Fibonacci Sequence
Fibonacci Sequence
Classic sequence where each number is the sum of the two preceding ones, demonstrating recursion, memoization, and dynamic programming.
Family: Dynamic Programming Status: ✅ Complete
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Overview
The Fibonacci sequence is a classic problem in computer science and mathematics that demonstrates the power of dynamic programming.
The sequence is defined as: each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
While this can be solved with simple recursion, the naive approach has exponential time complexity due to repeated calculations. Dynamic programming provides an elegant solution that reduces this to linear time complexity.
Mathematical Formulation¶
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Recurrence Relation
Mathematical Properties
Golden Ratio
F(n+1)/F(n) ≈ φ = 1.618033988749...
As n approaches infinity, the ratio approaches the golden ratio
Closed Form (Binet's Formula)
F(n) = (φ^n - (-φ)^(-n)) / √5
Direct calculation without recursion
Growth Rate
F(n) = O(φ^n)
Exponential growth with golden ratio base
Key Properties¶
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Optimal Substructure
Each Fibonacci number depends on the previous two
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Overlapping Subproblems
Same subproblems calculated multiple times in naive recursion
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Mathematical Beauty
Connects to golden ratio, nature, and art
Implementation Approaches¶
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Recommended approach - linear time, constant space
Complexity:
- Time: O(n)
- Space: O(1)
Advantages
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Optimal time and space complexity
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No recursion stack issues
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Easy to understand and implement
Disadvantages
- Less intuitive than recursive approach
Top-down approach with caching - demonstrates memoization pattern
Complexity:
- Time: O(n)
- Space: O(n)
Advantages
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Natural recursive structure
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Only computes needed subproblems
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Good for learning memoization
Disadvantages
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Uses more memory
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Potential stack overflow for large n
Produces Fibonacci sequence indefinitely
Complexity:
- Time: O(1) per number
- Space: O(1)
Advantages
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Memory efficient for large sequences
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Lazy evaluation
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Can generate infinite sequence
Disadvantages
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Not suitable for random access
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Requires iteration to get specific numbers
Simple but inefficient recursive approach
Complexity:
- Time: O(2^n)
- Space: O(n)
Advantages
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Simple and intuitive
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Direct translation of mathematical definition
Disadvantages
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Exponential time complexity
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Impractical for n > 40
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Repeated calculations
Complete Implementation
The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:
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Main implementation with all variants:
src/algokit/dynamic_programming/fibonacci.py -
Comprehensive test suite:
tests/unit/dynamic_programming/test_fibonacci.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Naive Recursion | O(2^n) | O(n) | Exponential due to repeated calculations |
Performance Considerations
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Naive recursion becomes impractical for n > 40
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Iterative approach is optimal for most use cases
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Memoization useful when you need multiple Fibonacci numbers
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Generator pattern best for sequence generation
Use Cases & Applications¶
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Application Categories
Computer Science
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Algorithm Analysis: Demonstrates dynamic programming concepts
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Recursion Examples: Teaching recursive vs iterative approaches
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Performance Testing: Benchmarking different implementation strategies
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Memory Management: Understanding space-time trade-offs
Finance & Economics
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Fibonacci Retracements: Technical analysis in trading
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Growth Models: Population and economic growth patterns
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Risk Assessment: Modeling exponential growth scenarios
Biology & Nature
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Population Growth: Rabbit breeding models
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Spiral Patterns: Shells, flowers, and natural structures
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Genetic Sequences: DNA pattern analysis
Design & Architecture
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Golden Ratio: Aesthetic proportions in design
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Spiral Structures: Architectural and artistic applications
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Harmonic Balance: Musical scales and compositions
Educational Value
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Dynamic Programming: Perfect introduction to memoization
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Algorithm Design: Shows importance of avoiding repeated work
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Mathematical Induction: Demonstrates recursive problem-solving
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Optimization: Illustrates space-time complexity trade-offs
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Pattern Recognition: Understanding overlapping subproblems
References & Further Reading¶
:material-book: Core Textbooks
:material-history: Historical & Cultural
: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.
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Navigation¶
Related Algorithms in Dynamic Programming:
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Coin Change Problem - Find the minimum number of coins needed to make a given amount using dynamic programming.
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0/1 Knapsack Problem - Optimize value within weight constraints using dynamic programming for resource allocation.