动态规划~~矩阵链乘法

基于自底向上的动态规划
 1 #include <iostream>
#include <cstdio>
#include <algorithm>
#include <vector>
#include <cstring>
#include <string>

#define MAX 1000
using namespace std;


int a[MAX] = {0}, dp[MAX][MAX] = {0},path[MAX][MAX] = {0}, n;

void matrix_chain_order(){
    for(int r = 2; r <= n; r ++ ){
        for(int i = 1; i <= n-r; i ++){
            int j = i+r-1;
            dp[i][j]  =dp[i][i]+dp[i+1][j]+a[i]*a[i+1]*a[j+1];
            path[i][j] = i;
            for(int k=i+1; k < j; k ++){
                if( dp[i][j] > dp[i][k] + dp[k+1][j] + a[i]*a[k+1]*a[j+1] ){
                    dp[i][j] = dp[i][k] + dp[k+1][j] + a[i]*a[k+1]*a[j+1];
                    path[i][j] = k;
                }
            }
        }
    }
}

void print_optimal_parens(int i, int j){
    if( i == j ) {cout<<'A'<<i;return;}
    else{
        cout<<"(";
        print_optimal_parens(i,path[i][j]);
        print_optimal_parens(path[i][j]+1,j);
        cout<<")";
    }
}


int main(){
    int T;
    cin >> T;
    while(T--){
        cin >> n ;
        memset(a,0,sizeof(a));
        for(int i = 1; i <= n; i ++ ) cin >>a[i];
        memset(dp,0,sizeof(dp));
        memset(path,0,sizeof(path));
        matrix_chain_order();

        cout<<"(";
        print_optimal_parens(1,n-1);
        cout<<")"<<endl;
    }
    return 0;
}

 基于备忘录的自顶向下的动态规划
 1 #include <iostream>
#include <cstdio>
#include <algorithm>
#include <vector>
#include <cstring>
#include <string>

#define MAX 1000
#define INF 1000000
using namespace std;


int a[MAX] = {0}, dp[MAX][MAX] = {0},path[MAX][MAX] = {0}, n;

int lookup_chain(int i,int j){
    if(dp[i][j] < INF) return dp[i][j];
    if(i == j ) return 0;
    int u = lookup_chain(i,i)+lookup_chain(i+1,j) + a[i]*a[i+1]*a[j+1];
    path[i][j] = i;
    for(int k = i+1; k < j; k ++ ){
        int t = lookup_chain(i,k)+lookup_chain(k+1,j)+a[i]*a[k+1]*a[j+1];
        if(t < u){
            u = t;
            path[i][j] = k;
        }
    }
    dp[i][j] = u;
    return u;
}

void print_optimal_parens(int i, int j){
    if( i == j ) {cout<<'A'<<i;return;}
    else{
        cout<<"(";
        print_optimal_parens(i,path[i][j]);
        print_optimal_parens(path[i][j]+1,j);
        cout<<")";
    }
}


int main(){
    int T;
    cin >> T;
    while(T--){
        cin >> n ;
        memset(a,0,sizeof(a));
        for(int i = 1; i <= n; i ++ ) cin >>a[i];
        memset(dp,INF,sizeof(dp));
        memset(path,0,sizeof(path));
        lookup_chain(1,n);

        cout<<"(";
        print_optimal_parens(1,n-1);
        cout<<")"<<endl;
    }
    return 0;
}