Graph Algorithms
Master 3 graph algorithms with side-by-side comparisons. From Minimum Spanning Trees to topological sorting, understand when to use each one, explore real-world applications, and prepare for technical interviews.
What Is a Graph? (A Model for Relationships)
A graph is a data structure made of:
- Vertices (nodes) representing entities
- Edges representing relationships between them
Graphs are powerful because they let you model connections, not just data. Those connections can represent:
- Roads between cities
- Dependencies between tasks
- Links between webpages
- Relationships between users
Core idea: Graphs are how computers understand networks. If you are modeling relationships, you are probably building a graph.
Where Graphs Quietly Run the World
Graphs appear whenever things are connected.
Networks & Infrastructure
- Internet routing
- Power grids
- Social networks
Mapping & Logistics
- Cities connected by roads
- Delivery networks
- Cost-optimized routes
Build Systems & Compilers
- Dependency graphs
- Task scheduling
- Build order resolution
Data & Recommendations
- Friend suggestions
- Content recommendations
- Knowledge graphs
Key takeaway: If you are modeling relationships, you are probably building a graph.
Why Graph Problems Get Tricky Fast
Graphs grow complex quickly because:
- Nodes can connect to many other nodes
- Cycles introduce infinite paths
- Choices compound exponentially
Small graphs are intuitive. Large graphs demand careful algorithms.
Reduce Search Space
Prune paths that cannot lead to optimal solutions
Enforce Structure
DAGs, trees, and constraints simplify problems
Greedy Choices
Local optimums can lead to global optimums
What Is a Minimum Spanning Tree (MST)?
A Minimum Spanning Tree connects all nodes in a graph:
- Without cycles
- With minimum total edge weight
The cheapest way to connect everything.
MSTs appear in:
- Network design
- City planning
- Clustering problems
How MST Algorithms Build Connections
Prim's and Kruskal's solve the same problem but think differently.
| Algorithm | Strategy |
|---|---|
| Prim's | Grows outward from a starting node |
| Kruskal's | Sorts edges and connects smallest first |
Prim's is vertex-focused
Like growing a tree from a seed. It expands from one node, always picking the cheapest edge to a new node.
Kruskal's is edge-focused
Like a bargain hunter. It sorts all edges by cost and picks the cheapest ones that connect new components.
Engineering insight: Dense graphs favor Prim's. Sparse graphs favor Kruskal's.
What Is Topological Sorting?
Topological sorting orders nodes so that:
Every dependency comes before what depends on it.
This only works on Directed Acyclic Graphs (DAGs).
Real-world examples:
- Build pipelines
- Course prerequisites
- Task scheduling systems
Mental model: You cannot build the roof before the foundation.
How Kahn's Algorithm Resolves Dependencies
Kahn's algorithm:
- 1.Finds nodes with no incoming edges
- 2.Processes them first
- 3.Removes their edges
- 4.Repeats until done
If nodes remain with dependencies, a cycle exists.
Why it matters: Kahn's algorithm detects impossible schedules automatically. If the output length does not match the number of nodes, you have a cycle.
Complexity Comparison
| Algorithm | Time | Space | Output | Data Structure |
|---|---|---|---|---|
| Prim's Algorithm | O(E log V) | O(V + E) | Minimum Spanning Tree (MST) | Priority Queue (Min-Heap) |
| Kruskal's Algorithm | O(E log E) | O(V + E) | Minimum Spanning Tree (MST) | Union-Find (Disjoint Set Union) |
| Kahn's Algorithm | O(V + E) | O(V) | Topological Ordering (Linear Sequence) | Queue + Indegree Array |
Click any algorithm name to read its full article with code examples, history, and use cases.
Quick Decision Guide
Need a Minimum Spanning Tree?
Graph is dense (many edges)?
Graph is sparse (few edges)?
Need topological ordering (DAG)?
Building a network or connecting cities?
Resolving build dependencies?
Common Interview Questions
Why does an MST never contain cycles?
Because removing any edge in a cycle reduces total cost without disconnecting the graph. A cycle means there's a redundant edge.
When should you prefer Kruskal over Prim?
When the graph is sparse or edge-heavy. Kruskal's sorts edges once, while Prim's uses a priority queue that gets expensive with dense graphs.
Also prefer Kruskal when you already have edges in a sorted list.
Why can't topological sorting work on cyclic graphs?
Because dependencies become circular and no valid order exists. If A depends on B and B depends on A, neither can go first.
What data structure makes Kruskal efficient?
Union-Find (Disjoint Set Union). It detects cycles in near-constant time using path compression and union by rank.
This is often the real interview question: implement Union-Find.
What happens if Kahn's algorithm processes fewer nodes than exist?
A cycle is present in the graph. The remaining nodes are all part of circular dependencies.
Myths and Quick Facts
Graph Myths
"Graphs are just trees"
Trees are a special case of graphs (connected, acyclic)
"MST gives shortest paths"
It minimizes total cost, not individual paths
"Topological sort is just sorting"
It enforces dependency constraints, not value ordering
"Dense graphs are rare"
Social networks prove otherwise
Did You Know?
- Many real graphs are sparse
- Cycles are often bugs in dependency systems
- MST algorithms are greedy but correct
- Graph problems scale faster than most data structures
Algorithm Personalities
Prim's builds patiently from home. Kruskal's shops for the cheapest deals. Kahn's refuses to work without prerequisites.
See Graph Algorithms in Action
Build graphs, adjust edge weights, and watch how algorithms connect nodes and resolve dependencies. Some grow carefully. Some choose greedily. Some fail loudly when constraints are impossible. Visualizing graphs turns abstraction into intuition.
Open Graph Visualizer