经典记忆化搜索
2017-10-05 20:16
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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 12 2 2
10 4 6
50 50 50
-1 7 18
-1 -1 -1
Sample Output
w(1, 1, 1) = 2w(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; }
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