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Dynamic Programming•Pattern: Interval DP•Medium

Matrix chain multiplication (minimum cost)

Problem Intuition & Objective

Find the optimal parenthesization of matrix multiplications to minimize scalar multiplications.

Time ComplexityO(n³)
Optimal
Space ComplexityO(n²)
Memory Bounds
Optimal Algorithmic Approach

Interval DP: fill table by subproblem length.

Core Invariants & Edge Cases

  • dp[i][j] = min cost to multiply matrices i through j
  • Try all splits k: dp[i][j] = min(dp[i][k] + dp[k+1][j] + p[i-1]*p[k]*p[j])
  • Fill by increasing chain length
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System Design & Architecture Roadmap

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