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DP Fundamentals

Understanding the core concepts of dynamic programming is essential for recognizing when and how to apply this powerful technique.

Overlapping subproblems​

  • Definition: Same subproblems are solved multiple times in recursive approach
  • Identification: Look for repeated calculations in recursive tree
  • Solution: Store results of subproblems to avoid recalculation
  • Example: Fibonacci sequence where F(n) = F(n-1) + F(n-2)

Optimal substructure​

  • Definition: Optimal solution contains optimal solutions to subproblems
  • Identification: Problem can be broken down into smaller similar problems
  • Solution: Build optimal solution from optimal solutions to subproblems
  • Example: Shortest path problem where optimal path contains optimal subpaths

Memoization vs tabulation​

  • Memoization: Top-down approach, recursive with caching
  • Tabulation: Bottom-up approach, iterative with table filling
  • Trade-offs: Memoization is more intuitive, tabulation is more space-efficient
  • When to use: Memoization for sparse problems, tabulation for dense problems

State definition​

  • State Variables: Parameters that define a subproblem
  • State Space: All possible combinations of state variables
  • State Transition: How to move from one state to another
  • Base Cases: Initial states with known solutions

Transition equations​

  • Recurrence Relation: Mathematical relationship between states
  • Boundary Conditions: Initial values for the recurrence
  • Optimization: Choosing best option among multiple choices
  • Implementation: Converting recurrence to code