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

Problem Definition

This is a problem of finding the minimum number of operations required to multiply a given N matrices.
The number of operations for matrix multiplication is calculated as follows:
For two matrices A and B, they can be multiplied only if the number of columns in A is equal to the number of rows in B.
The number of operations is calculated as:

Operations count = A's row count * A's column count * B's column count

N matrices can be multiplied in sequence, but the total number of operations varies depending on the order of multiplication.
Therefore, we must find the optimal multiplication order, for which we will use Dynamic Programming techniques.

Input Format

The first line contains the number of matrices N (1 ≤ N ≤ 30).
The second line contains N integers M1, M2, …, MN (1 ≤ Mi ≤ 100) representing the size of each matrix.
Each integer represents the number of rows and columns of the matrix.

Output Format

Output the minimum number of operations required to multiply the matrices.

Example Input

3
10 20 30

Example Output

Approach to Problem Solving

This problem can be solved using Dynamic Programming.
The problem of determining the order of matrix multiplication has the following recursive relationship.

1. Definition of Dynamic Programming

We define a DP array dp[i][j] that represents the minimum number of operations required to multiply the matrices from i to j.
Therefore, the goal is to compute dp[0][N-1].

2. Recursive Relationship

dp[i][j] = min(dp[i][k] + dp[k+1][j] + M[i] * M[k+1] * M[j+1]) (i ≤ k < j)
k can be set as another matrix between i and j, thereby considering all possible combinations to find the optimal multiplication order.

C# Code Implementation

Now we will implement the entire algorithm in C#.
The code below reads the matrix sizes through input and calculates the minimum number of operations using Dynamic Programming.

using System;

    class Program
    {
        static void Main(string[] args)
        {
            int N = int.Parse(Console.ReadLine());
            int[] M = new int[N + 1];
            string[] dimensions = Console.ReadLine().Split();

            for (int i = 0; i < N; i++)
            {
                M[i] = int.Parse(dimensions[i]);
                if (i != 0) M[i + 1] = int.Parse(dimensions[i]);
            }

            int[,] dp = new int[N, N];
            for (int len = 2; len <= N; len++)
            {
                for (int i = 0; i <= N - len; i++)
                {
                    int j = i + len - 1;
                    dp[i, j] = int.MaxValue;
                    for (int k = i; k < j; k++)
                    {
                        int q = dp[i, k] + dp[k + 1, j] + M[i] * M[k + 1] * M[j + 1];
                        dp[i, j] = Math.Min(dp[i, j], q);
                    }
                }
            }

            Console.WriteLine(dp[0, N - 1]);
        }
    }

Execution Explanation

The above C# code first receives input for the number of matrices and their sizes.
After initializing the dp[i][j] array, it sets indices i and j for all combinations of matrices through two nested loops.
It also considers all possible splits for k to calculate the minimum number of operations.
Finally, dp[0][N-1] gives the minimum multiplication operations.

Time Complexity and Space Complexity

The time complexity of this algorithm is O(N^3).
Due to the presence of two nested loops and one internal loop, it has a worst-case complexity of O(N^3).
The space complexity is O(N^2), as it requires memory space to store the DP array.

Conclusion

The problem of finding the minimum number of operations for matrix multiplication can be effectively solved using Dynamic Programming.
I hope the algorithm and code described above help you in solving coding test problems.
Through practice, I hope you can solve more problems and become familiar with various algorithmic techniques.

References

– For more information on algorithm problems, check platforms like Google, Baekjoon, and LeetCode.
– For the basics of Dynamic Programming and various patterns, please refer to related books.