经典记忆化搜索

问题

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

题解

直接递归肯定会超时，我们要结合记忆化特点，才有dp操作，防止重复操作，用空间换取时间。时间复杂度O(n^3)。

#include<iostream>#include<cstdio>#include<cstring>#include<cmath>#include<algorithm>using namespace std;typedef long long LL;#define ms(a, b) memset(a, b, sizeof(a))int dp[57][57][57];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;    while(~scanf("%d%d%d",&a,&b,&c)){        if(a==-1&&b==-1&&c==-1){break;}        int ans=dfs(a,b,c);        printf("w(%d, %d, %d) = %d\n",a,b,c,ans);    }    return 0;}