Matrix Chain Multiplication

Logical Breakdown

• [ ] Subproblem Identification:
• [ ] Optimal Substructure:
• [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
• [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

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Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

\[ dp[i] = ... \]

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Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

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Complexity Analysis

Approach	Time Complexity	Space Complexity
Iterative DP	\(O(N)\)	\(O(1)\)

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Code Reference

package com.dsa.dynamic_programming;

import java.util.*;

/ * Problem: Matrix Chain Multiplication * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: Matrix Chain Multiplication * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given an array A representing dimensions of matrices. * Matrix i has dimensions A[i-1] x A[i]. * Find the minimum number of scalar multiplications needed to compute the product. * * --------------------- * 2. Logical Breakdown (Interval DP) * --------------------- * Let dp[i][j] be the minimum cost to multiply the chain of matrices from index i to j. * (Note: The chain length is N-1 matrices). * * Recurrence: * To compute the product of matrices M_i...M_j, we must split the chain at some index 'k' * (where i <= k < j). * Cost = Cost(Left Part) + Cost(Right Part) + Cost of Multiplying the two resulting matrices. * * dp[i][j] = min( dp[i][k] + dp[k+1][j] + (A[i-1] * A[k] * A[j]) ) * for all k from i to j-1. * * Base Case: * If i == j (Single matrix): Cost is 0. * * --------------------- * 3. Complexity * --------------------- * Time Complexity: O(N^3) - Three nested loops (Len, i, k). * Space Complexity: O(N^2) * * ============================================================================================ */ public class MatrixChainMultiplication { public int solve(int A) { // TODO: Implement solution return 0; }

public int solveMCM(int[] dims1) {
    return 1;
}

}