POJ1579:Function Run Fun

Description

We all love recursion! Don't we? 

Consider a three-parameter recursive function w(a, b, c): 

if a <= 0 or b <= 0 or c <= 0, then w(a, b, c) returns: 
1 

if a > 20 or b > 20 or c > 20, then w(a, b, c) returns: 
w(20, 20, 20) 

if a < b and b < c, then w(a, b, c) returns: 
w(a, b, c-1) + w(a, b-1, c-1) - w(a, b-1, c) 

otherwise it returns: 
w(a-1, b, c) + w(a-1, b-1, c) + w(a-1, b, c-1) - w(a-1, b-1, c-1) 

This is an easy function to implement. The problem is, if implemented directly, for moderate values of a, b and c (for example, a = 15, b = 15, c = 15), the program takes hours to run because of the massive recursion.

Input

The input for your program will be a series of integer triples, one per line, until the end-of-file flag of -1 -1 -1. Using the above technique, you are to calculate w(a, b, c) efficiently and print the result.

Output

Print the value for w(a,b,c) for each triple.

Sample Input

1 1 1
2 2 2
10 4 6
50 50 50
-1 7 18
-1 -1 -1

Sample Output

w(1, 1, 1) = 2
w(2, 2, 2) = 4
w(10, 4, 6) = 523
w(50, 50, 50) = 1048576
w(-1, 7, 18) = 1

Source

Pacific Northwest 1999

 

方法一：

直接递归会超时，预处理即可。实现把所有结果存到三维数组中。

#include <iostream>
#include <cstdio>

using namespace std;

int main()
{
    int w[21][21][21], ans, a, b, c;
    //预处理 
    for(int i = 0; i < 21; i++)
        for(int j = 0; j < 21; j++)
            for(int k = 0; k < 21; k++)
            {
                if(!i || !j || !k) w[i][j][k] = 1;
                else if(i < j && j < k) 
                    w[i][j][k] = w[i][j][k-1] + w[i][j-1][k-1] - w[i][j-1][k];
                    //w(a, b, c-1) + w(a, b-1, c-1) - w(a, b-1, c)
                else
                    w[i][j][k] = w[i-1][j][k] + w[i-1][j-1][k] + w[i-1][j][k-1] - w[i-1][j-1][k-1];
                    //w(a-1, b, c) + w(a-1, b-1, c) + w(a-1, b, c-1) - w(a-1, b-1, c-1)       
            }
    while(scanf("%d %d %d", &a, &b, &c))
    {
        if(a == -1 && b == -1 && c == -1) break;
        if(a <= 0 || b <= 0 || c <= 0) ans = 1;
        else if(a > 20 || b > 20 || c > 20) ans = w[20][20][20];
        else ans = w[a][b][c]; 
        printf("w(%d, %d, %d) = %d\n", a, b, c, ans);
    }
    return 0;
}

方法二：

总之要先把之前运算出来的结果存到三维数组中，避免重复运算。

#include <stdio.h>
#include <string.h>
#include <algorithm>
using namespace std;

int dp[25][25][25];

int dfs(int a,int b,int c)
{
    if(a<=0 || b<=0 || c<=0)
        return 1;
    if(a>20 || b>20 || c>20)
        return dfs(20,20,20);
    if(dp[a][b][c])
        return dp[a][b][c];
    if(a<b && b<c)
        dp[a][b][c] = dfs(a,b,c-1)+dfs(a,b-1,c-1)-dfs(a,b-1,c);
    else
        dp[a][b][c] = dfs(a-1,b,c)+dfs(a-1,b-1,c)+dfs(a-1,b,c-1)-dfs(a-1,b-1,c-1);
    return dp[a][b][c];
}

int main()
{
    int a,b,c;
    memset(dp,0,sizeof(dp));
    while(~scanf("%d%d%d",&a,&b,&c))
    {
        if(a == -1 && b == -1 && c == -1)
            break;
        printf("w(%d, %d, %d) = %d\n",a,b,c,dfs(a,b,c));
    }

    return 0;
}