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DP - Matrix chain multiplication (MCM)
Last Update: 6 March, 2009

Problem: Matrix chain multiplication is an optimization problem that can be solved using dynamic programming. Given a sequence of matrices, we want to find the most efficient way to multiply these matrices together. The problem is not actually to perform the multiplications, but merely to decide in which order to perform the multiplications.

We have many options because matrix multiplication is associative. In other words, no matter how we parenthesize the product, the result will be the same. For example, if we had four matrices A, B, C, and D, we would have:

(ABC)D = (AB)(CD) = A(BCD) = A(BC)D = ....

However, the order in which we parenthesize the product affects the number of simple arithmetic operations needed to compute the product, or the efficiency. For example, suppose A is a 10 x 30 matrix, B is a 30 x 5 matrix, and C is a 5 x 60 matrix. Then,

(AB)C = (10x30x5) + (10x5x60) = 1500 + 3000 = 4500 operations
A(BC) = (30x5x60) + (10x30x60) = 9000 + 18000 = 27000 operations.

Clearly the first method is the more efficient. Now that we have identified the problem, how do we determine the optimal parenthesization of a product of n matrices? We could go through each possible parenthesization (brute force), but this would require time O(2n), which is very slow and impractical for large n. The solution, as we will see, is to break up the problem into a set of related subproblems. By solving subproblems one time and reusing these solutions many times, we can drastically reduce the time required. This is known as dynamic programming.

My Code:

``````//C++ Code for  Matrix Chain Multiplicatio
//ACM 34

#include<cstdio>
#include<cmath>
#include<cstdlib>
#include<cstring>
#include<string>
#include<cctype>
#include<iostream>
#include<stack>
#include<queue>
#include<vector>
#include<algorithm>
#include<map>
using namespace std;
#define INF  1047483647
#define EPS  0.000000001
#define MAX(a,b)  ((a>b)?a:b)
#define MIN(a,b)  ((a<b)?a:b)
#define CLEAR(A,N)  (memset(A,0,(N)*sizeof(A)))
#define VI vector<int>

int mcm,path;

int findMCM(int i, int j, int P[]) {
int k,q;
if(mcm[i][j]<INF)
return mcm[i][j];
if(i==j)
mcm[i][j]=0;
else{
for(k=i;k<j;k++){
q = findMCM(i,k,P) + findMCM(k+1,j,P) + P[i-1]*P[k]*P[j];
if(q<mcm[i][j]){
mcm[i][j]=q;
path[i][j]=k;
}
}
}
return mcm[i][j];
}

void pathOfMCM(int i, int j) {
if(i==j){
printf("A%d",i);
return;
}
printf("(");
pathOfMCM(i,path[i][j]);
printf(" x ");
pathOfMCM(path[i][j]+1,j);
printf(")");
}

void initMatrix(int N){
int i,j;
for(i=0;i<=N;i++){
for(j=0;j<=N;j++){
mcm[i][j]=INF;
path[i][j]=0;
}
}
}

int main()
{
//freopen("in.txt","rt",stdin)
//freopen("out.txt","wt",stdout)

int P,a,b,i,j,N,t=0;

while(scanf("%d",&N)==1 && N!=0) {

j=0;
for(i=0;i<N;i++){
scanf("%d %d",&a,&b);
P[j++]=a;
}
P[j]=b;
initMatrix(N);

a=findMCM(1,N,P);
//cout<

printf("Case %d: ",++t);
pathOfMCM(1,N);
printf("\n");
}

// End of main function.......
return 0;
}

/***  Programmed by:
###########################
##   ~  ORONNO  ~		 ##