops outside of your paddy ?for the situation in Figure-3, what you can see is shown in Figure-4: 
 
From Figure-4, you can reconstruct all the possible paths which the frogs may have followed across your paddy. You are only interested in frogs which have landed on at least 3 of your rice plants in their voyage through the paddy. Such a path is said to be a frog path. In this case, that means that the three paths shown in Figure-3 are frog paths (there are also other possible frog paths). The vertical path down column 1 might have been a frog path with hop length 4 except there are only 2 plants flattened so we are not interested; and the diagonal path including the plants on row 2 col. 3, row 3 col. 4, and row 6 col. 7 has three flat plants but there is no regular hop length which could have spaced the hops in this way while still landing on at least 3 plants, and hence it is not a frog path. Note also that along the line a frog path follows there may be additional flattened plants which do not need to be landed on by that path (see the plant at (2, 6) on the horizontal path across row 2 in Figure-4), and in fact some flattened plants may not be explained by any frog path at all. 

Your task is to write a program to determine the maximum number of landings in any single frog path (where the maximum is taken over all possible frog paths). In Figure-4 the answer is 7, obtained from the frog path across row 6. 

Input

Your program is to read from standard input. The first line contains two integers R and C, respectively the number of rows and columns in your rice paddy, 1 <= R,C <= 5000. The second line contains the single integer N, the number of flattened rice plants, 3 <= N <= 5000. Each of the remaining N lines contains two integers, the row number (1 <= row number <= R) and the column number (1 <= column number <= C) of a flattened rice plant, separated by one blank. Each flattened plant is only listed once.

Output

Your program is to write to standard output. The output contains one line with a single integer, the number of plants flattened along a frog path which did the most damage if there exists at least one frog path, otherwise, 0.

Sample Input

6 7
14
2 1
6 6
4 2
2 5
2 6
2 7
3 4
6 1
6 2
2 3
6 3
6 4
6 5
6 7

Sample Output

7

Source

IOI 2002

 

题意：在一个row*col的矩阵里，留了n个青蛙的脚印。已知每只青蛙的步长是一定的，且都是从一个边界外，跳到了另一边的边界外，而且跳的是直线。每跳落地就留一个脚印在那个格子。现在问你那只在地里留下最多脚印的青蛙留下了多少只脚印(只要有可能就好)。

 

思路：离散的思想，先排序，然后枚举每两个脚印做直线，判断这条直线能否构成一只青蛙的跳动轨迹。一开始觉得这样的思想是0(n^3)，但是剪枝做得好话，应该可以接近0(n^2)。

 

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>

using namespace std;

const int maxn=5010;

int hash[maxn][maxn];
int row,col;

struct node{
    int r,c;
}frog[maxn];

int cmp(node a,node b){ //以纵坐标先比较，当c相同时再比较r，，因为我们frog是从左向右穿过。。。。。
    if(a.c==b.c)
        return a.r<b.r;
    return a.c<b.c;
}

int inmap(int r,int c){
    if(r>0 && r<=row && c>0 && c<=col)
        return 1;
    return 0;
}

int solve(int r,int c,int dr,int dc){
    int cnt=0;
    while(inmap(r,c)){
        if(!hash[r][c]) //有没有被frog踩过
            return 0;
        cnt++;
        r+=dr;
        c+=dc;
    }
    return cnt;
}

int main(){

    //freopen("input.txt","r",stdin);

    int n,dr,dc;
    while(~scanf("%d%d",&row,&col)){
        memset(hash,0,sizeof(hash));
        scanf("%d",&n);
        for(int i=0;i<n;i++){
            scanf("%d%d",&frog[i].r,&frog[i].c);
            hash[frog[i].r][frog[i].c]=1;
        }
        sort(frog,frog+n,cmp);
        int ans=2;  //踩过的点至少为3，才能输出，可以先设为2再比较
        for(int i=0;i<n-1;i++)
            for(int j=i+1;j<n;j++){
                dr=frog[j].r-frog[i].r; 
                dc=frog[j].c-frog[i].c;
                if(frog[i].c+ans*dc>col)    //剪枝一：有没有跳出去？
                    break;
                if(inmap(frog[i].r-dr,frog[i].c-dc))    //剪枝二：是不是从外面跳进来的
                    continue;
                if(!inmap(frog[i].r+ans*dr,frog[i].c+ans*dc))   //剪枝三：在剪枝一（没有跳出去）的情况下，这一步是不是青蛙走过的路
                    continue;
                ans=max(ans,solve(frog[i].r,frog[i].c,dr,dc));
            }
        if(ans<3)
            printf("0\n");
        else
            printf("%d\n",ans);
    }
    return 0;
}