Overview
A minimum spanning tree is the target structure found by algorithms such as Kruskal's and Prim's algorithms.
CoreSource checked
Minimum Spanning Tree
Connects all nodes in a weighted graph with the lowest total edge cost and no cycles.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Graph Traversal
- Difficulty
- Intermediate
- Family
- Graph Optimization
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 7
- Code example
- Ready
- Visualization
- Ready
Core Concepts
weighted graphsafe edgecycle avoidancetree weight
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
Depends on algorithm
Average
Depends on algorithm
Worst
Depends on algorithm
Space
O(V) to O(E)
When to Use
- Use when you need the cheapest way to connect all nodes without caring about route length between pairs.
- Use when network design is a central requirement.
When Not to Use
- Avoid when you need shortest paths from a source rather than one low-cost connecting structure.
- Avoid when the graph changes faster than the traversal result can be reused.
What It Does
- Connects every node
- Avoids cycles
- Minimizes total selected edge weight
Use Cases
- Network design
- Clustering by edge weight
- Approximation and graph preprocessing
Inputs
- Connected weighted undirected graph
Outputs
- Set of tree edges
- Total tree weight
Key Formulas (1)
Tree weight
totalweight = ∑(weight(edge) for edge in selectededges)
total_weight = sum(weight(edge) for edge in selected_edges)Measures the objective minimized by a spanning tree algorithm.
Pseudocode
- 1.Start with no selected edges
- 2.Choose edges according to a safe-edge rule
- 3.Reject choices that create cycles
- 4.Stop when all nodes are connected
Canonical Example
- Input
- edges = A-B:1, B-C:2, A-C:4
- Output
- tree = [A-B, B-C], weight = 3
- Explanation
- The cheaper two edges connect all three nodes without using the expensive edge A-C.
Implementation Notes
- Track the selected edge set and verify it contains exactly V - 1 edges for a connected graph.
- Use explicit node ids and keep traversal state separate from graph storage.
- Keep a tiny example with expected output beside the implementation.
Common Pitfalls
- Confusing minimum spanning trees with shortest path trees leads to wrong route assumptions.
- Forgetting to track visited or finalized nodes can create repeated work or cycles.
- Not defining boundary, empty, or degenerate inputs before implementation.
Interactive Mechanism Walkthrough
Graph Frontier & Edge RelaxationHighlight selected low-cost edges until a cycle-free spanning tree covers the graph.
Step 1 of 4: Start with no selected edges
graph-traversalNotes
The graph must be connected for one tree to span every node; otherwise the result is a forest.
Sources (2)
What each reference supports for this entry.
Introduction to Algorithms (CLRS)
BookTextbook support for Minimum Spanning Tree's standard algorithm design, invariants, and complexity analysis.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Minimum Spanning Tree's problem context, known alternatives, and practical selection tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Used byBoruvka's Algorithm
Graph Traversal · Core
Boruvka's Algorithm uses Minimum Spanning Tree as a subroutine, representation, or supporting idea while growing many tree components at once and repeatedly adding each component's cheapest outgoing edge.
Compare with thisUsed byChristofides Algorithm
Optimization · Core
The first construction step is a minimum spanning tree of the metric graph.
Compare with thisUsed byReverse-Delete Algorithm
Graph Traversal · Core
Reverse-Delete Algorithm uses Minimum Spanning Tree as a subroutine, representation, or supporting idea while removing expensive edges when their removal does not disconnect the graph.
Compare with thisRelationships
View full relationship pageGeneralizesKruskal's Algorithm
Kruskal's Algorithm is one method for finding a minimum spanning tree.
GeneralizesPrim's Algorithm
Prim's Algorithm is one method for finding a minimum spanning tree.
Related Algorithms
Tags
GraphWeighted GraphMinimum Spanning TreeGreedy