Matrix Chain Multiplication Algorithm

Matrix Chain Multiplication Algorithm is a fundamental problem in computer science and is used in many applications such as optimization, machine learning, and computer graphics. It involves finding the most efficient way to multiply a sequence of matrices. Although the problem is not to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.

The Matrix Chain Multiplication Algorithm is an optimization algorithm that solves the Matrix Chain Multiplication problem. It is a dynamic programming algorithm that uses the optimal substructure property to find the optimal solution. The algorithm has a time complexity of O(n^3) and a space complexity of O(n^2), where n is the number of matrices in the sequence.

The Matrix Chain Multiplication Algorithm is widely used in computer science and is an essential tool for solving many problems. It is used in many applications such as computer graphics, machine learning, and optimization. The algorithm is easy to implement and has a relatively low time complexity, making it an efficient solution for many problems.

What is Matrix Chain Multiplication?

Matrix Chain Multiplication is a dynamic programming algorithm used to find the most efficient way to multiply a sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.

Matrix multiplication is associative, meaning that the product of three or more matrices can be calculated in different orders. However, the order in which the matrices are multiplied can affect the number of arithmetic operations required to calculate the final product. The goal of the Matrix Chain Multiplication algorithm is to find the order of matrix multiplications that requires the least amount of arithmetic operations.

The Matrix Chain Multiplication problem can be represented as follows: given a sequence of matrices A₁, A₂, …, A_n, where the dimensions of matrix A_i are p_i-1 x p_i, find the order in which to multiply the matrices to minimize the number of arithmetic operations required.

The Naive Approach

When faced with the matrix chain multiplication problem, the most straightforward approach is to try out all possible ways of multiplying the matrices. This method is called the naive approach, and it involves generating all possible parenthesizations of the matrix chain and computing the cost of each one. The cost of a parenthesization is the number of scalar multiplications needed to compute the product.

For example, suppose we have a matrix chain A1, A2, A3, and A4, with dimensions 10×20, 20×30, 30×40, and 40×30, respectively. The naive approach would involve computing the cost of all possible parenthesizations of this chain, which are:

From the above table, we can see that the optimal parenthesization is ((A1A2)A3)A4, which requires 1,620,000 scalar multiplications.

However, the naive approach is not practical for large matrix chains, as the number of possible parenthesizations grows exponentially with the number of matrices. For a chain of n matrices, there are (n-1)! possible parenthesizations. For example, a chain of 10 matrices has over 34 million possible parenthesizations.

The Dynamic Programming Approach

When it comes to solving the Matrix Chain Multiplication problem, Dynamic Programming is the most efficient approach. In this section, we will explore the Dynamic Programming approach to solving the Matrix Chain Multiplication problem. We will start by defining the problem, then we will discuss the Matrix Chain Multiplication Algorithm, and finally, we will provide the Pseudo Code for the algorithm.

Defining the Problem

The Matrix Chain Multiplication problem involves finding the most efficient way to multiply a sequence of matrices. Given a sequence of matrices A1, A2, A3, …, An, the goal is to find the order in which to multiply these matrices that requires the least number of scalar multiplications.

The Matrix Chain Multiplication Algorithm

The Matrix Chain Multiplication Algorithm is a Dynamic Programming algorithm that solves the Matrix Chain Multiplication problem. The algorithm works by breaking down the problem into subproblems and solving each subproblem only once. The solution to each subproblem is then saved and used to solve larger subproblems until the entire problem is solved.

The Matrix Chain Multiplication Algorithm uses a table to store the minimum number of scalar multiplications required to multiply each subsequence of matrices. The table is filled in diagonally, starting with the subsequence of length 2 and working up to the entire sequence of matrices.

PseudoCode for Matrix Chain Multiplication

function MatrixChainMultiplication(dimensions):
    n = length(dimensions) - 1  // number of matrices
    table = createTable(n)  // create a table to store minimum scalar multiplications
    
    // Initialize diagonal elements of the table to 0
    for i = 1 to n:
        table[i][i] = 0
    
    // Fill in the table diagonally
    for length = 2 to n:
        for i = 1 to n - length + 1:
            j = i + length - 1
            table[i][j] = infinity
            
            for k = i to j - 1:
                cost = table[i][k] + table[k + 1][j] + dimensions[i - 1] * dimensions[k] * dimensions[j]
                if cost < table[i][j]:
                    table[i][j] = cost
    
    // Return the minimum number of scalar multiplications required to multiply the entire sequence
    return table[1][n]


Let’s go through the code step by step:

1. The MatrixChainMultiplication function takes an array dimensions as input, which represents the dimensions of the matrices to be multiplied.
2. The variable n is set to length(dimensions) - 1, which represents the number of matrices in the sequence.
3. A table table is created to store the minimum scalar multiplications required to multiply each subsequence of matrices. The table has dimensions n x n, representing the subsequence from matrix i to j.
4. The diagonal elements of the table are initialized to 0 because multiplying a single matrix does not require any scalar multiplications.
5. The table is filled diagonally, starting with subsequences of length 2 and working up to the entire sequence. The variable length represents the length of the subsequence being considered.
6. For each subsequence length, the nested loops iterate over all possible starting indices i and calculate the ending index j.
7. The minimum number of scalar multiplications for the subsequence i to j is initially set to infinity.
8. Another loop iterates over the possible partition points k within the subsequence. The cost of multiplying the subsequence is calculated as the sum of the costs for the two partitions plus the cost of multiplying the resulting matrices.
9. If the calculated cost is lower than the current minimum cost in the table, it is updated with the new cost.
10. After all the calculations, the minimum number of scalar multiplications required to multiply the entire sequence of matrices is stored in table[1][n].
11. The function returns the minimum number of scalar multiplications.

Note that the createTable function is not shown in the pseudocode. It would be responsible for creating a 2D table of the given dimensions to store the minimum scalar multiplications. The pseudocode assumes that this function is defined separately.

Optimizations

In order to improve the efficiency of the Matrix Chain Multiplication Algorithm, we can use a few different optimizations. Two of the most common optimizations are Parenthesization and Memoization.

Parenthesization

Parenthesization refers to the way that we group the matrices together in order to perform the multiplication. By choosing the optimal parenthesization, we can reduce the number of scalar multiplications that we need to perform.

One way to find the optimal parenthesization is to use dynamic programming. We can create a table that stores the minimum number of scalar multiplications required to compute the product of a sequence of matrices of various lengths. By filling in this table, we can determine the optimal parenthesization.

Memoization

Memoization is a technique used to avoid redundant calculations by storing the results of expensive function calls and returning the cached result when the same inputs occur again.

We can use memoization to optimize the Matrix Chain Multiplication Algorithm by storing the results of subproblems that we have already solved. By doing so, we can avoid redundant calculations and improve the overall efficiency of the algorithm.

• For example, if we need to compute the product of matrices A, B, and C, and we have already computed the product of matrices B and C, we can store the result of that calculation and use it when we compute the product of matrices A, B, and C.
• Another example is when we need to compute the product of matrices A, B, C, and D, and we have already computed the product of matrices B, C, and D. We can store the result of that calculation and use it when we compute the product of matrices A, B, C, and D.

Applications

The Matrix Chain Multiplication Algorithm has a wide range of applications in various fields, including computer science, engineering, and physics. In this section, we will discuss two of the most common applications of the algorithm.

Matrix Chain Order

One of the most important applications of the Matrix Chain Multiplication Algorithm is in determining the optimal order of matrix multiplication. Given a sequence of matrices, there are multiple ways to multiply them together, and the order in which they are multiplied can have a significant impact on the efficiency of the computation. By using the Matrix Chain Multiplication Algorithm, we can determine the optimal order of multiplication, which minimizes the number of scalar multiplications required.

For example, suppose we have three matrices A, B, and C, with dimensions 10 x 20, 20 x 30, and 30 x 40, respectively. If we multiply them in the order (A x B) x C, we will need to perform 10 x 20 x 30 + 10 x 30 x 40 = 18,000 scalar multiplications. However, if we multiply them in the order A x (B x C), we will only need to perform 20 x 30 x 40 + 10 x 20 x 40 = 24,000 scalar multiplications. By using the Matrix Chain Multiplication Algorithm, we can determine that the optimal order of multiplication is A x (B x C).

Optimal Binary Search Trees

Another important application of the Matrix Chain Multiplication Algorithm is in the construction of optimal binary search trees. In computer science, binary search trees are a common data structure used for searching and sorting data. An optimal binary search tree is one that minimizes the total cost of searching, which is the sum of the probabilities of accessing each node multiplied by its depth.

By using the Matrix Chain Multiplication Algorithm, we can determine the optimal order of accessing the nodes in the binary search tree, which minimizes the total cost of searching. This is done by treating the probabilities of accessing each node as the dimensions of the matrices, and using the Matrix Chain Multiplication Algorithm to determine the optimal order of multiplication.

For example, suppose we have the following probabilities of accessing nodes in a binary search tree:

• A: 0.3
• B: 0.2
• C: 0.1
• D: 0.15
• E: 0.25

If we construct the binary search tree in the order A, B, C, D, E, the total cost of searching will be:

• A: 0.3 x 1 = 0.3
• B: 0.2 x 2 = 0.4
• C: 0.1 x 3 = 0.3
• D: 0.15 x 4 = 0.6
• E: 0.25 x 5 = 1.25

The total cost of searching is 0.3 + 0.4 + 0.3 + 0.6 + 1.25 = 2.85. However, if we construct the binary search tree in the optimal order C, B, A, E, D, the total cost of searching will be:

• C: 0.1 x 1 = 0.1
• B: 0.2 x 2 = 0.4
• A: 0.3 x 3 = 0.9
• E: 0.25 x 4 = 1
• D: 0.15 x 5 = 0.75

The total cost of searching is 0.1 + 0.4 + 0.9 + 1 + 0.75 = 3.15. By using the Matrix Chain Multiplication Algorithm, we can determine that the optimal order of accessing the nodes in the binary search tree is C, B, A, E, D, which minimizes the total cost of searching.

Matrix Chain Multiplication Code

Here’s an implementation of the Matrix Chain Multiplication algorithm in Python.

def matrix_chain_order(dimensions):
    n = len(dimensions) - 1
    table = [[0 for i in range(n)] for j in range(n)]

    for i in range(n):
        table[i][i] = 0

    for length in range(2, n + 1):
        for i in range(n - length + 1):
            j = i + length - 1
            table[i][j] = float('inf')

            for k in range(i, j):
                cost = table[i][k] + table[k + 1][j] + dimensions[i] * dimensions[k + 1] * dimensions[j + 1]
                if cost < table[i][j]:
                    table[i][j] = cost

    return table[0][n - 1]

dimensions = [10, 20, 30, 40, 30]
result = matrix_chain_order(dimensions)
print("Minimum number of scalar multiplications:", result)


This Python code implements the matrix_chain_order() function, which takes a list of matrix dimensions as input and returns the minimum number of scalar multiplications required to multiply the given sequence of matrices.

In this example, we have a sequence of matrices with dimensions [10, 20, 30, 40, 30]. The matrix_chain_order() function will return the minimum number of scalar multiplications required, which is 26000 in this case.