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Luke Sun

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Graph Theory: Overview

| , 2 minutes reading.

Graph Theory: The Hidden Structure of the World

The Map is Not the Territory

When we learn Graph Theory in school, we see circles (Nodes) and lines (Edges). It feels abstract and dry. But in software engineering, a graph is never just a graph.

  • It is a City (Google Maps).
  • It is a Social Circle (Facebook Friends).
  • It is a Dependency Tree (Webpack/NPM).
  • It is a State Machine (Game Logic).

The “Algorithms” we discuss in this section are not just math formulas. They are Strategies for Navigation. They are the tools we use to answer fundamental questions about relationships:

  • “How do I get there?” (Pathfinding)
  • “Who is important?” (Centrality)
  • “Are we connected?” (Connectivity)
  • “What must happen first?” (Dependency)

The Landscape of Algorithms

In this section, we explore the seven most critical strategies for navigating these structures.

AlgorithmThe Soul / MetaphorBest For

DijkstraThe Water Spreader
Cautiously explores the closest options first.		GPS / Navigation
When you need the absolute shortest path in a physical space.
Bellman-FordThe Auditor
Iteratively checks for mistakes and negative costs.		Finance / Arbitrage
When costs can be negative (rewards) or systems are unstable.
A Search*The Visionary
Balances current cost with a future goal (Heuristic).		Games / Grid Maps
When you know where you are going and want speed.
Floyd-WarshallThe Diplomat
Sits back and negotiates consensus between all pairs.		Dense Networks
When you need to know distance between everyone (small scale).
Topological SortThe Project Manager
Unlocks tasks only when prerequisites are met.		Build Systems / Compilers
Resolving dependencies (npm, make, webpack).
Union-FindThe Bureaucrat
Flattens hierarchies to instantly find the “Boss”.		Social Groups / Image Segments
Grouping dynamic data together efficiently.
LCAThe Mediator
Jumps back in time to find the common origin.		Git / Version Control
Finding the merge base or biological ancestor.

Engineering Mindset: Modeling the Problem

The hardest part of graph theory isn’t writing the code (libraries exist for that). It’s Modeling.

  • Is your problem a “Shortest Path” problem? (e.g., Minimizing latency)
  • Is it a “Connectivity” problem? (e.g., Access control lists)
  • Is it a “Flow” problem? (e.g., Max bandwidth)

Once you map your messy real-world problem to one of these classic models, the solution often becomes trivial.

Summary

Don’t memorize the code. Understand the Personality of each algorithm.

  • Need speed with direction? Call the Visionary (A*).
  • Need absolute truth in a messy world? Call the Auditor (Bellman-Ford).
  • Need to organize chaos? Call the Project Manager (Topological Sort).

Let’s begin our journey through the map.