💰

Understanding Dynamic Programming: Coin Change Problem

Aug 9, 2024

Problem-Solving Skills with Dynamic Programming - Coin Change Problem

Introduction to Dynamic Programming (DP)

  • Dynamic Programming: One of the hardest problem-solving concepts, often misunderstood.
  • Key Terms: Recursion, iteration, top-down, bottom-up, memorization, tabulation.
  • Essence of DP: Not about removing duplicate computations, but realizing they never needed to exist.

Misconceptions and Correct Perspective

  • Misconception: DP is about removing duplicate nodes/trees.
  • Correct View: Recurrence relations are definitions, not tasks. They simplify the problem.
  • Implementations: Recursive top-down memorization and iterative bottom-up tabulation.

Example: Fibonacci Sequence

  • Common Visualization: As a tree with duplicate nodes.
  • Core Idea: Simplify problems with recurrence relations, avoiding naïve recursion.

Case Study: Coin Change Problem

  • Problem Statement: Given coins of different denominations and an amount, return the number of combinations to make that amount. If impossible, return zero.
  • Assumptions: Infinite number of each coin, fits in a 32-bit integer.

Steps to Solve

1. Understanding Examples

  • Example 1: Amount is 5, coins are [1, 2, 5]. Four ways to make up the amount.
  • Example 2: Amount is 3, coins are [2]. No way to make up the amount.
  • Example 3: Similar analysis.

2. Observations and Constraints

  • Input Constraints: Length of coins < 300, each coin and amount < 5000, coins are unique and positive.
  • Problem Requirements: Avoid duplicates, count unique combinations.

3. Problem Breakdown

  • Combination Counting: Avoid duplicate combinations by sticking to any ordering.
  • Action Effects: Taking a coin decreases the amount, skipping a coin moves to the next denomination.
  • Recursive Approach: Can be framed as a recurrence relation.

4. Formulating the Solution

  • Function Definition: Recursive function with parameters for current coin index and remaining amount.
  • Base Cases: Amount reaches zero (one way), no coins left but amount > 0 (zero ways).
  • Recursive Calls: Count ways by either taking or skipping the current coin.

Time Complexity Analysis

  • Unique States: Product of number of coins and the amount (O(coins * amount)).
  • Cached Complexity: Each state computation is O(1).
  • Total Complexity: O(coins * amount).

Additional Concepts

1. Recurrence Relations

  • Examples: Fibonacci, Tribonacci, Catalan numbers.
  • Important: Recurrence relations define states as values, not tasks.

2. Backtracking vs. DP

  • Example: Sudoku solver uses backtracking, not DP.

Final Review and Summary

  • DP Definition: Decision parameterization; simplifies problem-solving by focusing on key decisions with minimal parameters.
  • Key Takeaways: Understand and break down the problem, ensure observations are thorough, and verify with correct base cases.
  • Future Topics: Iterative conversion, space optimization.

Conclusion

  • Dynamic Programming: Encourages thoughtful problem-solving over brute force memorization.
  • Next Steps: Share knowledge, focus on understanding rather than just memorization.

Note: The above notes provide an organized summary for studying the key topics and solving the coin change problem using dynamic programming.

Full transcript