Project 10: Graph Algorithms Suite

Build a toolkit of core graph algorithms with visualizations and tests.

Quick Reference

Attribute	Value
Difficulty	Level 3: Advanced
Time Estimate	3-4 weeks
Language	Python (Alternatives: C++, Java, Go)
Prerequisites	Projects 6 and 9, recursion, priority queue
Key Topics	Graphs, BFS/DFS, shortest paths, MST

1. Learning Objectives

By completing this project, you will:

Implement graph representations (adjacency list/matrix).
Implement BFS, DFS, Dijkstra, Bellman-Ford, and MST algorithms.
Visualize graph traversal and shortest paths.
Analyze complexity and choose appropriate algorithms.

2. Theoretical Foundation

2.1 Core Concepts

Graph representation: Lists are efficient for sparse graphs; matrices for dense.
Traversal: BFS explores levels, DFS explores depth.
Shortest paths: Dijkstra for non-negative weights, Bellman-Ford for negatives.
MST: Kruskal and Prim minimize total edge weight.

2.2 Why This Matters

Graphs model real-world relationships: networks, maps, dependencies, social graphs. Algorithms here are foundational for routing and optimization.

2.3 Historical Context / Background

Dijkstra (1950s) introduced shortest paths; Kruskal and Prim developed MST algorithms soon after.

2.4 Common Misconceptions

Misconception: BFS works for weighted graphs. Reality: Only for unweighted graphs.
Misconception: Dijkstra handles negative edges. Reality: Use Bellman-Ford for negatives.

3. Project Specification

3.1 What You Will Build

A CLI and library that can load graphs, run traversals, compute shortest paths, and output paths and costs. Optional graphviz output for visualization.

3.2 Functional Requirements

Representations: Adjacency list and matrix.
Traversals: BFS and DFS with visitation order.
Shortest paths: Dijkstra and Bellman-Ford.
MST: Prim or Kruskal.
Visualization: Export to DOT format.

3.3 Non-Functional Requirements

Performance: Use heap-based priority queue for Dijkstra.
Reliability: Detect negative cycles in Bellman-Ford.
Usability: CLI flags for algorithm selection.

3.4 Example Usage / Output

$ ./graph-tool --algo dijkstra --start A
Path A->D: A B D (cost 7)
$ ./graph-tool --algo mst
MST cost: 12

3.5 Real World Outcome

You will load a graph from a text file, run algorithms, and see traversal order or shortest path results. DOT exports let you view the graph with Graphviz.

4. Solution Architecture

4.1 High-Level Design

┌───────────┐     ┌──────────────┐     ┌──────────────┐
│ File/CLI  │────▶│ Graph Core   │────▶│ Algorithms   │
└───────────┘     └──────┬───────┘     └──────┬───────┘
                            │                │
                      ┌─────▼─────┐    ┌──────▼──────┐
                      │ Visualizer│    │ Reporter    │
                      └───────────┘    └─────────────┘

4.2 Key Components

Component	Responsibility	Key Decisions
graph.py	Data structure	List vs matrix
algorithms.py	BFS/DFS/Dijkstra/MST	Shared adjacency API
io.py	Load graph from file	Simple text format
viz.py	DOT export	Optional dependency

4.3 Data Structures

# adjacency list
adj = {
    "A": [("B", 3), ("C", 5)],
    "B": [("D", 4)],
}

4.4 Algorithm Overview

Key Algorithm: Dijkstra (non-negative weights)

Initialize distances to infinity.
Use min-heap to pick closest node.
Relax edges.

Complexity Analysis:

Time: O((V + E) log V)
Space: O(V)

5. Implementation Guide

5.1 Development Environment Setup

python3 -m venv .venv
. .venv/bin/activate
pip install networkx graphviz

5.2 Project Structure

graph-tool/
├── src/
│   ├── graph.py
│   ├── algorithms.py
│   ├── io.py
│   ├── viz.py
│   └── cli.py
├── tests/
└── README.md

5.3 The Core Question You’re Answering

“Which graph algorithm fits which problem, and why?”

5.4 Concepts You Must Understand First

Stop and research these before coding:

Graph representations
Edge relaxation
Union-find for Kruskal

5.5 Questions to Guide Your Design

How will you represent directed vs undirected graphs?
How will you parse weighted edges from input?
How will you export paths for visualization?

5.6 Thinking Exercise

Given a weighted graph with 5 nodes, run Dijkstra by hand and record the priority queue at each step.

5.7 The Interview Questions They’ll Ask

Why does BFS find shortest path in unweighted graphs?
When should you use Bellman-Ford?
Explain MST vs shortest path tree.

5.8 Hints in Layers

Hint 1: Start with adjacency list It is simplest and handles sparse graphs well.

Hint 2: Use heapq Python’s heapq makes Dijkstra easy.

Hint 3: Separate graph core from algorithms Keeps code reusable.

5.9 Books That Will Help

Topic	Book	Chapter
Graphs	“Algorithms, Fourth Edition”	Ch. 4.1
BFS/DFS	“Introduction to Algorithms”	Ch. 22
Shortest paths	“Introduction to Algorithms”	Ch. 24
MST	“Introduction to Algorithms”	Ch. 23

5.10 Implementation Phases

Phase 1: Foundation (5-7 days)

Goals:

Build graph representations
Implement BFS and DFS

Tasks:

Implement adjacency list and matrix.
Add BFS/DFS traversal output.

Checkpoint: Traversal order matches expected.

Phase 2: Shortest Paths (7-10 days)

Goals:

Implement Dijkstra and Bellman-Ford
Add path reconstruction

Tasks:

Implement priority queue-based Dijkstra.
Add negative cycle detection.

Checkpoint: Shortest paths match known examples.

Phase 3: MST and Visualization (5-7 days)

Goals:

Implement MST
Add DOT export

Tasks:

Implement Prim or Kruskal.
Export graph and MST to DOT.

Checkpoint: MST cost matches expected.

5.11 Key Implementation Decisions

Decision	Options	Recommendation	Rationale
Graph format	list, matrix	list	Sparse-friendly
MST	Prim, Kruskal	Kruskal	Clean with union-find
Visualization	optional	yes	Helps intuition

6. Testing Strategy

6.1 Test Categories

Category	Purpose	Examples
Unit Tests	Core algorithms	Known graphs
Integration Tests	CLI + IO	Load from file
Edge Case Tests	Disconnected graphs	Multiple components

6.2 Critical Test Cases

Graph with negative edges but no negative cycles.
Disconnected graph for BFS/DFS.
Graph with multiple equal shortest paths.

6.3 Test Data

A B 3
A C 5
B D 4

7. Common Pitfalls and Debugging

7.1 Frequent Mistakes

Pitfall	Symptom	Solution
Using BFS on weighted graph	Incorrect distances	Use Dijkstra
Forgetting to relax edges	Wrong path cost	Check relaxation loop
Missing negative cycle detection	Infinite updates	Add final pass

7.2 Debugging Strategies

Print distances after each relaxation.
Compare with known examples from textbooks.

7.3 Performance Traps

Using a matrix for sparse graphs leads to wasted memory and slow scans.

8. Extensions and Challenges

8.1 Beginner Extensions

Add topological sort for DAGs.
Add path reconstruction for BFS.

8.2 Intermediate Extensions

Implement A* search.
Add dynamic graph updates.

8.3 Advanced Extensions

Implement Johnson’s algorithm for all-pairs shortest paths.
Add visualization of algorithm steps.

9. Real-World Connections

9.1 Industry Applications

Routing and navigation systems
Dependency resolution in build systems

networkx: Python graph algorithms.
Boost Graph Library: C++ graph toolkit.

9.3 Interview Relevance

Graph problems are common in interviews, especially BFS/DFS and shortest path variants.

10. Resources

10.1 Essential Reading

“Algorithms, Fourth Edition” by Sedgewick - Ch. 4.1
“Introduction to Algorithms” - Ch. 22-24

10.2 Video Resources

Graph algorithm visualizations on YouTube

10.3 Tools and Documentation

Graphviz: https://graphviz.org
networkx: https://networkx.org

11. Self-Assessment Checklist

11.1 Understanding

I can explain BFS vs DFS use cases.
I can explain why Dijkstra needs non-negative edges.
I can explain MST vs shortest path.

11.2 Implementation

Algorithms pass tests on known graphs.
CLI outputs are correct.

11.3 Growth

I can choose the right graph algorithm for a problem.

12. Submission / Completion Criteria

Minimum Viable Completion:

BFS, DFS, Dijkstra
Graph loader

Full Completion:

Bellman-Ford and MST
DOT export

Excellence (Going Above and Beyond):

A* and all-pairs shortest paths
Step-by-step visual playback

This guide was generated from LEARN_ALGORITHMS_DEEP_DIVE.md. For the complete learning path, see the parent directory.

Project 10: Graph Algorithms Suite

Quick Reference

1. Learning Objectives

2. Theoretical Foundation

2.1 Core Concepts

2.2 Why This Matters

2.3 Historical Context / Background

2.4 Common Misconceptions

3. Project Specification

3.1 What You Will Build

3.2 Functional Requirements

3.3 Non-Functional Requirements

3.4 Example Usage / Output

3.5 Real World Outcome

4. Solution Architecture

4.1 High-Level Design

4.2 Key Components

4.3 Data Structures

4.4 Algorithm Overview

5. Implementation Guide

5.1 Development Environment Setup

5.2 Project Structure

5.3 The Core Question You’re Answering

5.4 Concepts You Must Understand First

5.5 Questions to Guide Your Design

5.6 Thinking Exercise

5.7 The Interview Questions They’ll Ask

5.8 Hints in Layers

5.9 Books That Will Help

5.10 Implementation Phases

Phase 1: Foundation (5-7 days)

Phase 2: Shortest Paths (7-10 days)

Phase 3: MST and Visualization (5-7 days)

5.11 Key Implementation Decisions

6. Testing Strategy

6.1 Test Categories

6.2 Critical Test Cases

6.3 Test Data

7. Common Pitfalls and Debugging

7.1 Frequent Mistakes

7.2 Debugging Strategies

7.3 Performance Traps

8. Extensions and Challenges

8.1 Beginner Extensions

8.2 Intermediate Extensions

8.3 Advanced Extensions

9. Real-World Connections

9.1 Industry Applications

9.2 Related Open Source Projects

9.3 Interview Relevance

10. Resources

10.1 Essential Reading

10.2 Video Resources

10.3 Tools and Documentation

10.4 Related Projects in This Series

11. Self-Assessment Checklist

11.1 Understanding

11.2 Implementation

11.3 Growth

12. Submission / Completion Criteria