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 arrayarr
(size n+1) Eacharr[i]
represents the number of rows and columns of thei
th matrix. (In other words, matrixA
has the sizearr[i-1] * arr[i]
. Here,arr[0]
is the number of rows of the first matrix, andarr[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 indexi
toj
. - 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.