Dynamic Programming

An overview of dynamic programming, a method for solving problems by breaking them down into overlapping sub-problems and storing the results of solved sub-problems to avoid redundant calculations.

Definition

A method for solving problems by breaking them down into overlapping sub-problems and storing the results of solved sub-problems to avoid redundant calculations.

Steps

  1. Characterize the problem: Identify sub-problems and overlapping nature.
  2. Define the state: Determine what each sub-problem represents.
  3. State transition: Find a relation between smaller sub-problems.
  4. Base cases: Identify and solve trivial cases directly.
  5. Optimization: Build the solution bottom-up or top-down (memoization).

Examples

  • Fibonacci Sequence: Uses memoization to store previously computed values.
  • Knapsack Problem: Finds the optimal combination of items to maximize value within a weight limit.
  • Longest Common Subsequence (LCS): Finds the longest subsequence common to two sequences.

Advantages

  • Avoids redundant computations.
  • Efficient for optimization problems.

Limitations

  • Requires extra memory for storing intermediate results.
  • May not be applicable if sub-problems don't overlap.