Problem 039——UVa 442 - Matrix Chain Multiplication

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本文探讨了矩阵链乘法的优化策略，通过分析不同计算顺序对元素乘法操作次数的影响，展示了如何选择最优的计算顺序来减少所需的基本乘法操作数。包括实例解析和代码实现。

Matrix Chain Multiplication

Suppose you have to evaluate an expression like A*B*C*D*E where A,B,C,D and E are matrices. Since matrix multiplication is associative, the order in which multiplications are performed is arbitrary. However, the number of elementary multiplications needed strongly depends on the evaluation order you choose.

For example, let A be a 50*10 matrix, B a 10*20 matrix and C a 20*5 matrix. There are two different strategies to compute A*B*C, namely (A*B)*C and A*(B*C).

The first one takes 15000 elementary multiplications, but the second one only 3500.

Your job is to write a program that determines the number of elementary multiplications needed for a given evaluation strategy.

Input Specification

Input consists of two parts: a list of matrices and a list of expressions.

The first line of the input file contains one integer n ( ), representing the number of matrices in the first part. The next n lines each contain one capital letter, specifying the name of the matrix, and two integers, specifying the number of rows and columns of the matrix.

The second part of the input file strictly adheres to the following syntax (given in EBNF):

SecondPart = Line { Line } <EOF>
Line       = Expression <CR>
Expression = Matrix | "(" Expression Expression ")"
Matrix     = "A" | "B" | "C" | ... | "X" | "Y" | "Z"

Output Specification

For each expression found in the second part of the input file, print one line containing the word "error" if evaluation of the expression leads to an error due to non-matching matrices. Otherwise print one line containing the number of elementary multiplications needed to evaluate the expression in the way specified by the parentheses.

Sample Input

9
A 50 10
B 10 20
C 20 5
D 30 35
E 35 15
F 15 5
G 5 10
H 10 20
I 20 25
A
B
C
(AA)
(AB)
(AC)
(A(BC))
((AB)C)
(((((DE)F)G)H)I)
(D(E(F(G(HI)))))
((D(EF))((GH)I))

Sample Output

#include<iostream>
#include<string>
#include<stack>
using namespace std;

struct MCM
{
    int h;
    int l;
}s[36];

int main()
{
    int n;
    cin >> n;
    char ch;
    for(int i=0;i<n;i++)
    {
        cin >> ch;
        int k=ch-'A';
        cin >> s[k].h >> s[k].l;
    }

    string tt;
    while(cin >> tt)
    {
        stack<struct MCM> ss;
        unsigned long long int sum=0;
        int k=1;
        for(int i=0;i<tt.length();i++)
        {
            int ch=tt[i]-'A';
            if(tt[i]>='A' && tt[i]<='Z') ss.push(s[ch]);
            else if(tt[i]==')')
            {
                struct MCM t2=ss.top(); ss.pop();
                struct MCM t1=ss.top(); ss.pop();
                struct MCM t3;
                t3.h=t1.h,t3.l=t2.l;
                if(t1.l==t2.h) {sum+=t1.h*t1.l*t2.l;ss.push(t3);}
                else{ cout << "error\n"; k=0; break;}
            }
        }
        if(k) cout << sum << "\n";
    }
    return 0;
}