ZOJ Problem Set - 1094 Matrix Chain Multiplication

Matrix Chain Multiplication

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Time Limit: 2 Seconds      Memory Limit: 65536 KB

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Matrix multiplication problem is a typical example of dynamical programming.

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 (1 <= n <= 26), 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

0
0
0
error
10000
error
3500
15000
40500
47500
15125

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Source: University of Ulm Local Contest 1996

題目大意：

給定矩陣數，具體每個矩陣的引數，判斷每一種連乘方式需要連乘的次數

解題思路：

用stack棧來模擬矩陣連乘

AC：

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


struct Node//建立結構體，儲存矩陣資訊
{
	int row, col;
};

int main()
{
	int n;//矩陣數量
	char name;//矩陣名稱
	map<char, Node> matrix;//矩陣引數
	cin >> n;
	for(int i=0; i<n; i++)
	{
		cin>>name;
		cin>>matrix[name].row >> matrix[name].col;
	}
	string exp;
	while(cin >> exp)
	{
		int i;
		int count = 0;		//矩陣做乘法的次數
		stack<Node> array;	//模擬矩陣的乘法
		for(i=0; i<exp.size(); i++)
		{
			if(exp[i] == '(') continue;
			if(exp[i] == ')')	//遇到右括號則把，棧頂的兩個矩陣相乘，再壓入堆疊
			{
				Node b = array.top();
				array.pop();
				Node a = array.top();
				array.pop();
				if(a.col != b.row)
				{
					cout<<"error"<<endl;
					break;
				}
				count += a.row*b.row*b.col;
				Node tmp = {a.row, b.col};
				array.push(tmp);
			}
			else
				array.push(matrix[exp[i]]);
		}
		if(i == exp.size())
			cout<<count<<endl;
	}
}