Matrix Extensions Algorithm
The Matrix Extensions Algorithm (MEA) is a computational method for solving combinatorial optimization problems, particularly those involving partial orders or precedence constraints. It is based on the idea of extending a given partial order into a complete order by adding new elements to the existing partial order matrix, while maintaining the original partial order constraints. This algorithm is known for its ability to generate all possible linear extensions of a given partial order, providing solutions to problems such as sequencing, scheduling, and ranking. MEA has been successfully applied to various real-world problems, including project scheduling, DNA sequence assembly, and sports tournament scheduling.
The MEA starts by initializing an empty matrix and iteratively extends it by adding new rows and columns, representing tasks or elements, while respecting the precedence constraints of the problem. At each step, the algorithm generates a set of potential extensions, checks for any violations of the partial order constraints, and selects the best extension based on a predefined criterion. The process is repeated until all elements have been added to the matrix, resulting in a complete linear order that satisfies the initial partial order constraints. The algorithm can then backtrack to explore alternative extensions and generate multiple solutions for the problem at hand. This backtracking and branching mechanism allows MEA to efficiently traverse the solution space and identify the optimal or near-optimal solutions, making it an effective tool for solving various combinatorial optimization problems.
using System;
namespace Utilities.Extensions
{
public static class MatrixExtensions
{
public static double[,] Multiply(this Array source, Array operand)
{
if ((source.Rank != 2) || (operand.Rank != 2))
{
throw new ArgumentException("Rank of both operands should be equal 2!");
}
if (source.GetLength(1) != operand.GetLength(0))
{
throw new InvalidOperationException("Width of a first operand should match height of a second!");
}
double[,] result = new double[source.GetLength(0), operand.GetLength(1)];
for (var i = 0; i < result.GetLength(0); i++)
{
for (var j = 0; j < result.GetLength(1); j++)
{
double elementProduct = 0;
for (var k = 0; k < source.GetLength(1); k++)
{
elementProduct += (double)source.GetValue(i, k) * (double)operand.GetValue(k, j);
}
result[i, j] = elementProduct;
}
}
return result;
}
}
}
