Swift Coding Test Course, Finding the Minimum Number of Matrix Multiplication Operations

In this lecture, we will solve an algorithm problem that finds the minimum number of multiplication operations required for matrix multiplication using Swift. This problem is a good example of utilizing dynamic programming techniques and is a type of problem that can frequently appear in practical coding tests.

Problem Description

The problem is to calculate the minimum number of multiplication operations required to multiply the given matrices in order. Since the size of the matrices affects the multiplication operations, we need to find the optimal multiplication order.

Problem Specification

   Input: Integer array arr (size n+1)
   Each arr[i] represents the number of rows and columns of the ith matrix. 
   (In other words, matrix A has the size arr[i-1] * arr[i]. Here, arr[0] is the number of rows of the first matrix, and arr[n] is the number of columns of the last matrix)

   Output: Minimum number of multiplication operations required for matrix multiplication

Example

Input: arr = [10, 20, 30, 40]
Output: 3000
Description: To achieve the minimum number of operations, we need to perform multiplication in the optimal order: (A1(10×20) * A2(20×30)) * A3(30×40) = 10 * 20 * 40 + (A1 * A2) * A3.

Problem Solving Approach

This problem can be solved using dynamic programming.
The basis of the algorithm is as follows.

  • Understand the optimal structure of matrix multiplication.
  • Define the recurrence relation: Define dp[i][j] as the minimum multiplication cost of multiplying matrices from index i to j.
  • Derive the recurrence relation for minimum operations and fill the dp array.
  • Return the result.

Recurrence Relation

Considering the multiplication from matrix A to B, and setting K as the point that divides A and B, we can establish the following equation:

   dp[i][j] = min(dp[i][k] + dp[k+1][j] + arr[i]*arr[k+1]*arr[j+1])

This equation represents the process of partitioning into several parts to find the optimal number of multiplication operations, where k is any possible splitting point between i and j.

Implementation

Now let’s write Swift code to solve the problem.


func matrixChainOrder(arr: [Int]) -> Int {
    let n = arr.count - 1
    var dp = Array(repeating: Array(repeating: 0, count: n), count: n)

    for length in 2...n { // Loop through the length of the matrices
        for i in 0..< (n - length + 1) {
            let j = i + length - 1
            dp[i][j] = Int.max
            for k in i..< (j) {
                dp[i][j] = min(dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i] * arr[k + 1] * arr[j + 1])
            }
        }
    }
    
    return dp[0][n - 1]
}

// Example usage
let arr = [10, 20, 30, 40]
let result = matrixChainOrder(arr: arr)
print("Minimum number of multiplication operations: \(result)")

Code Explanation

The above code is a function that calculates the minimum number of operations for matrix multiplication.

  • First, it initializes the dp array based on the number of matrices received as input.
  • Next, it uses a double loop to calculate the number of operations for all possible combinations of matrices.
  • To find the smallest value, it explores possible split points in the third loop.
  • Finally, it returns dp[0][n - 1] to output the minimum value for all matrix multiplications.

Complexity Analysis

The time complexity of this algorithm is O(n^3). This indicates that the number of operations increases proportional to the number of matrices. The space complexity is O(n^2), which increases due to the dp array used following the recurrence relation.

Conclusion

In this lecture, we explained the problem of finding the minimum number of multiplication operations for matrices. We learned how to derive the optimal solution using dynamic programming techniques and how to implement it in Swift. This algorithm is very useful for effectively solving complex problems and is a frequently presented topic in future coding tests.

We hope this problem encourages you to research various algorithms and deterministic problem-solving methods, and to enhance your coding skills further.