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Optimize Matrix-Chain Multiplication

GsaData Scientist · Technical Screen
HardData ScientistTechnical ScreenCoding & Algorithms

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This problem tests your ability to solve a classic dynamic programming challenge, focusing on finding the optimal order of matrix multiplications to minimize the total number of scalar operations.

Problem Description

You are given a list of integers dimensions, where each matrix A_i has dimensions dimensions[i] x dimensions[i+1]. The cost of multiplying a matrix of size p x q by a matrix of size q x r is p * q * r scalar operations. Your goal is to find the minimum number of scalar multiplications needed to compute the product of the entire chain of matrices.

Function Signature

def minimum_matrix_chain_cost(dimensions: List[int]) -> int:

Constraints

  • 2 <= len(dimensions) <= 500
  • 1 <= dimensions[i] <= 1000

Examples

Example 1: Input: dimensions = [10, 30, 5, 60] Output: 4500 Explanation:

  • (A1A2) first: (10305) + (105*60) = 1500 + 3000 = 4500
  • A1*(A2A3) first: (30560) + (1030*60) = 9000 + 18000 = 27000 Minimum is 4500.

Example 2: Input: dimensions = [1, 2, 3] Output: 6

Edge Cases

  • A list containing only two dimensions (a single matrix), which should return 0.
  • A chain of three matrices.

Hints

Hint 1

Define a 2D DP table `dp[i][j]` that represents the minimum cost to multiply the sequence of matrices from index `i` to `j`.

Hint 2

The recurrence relation is `dp[i][j] = min(dp[i][k] + dp[k+1][j] + dimensions[i] * dimensions[k+1] * dimensions[j+1])` for all valid `k` between `i` and `j-1`.

Hint 3

Use a nested loop structure to fill the DP table based on the increasing length of the matrix chains.

Follow-up questions

  • How would you modify your solution to reconstruct and print the optimal parenthesization of the matrices?
  • What is the time and space complexity of this approach, and can the space complexity be improved?
  • If the number of matrices were to grow significantly (e.g., n=10,000), would this dynamic programming approach still be feasible? What are the alternatives?