Coin Change Problem
Coin Change Problem
Find the minimum number of coins needed to make a given amount using dynamic programming.
Family: Dynamic Programming Status: 📋 Planned
Need Help Understanding This Algorithm?
Overview
The Coin Change problem is a classic dynamic programming challenge that asks: given a set of coin denominations
and a target amount, what is the minimum number of coins needed to make up that amount? This problem demonstrates the power of dynamic programming in solving optimization problems with overlapping subproblems.
While a greedy approach might work for some coin sets (like US coins), it fails for arbitrary denominations. Dynamic programming provides an elegant solution that guarantees the optimal result by building solutions from smaller subproblems.
Mathematical Formulation¶
🧮 Ask ChatGPT about Mathematical Formulation
Problem Definition
Given:
- A set of coin denominations: C = {c₁, c₂, ..., cₙ}
- A target amount: A
Find the minimum number of coins needed to make amount A:
min Σᵢ₌₁ⁿ xᵢ subject to Σᵢ₌₁ⁿ cᵢxᵢ = A
Where xᵢ represents the number of coins of denomination cᵢ.
Key Properties
Optimal Substructure
The optimal solution for amount A contains optimal solutions for amounts A - cᵢ
Each optimal solution builds on smaller optimal solutions
Overlapping Subproblems
The same subproblems are solved multiple times
Dynamic programming avoids redundant calculations
State Transition
dp[A] = min(dp[A - cᵢ] + 1) for all valid coins cᵢ
Recurrence relation for building solutions
Key Properties¶
🔑 Ask ChatGPT about Key Properties
-
Optimal Substructure
The optimal solution for amount A contains optimal solutions for amounts A - cᵢ
-
Overlapping Subproblems
The same subproblems are solved multiple times
-
Greedy vs DP
Greedy works for some coin sets but fails for arbitrary denominations
Implementation Approaches¶
💻 Ask ChatGPT about Implementation
Optimal solution using bottom-up DP approach
Complexity:
- Time: O(amount × coins)
- Space: O(amount)
Advantages
-
Optimal solution, guaranteed correct
-
Handles arbitrary coin denominations
-
Clear and straightforward implementation
Disadvantages
-
Requires building solutions for all amounts up to target
-
Can be memory-intensive for large amounts
Top-down approach with caching
Complexity:
- Time: O(amount × coins)
- Space: O(amount)
Advantages
-
Natural recursive structure
-
Only computes needed subproblems
-
Good for learning memoization
Disadvantages
-
Uses more memory for memoization
-
Potential stack overflow for large amounts
Fast but may not be optimal for arbitrary coin sets
Complexity:
- Time: O(coins log coins)
- Space: O(1)
Advantages
-
Very fast execution
-
Simple implementation
-
Works for standard coin sets (US coins)
Disadvantages
-
May not give optimal results
-
Fails for arbitrary coin denominations
Complete Implementation
The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:
-
Main implementation with all variants:
src/algokit/dynamic_programming/coin_change.py -
Comprehensive test suite:
tests/unit/dynamic_programming/test_coin_change.py
Complexity Analysis¶
📊 Ask ChatGPT about Complexity
Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Dynamic Programming | O(amount × coins) | O(amount) | Optimal solution, guaranteed correct |
Performance Considerations
-
DP approach is optimal but requires building solutions for all amounts up to target
-
Memoization useful when you need solutions for multiple amounts
-
Greedy approach only works for coin sets with specific properties (like US coins)
-
Large amounts can make DP memory-intensive
Use Cases & Applications¶
🌍 Ask ChatGPT about Applications
Application Categories
Financial Systems
-
Vending Machines: Optimal coin dispensing
-
Cash Registers: Minimum coin change calculation
-
Banking: ATM cash withdrawal optimization
-
Payment Processing: Efficient change distribution
Computer Science
-
Algorithm Design: Dynamic programming principles
-
Optimization Problems: Resource allocation
-
Game Development: Score systems and rewards
-
Data Structures: Understanding state management
Real-World Scenarios
-
Retail: Cashier change optimization
-
Transportation: Fare collection systems
-
Gaming: Point systems and achievements
-
Manufacturing: Part quantity optimization
Educational Value
-
Dynamic Programming: Understanding optimal substructure
-
State Transitions: Learning recurrence relations
-
Optimization: Comparing greedy vs optimal approaches
-
Problem Solving: Breaking complex problems into subproblems
Educational Value
-
Dynamic Programming: Perfect example of optimal substructure and overlapping subproblems
-
Algorithm Design: Shows when greedy approaches fail and DP succeeds
-
State Management: Demonstrates building solutions from smaller subproblems
-
Optimization: Illustrates the trade-off between correctness and efficiency
References & Further Reading¶
:material-book: Core Textbooks
:material-library: Dynamic Programming
:material-web: Online Resources
:material-code-tags: Implementation & Practice
Interactive Learning
Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.
Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.
Need More Help? Ask ChatGPT!
Navigation¶
Related Algorithms in Dynamic Programming:
-
Fibonacci Sequence - Classic sequence where each number is the sum of the two preceding ones, demonstrating recursion, memoization, and dynamic programming.
-
0/1 Knapsack Problem - Optimize value within weight constraints using dynamic programming for resource allocation.