递归问题

问题：

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

 

回答：典型的记忆化递归问题

    #include <stdio.h>  
    #include <limits.h>  
      
    const int MAX_N = 21;  
    int W[MAX_N][MAX_N][MAX_N];  
      
    int getValue(int a, int b, int c)  
    {  
        if (a <= 0 || b <= 0 || c <= 0) return W[0][0][0] = 1;  
        if (a >= MAX_N || b >= MAX_N || c >= MAX_N)  
            return getValue(MAX_N-1, MAX_N-1, MAX_N-1);  
      
        if (W[a][b][c]) return W[a][b][c];  
      
        if (a < b && b < c)  
        {  
            W[a][b-1][c-1] = getValue(a, b-1, c-1);  
            W[a][b][c-1] = getValue(a, b, c-1);  
            W[a][b-1][c] = getValue(a, b-1, c);  
            return W[a][b][c] = W[a][b][c-1] + W[a][b-1][c-1] - W[a][b-1][c];  
        }  
      
        W[a-1][b-1][c-1] = getValue(a-1, b-1, c-1);  
        W[a-1][b-1][c] = getValue(a-1, b-1, c);  
        W[a-1][b][c-1] = getValue(a-1, b, c-1);  
        W[a-1][b][c] = getValue(a-1, b, c);  
      
        return W[a][b][c] = W[a-1][b][c] + W[a-1][b-1][c]  
        + W[a-1][b][c-1] - W[a-1][b-1][c-1];  
    }  
      
    int main()  
    {  
        int a, b, c;  
        while (~scanf("%d %d %d", &a, &b, &c)&& !(a == -1 && b == -1 && c == -1))  
        {  
            printf("w(%d, %d, %d) = %d\n", a, b, c, getValue(a, b, c));  
        }  
        return 0;  
    }