Overview
Euler tour traversal captures the order in which nodes are entered and left, which supports subtree queries, ancestor checks, and range-based tree processing.
CoreSource checked
Euler Tour
Records entry and exit order while traversing a tree or graph.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Graph Traversal
- Difficulty
- Intermediate
- Family
- Tree Traversal
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
Core Concepts
entry timeexit timedfs ordersubtree interval
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(V + E)
Average
O(V + E)
Worst
O(V + E)
Space
O(V)
When to Use
- Use when analyzing graph structure needs entry time.
- Use when the input matches the assumptions behind tree Traversal.
When Not to Use
- Avoid when the graph is so large that it needs streaming or external-memory handling.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Records entry and exit order while traversing a tree or graph
- Uses entry time and exit time to guide the work.
- Produces visited nodes.
Use Cases
- Analyzing graph structure
- Finding connectivity patterns
- Preparing graph-derived orderings
Inputs
- Graph
- Start nodes or query nodes
- Traversal state
Outputs
- Visited nodes
- Derived graph relationships
Key Formulas (1)
Ancestor interval test
entry[u] ≤ entry[v] and exit[v] ≤ exit[u]
entry[u] <= entry[v] and exit[v] <= exit[u]Checks whether u is an ancestor of v in a DFS tree.
Pseudocode
- 1.Prepare the inputs and state needed for Euler Tour.
- 2.Apply the entry time step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return visited nodes.
Canonical Example
- Input
- Euler Tour input with a small representative case
- Output
- Visited nodes
- Explanation
- Euler Tour applies entry time to produce visited nodes.
Implementation Notes
- Keep visitation state and edge classification state separate.
- Keep the entry time step separate enough to test independently.
Common Pitfalls
- Mixing directed and undirected edge assumptions can produce incorrect graph relationships.
- Forgetting to check edge cases can make Euler Tour look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Graph Frontier & Edge RelaxationHighlight active nodes, discovered edges, and the graph structure produced by each step.
Step 1 of 4: Prepare the inputs and state needed for Euler Tour.
graph-traversal-mechanismNotes
Euler Tour should be used with the assumptions and tradeoffs of tree Traversal in mind.
Sources (2)
What each reference supports for this entry.
Introduction to Algorithms (CLRS)
BookTextbook support for Euler Tour's standard algorithm design, invariants, and complexity analysis.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Euler Tour's problem context, known alternatives, and practical selection tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
UsesDepth-First Search
Graph Traversal · Foundation
Euler-tour based tree traversals are commonly produced by depth-first traversal order.
Compare with thisUsed byFleury's Algorithm
Graph Traversal · Core
Fleury's Algorithm uses Euler Tour as a subroutine, representation, or supporting idea while walking unused edges while avoiding bridges unless no alternative remains.
Compare with thisUsed byChinese Postman Algorithm
Optimization · Extended
Chinese Postman solutions construct an Eulerian traversal after augmenting the graph.
Compare with thisRelationships
View full relationship pageEuler-tour based tree traversals are commonly produced by depth-first traversal order.
Related Algorithms
Tags
TreeTraversalLCADepth First Search