矩阵链乘法

1.C++实现：

#include<bits/stdc++.h>
using namespace std;
//int p[] = {30,35,15,5,10,20,25};
int p1[] = { 2,10,3,12,5,50,6};

void print(int s[][7], int i, int j){
	if(i == j){
		cout<<"A"<<i;
	}
	else{
		cout<<"(";
		print(s, i, s[i][j]);
		print(s, s[i][j]+1, j);
		cout<<")";
	}
}

void matrix_multiply(int p1[],int length){

	int n = length - 1;
	int m[n+1][n+1];
	int s[7][7];
	for( int i = 1; i <= n; i++){
		m[i][i] = 0;
	}

	for(int l = 2; l<= n; l++){
		for(int i = 1; i <= n-l+1; i++){
			int j = i+l-1;
			m[i][j] = 1e9;
			for(int k = i; k<=j-1; k++){
				int q = m[i][k] + m[k+1][j] + p1[i-1]*p1[k]*p1[j];
				if(q < m[i][j]){
					m[i][j] = q;
					s[i][j] = k;
				}
			}
		}
	}

	print(s,1,n);
	cout<<endl;
	cout<<m[1][n]<<endl;
}

int main(){
	matrix_multiply(p1, 7);
	return  0;
}

2.java实现（自底向上）：

public class 矩阵链乘法 {
	private static int p[] = {30,35,15,5,10,20,25};
	private static int p1[] = { 2,10,3,12,5,50,6};
	public static void main(String[] args) {
		// TODO Auto-generated method stub
		matrix_multiply(p);
		matrix_multiply(p1);
	}
	
	public	static void print(int s[][], int i, int j){
		if(i == j){
			System.out.print("A"+i);
		}
		else{
			System.out.print("(");
			print(s, i, s[i][j]);
			print(s, s[i][j]+1, j);
			System.out.print(")");
		}
	}
	
	public	static void matrix_multiply(int p[]){

		int n = p.length - 1;
		int m[][] = new int[n+1][n+1];
		int s[][] = new int[n+1][n+1];
		for( int i = 1; i <= n; i++){
			m[i][i] = 0;
		}

		for(int l = 2; l<= n; l++){
			for(int i = 1; i <= n-l+1; i++){
				int j = i+l-1;
				m[i][j] = (int) 1e9;
				for(int k = i; k<=j-1; k++){
					int q = m[i][k] + m[k+1][j] + p[i-1]*p[k]*p[j];
					if(q < m[i][j]){
						m[i][j] = q;
						s[i][j] = k;
					}
				}
			}
		}
		print(s,1,n);
		System.out.println();
		System.out.println(m[1][n]);
		
	}
}

3.自顶向下的纯递归：

public class 矩阵链乘法_纯递归 {
	private static int p[] = {30,35,15,5,10,20,25};
	private static int p1[] = { 2,10,3,12,5,50,6};
	public static void main(String[] args) {
		// TODO Auto-generated method stub
		System.out.println(recursive_matrix_chain(p, 1, 6)); 
		System.out.println(recursive_matrix_chain(p1, 1, 6)); 
	}
	
	public static int recursive_matrix_chain(int []p, int i, int j) {
		int n = p.length;
		int m[][] = new int[n][n];
		if(i == j) {
			return 0;
		}
		
		m[i][j] = (int) 1e9;
		for(int k = i; k<= j-1; k++) {
			int q = recursive_matrix_chain(p, i, k)+
					recursive_matrix_chain(p, k+1, j)+ p[i-1]*p[k]*p[j];
			if(q<m[i][j]) {
				m[i][j] = q;
			}
		}
		return m[i][j];
	}
}

4.备忘的自顶向下方法：

public class 备忘的自顶向下递归 {
	private static int p[] = {30,35,15,5,10,20,25};
	private static int p1[] = { 2,10,3,12,5,50,6};
	public static void main(String[] args) {
		// TODO Auto-generated method stub
		System.out.println(memoized_matrix_chian(p));
		System.out.println(memoized_matrix_chian(p1));
	}
	
	public static int memoized_matrix_chian(int p[]) {
		int n = p.length - 1;
		int m[][] = new int[n+1][n+1];
		for(int i = 0; i<=n; i++) {
			for(int j = 0; j<=n; j++) {
				m[i][j] = (int) 1e9;
			}
		}
		return recursive_matrix_chain(p, m, 1, n);
	}
	public static int recursive_matrix_chain(int []p,int m[][], int i, int j) {
		if( m[i][j] < 1e9) {
			return m[i][j];
		}
		if(i == j) {
			m[i][j] = 0;
		}
		else {
			for(int k = i; k<= j-1; k++) {
				int q = recursive_matrix_chain(p, m, i, k)+
						recursive_matrix_chain(p, m, k+1, j)+ p[i-1]*p[k]*p[j];
				if(q<m[i][j]) {
					m[i][j] = q;
				}
			}
		}
		return m[i][j];
	}
	
}