PKU ACM 1579

 

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.


所谓动态规划，是因为递归中产生的重叠子问题，导致低效率，为了提高效率，有两种方法:记忆化搜索和自底向上递推。
这里显然可以用一个三维数组自底向上的递推,另外轮换对称的性质（w(a,b,c-1)==w(a,b-1,c)）也可以简化计算。注意判断条件的先后顺序，否则wa
#include <stdio.h>#include <vector>using namespace std;int v[21][21][21];struct input{ int a; int b; int c;};int main(){ int i,j,k; vector<struct input> p; for(i=0;i<21;i++) for(j=0;j<21;j++) for(k=0;k<21;k++) { if(i==0||j==0||k==0) v[i][j][k]=1; else if(i<j&&j<k) v[i][j][k]=v[i][j-1][k-1];//w(a, b, c-1) + w(a, b-1, c-1) - w(a, b-1, c) else v[i][j][k]=v[i-1][j][k]+v[i-1][j-1][k]+v[i-1][j][k-1]-v[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(1) { scanf("%d %d %d",&i,&j,&k); if(i==-1&&j==-1&&k==-1) break; struct input tmp; tmp.a=i; tmp.b=j; tmp.c=k; p.push_back(tmp); } vector<input>::iterator f=p.begin(); for(;f!=p.end();f++) if(f->a<=0||f->b<=0||f->c<=0) printf("w(%d, %d, %d) = %d/n",f->a,f->b,f->c,v[0][0][0]); else if(f->a>20||f->b>20||f->c>20) printf("w(%d, %d, %d) = %d/n",f->a,f->b,f->c,v[20][20][20]); else printf("w(%d, %d, %d) = %d/n",f->a,f->b,f->c,v[f->a][f->b][f->c]); return 0;}