Dynamic Programming Part 3
Part 3 covers advanced dynamic programming topics including problems that involve partitioning (divide-and-conquer strategies with DP), as well as DP applied to trees and grid-based problems. These are typically more complex and involve multiple dimensions or creative state definitions, solidifying a deep understanding of DP techniques.
This pattern involves breaking problems into two parts and trying all possible partition points. Matrix Chain Multiplication (MCM) is a classic example: it determines the most efficient way to parenthesize matrix multiplications. The recursive solution tries every split point, and DP is used to avoid recomputation. Many hard problems follow this pattern, such as minimizing palindrome partition cuts, evaluating boolean expressions with different parenthesizations, scrambled string checks, and the egg dropping problem.
Example Problems:
Matrix Chain MultiplicationPalindrome Partitioning (Minimum Cuts)Boolean ParenthesizationScrambled StringEgg Dropping Problem
Solution Spotlight: Matrix Chain Multiplication
This solution computes the minimum multiplication cost for a chain of matrices using the Matrix Chain Multiplication DP approach. Using
dimensions (where matrix i has dimensions dimensions[i-1] x dimensions[i]), it fills a table dp[i][j] for the minimum cost of multiplying matrices i through j. The code tries every possible split k between i and j, using the costs of solving subchains [i..k] and [k+1..j], plus the cost of multiplying the two results. By building up for chains of length 2 to n-1, the algorithm finds the optimal cost for the full chain. Similar partitioning strategies are applied in Palindrome Partitioning (where splits are at cut positions in a string), Boolean Parenthesization (splitting expression at operators), Scrambled String, and Egg Dropping (where splits represent dropping an egg from a floor).MatrixChainMultiplication.java
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