Overview
Capacity Scaling Max Flow is a network flow method for maximum flow. It works by solving flow on large-capacity edges first and lowering the scale threshold.
ExtendedSource checked
Capacity Scaling Max Flow
Capacity Scaling Max Flow addresses maximum flow by solving flow on large-capacity edges first and lowering the scale threshold.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Graph Traversal
- Difficulty
- Intermediate
- Family
- Network Flow
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 0
- Relationships
- 6
- Code example
- Ready
- Visualization
- Ready
Core Concepts
maximum flownetwork flowresidual graphgraph traversal
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(E)
Average
O(V^2 E)
Worst
O(V^2 E)
Space
O(V + E)
When to Use
- Use Capacity Scaling Max Flow when solving maximum flow and you want to benefit from solving flow on large-capacity edges first and lowering the scale threshold.
- Choose it when O(V^2 E) average/worst-case time and O(V + E) memory match your workload profile.
- Prefer it over Push-Relabel or Gomory-Hu Tree when the problem structure favors maximum flow.
When Not to Use
- Avoid when the graph model does not match the method's direction, weight, or connectivity assumptions.
- Avoid when a simpler traversal gives the same information with less bookkeeping.
What It Does
- Computes maximum flow within Network Flow workloads (O(V^2 E) time, O(V + E) space).
- Works by solving flow on large-capacity edges first and lowering the scale threshold.
- Maintains key invariants across maximum flow, network flow, residual graph.
Use Cases
- Production and research workloads requiring maximum flow via solving flow on large-capacity edges first and lowering the scale threshold.
- Network Flow pipelines where O(V^2 E) time and O(V + E) space fit system constraints.
- Engineering comparisons against Push-Relabel and Gomory-Hu Tree when tuning maximum flow.
Inputs
- Directed capacitated graph G = (V, E, c) with source s and sink t
Outputs
- Maximum s-t flow assignment f(u, v) on each edge
- Residual network and minimum-cut partition
Pseudocode
- 1.Construct the graph representation G = (V, E) and initialize auxiliary structures for maximum flow (network flow).
- 2.Seed the initial frontier, partition, or residual capacities according to Capacity Scaling Max Flow's invariants.
- 3.Iteratively process vertices or edges by solving flow on large-capacity edges first and lowering the scale threshold.
- 4.Maintain structural invariants (maximum flow, network flow, residual graph) and prune or contract resolved subproblems.
- 5.Extract and return the final maximum flow structure from the converged graph state.
Canonical Example
- Input
- Directed capacitated graph G = (V, E, c) with source s and sink t configured for a representative maximum flow test case
- Output
- Maximum s-t flow assignment f(u, v) on each edge satisfying Capacity Scaling Max Flow invariants
- Explanation
- Capacity Scaling Max Flow processes the input by solving flow on large-capacity edges first and lowering the scale threshold, tracking maximum flow and network flow to produce the verified maximum flow result.
Implementation Notes
- Represent vertices and edges explicitly before optimizing storage.
- Keep graph direction, weights, and multiedges clear in tests.
- Record tie-breaking when multiple valid outputs exist.
Common Pitfalls
- Using the method on the wrong graph type, such as directed versus undirected or weighted versus unweighted.
- Forgetting disconnected components or isolated vertices.
- Letting tie-breaking hide nondeterministic output in tests.
Interactive Mechanism Walkthrough
Graph Frontier & Edge RelaxationShow solving flow on large-capacity edges first and lowering the scale threshold on a small maximum flow example.
Step 1 of 5: Construct the graph representation G = (V, E) and initialize auxiliary structures for maximum flow (network flow).
graph-traversal-mechanismNotes
Capacity Scaling Max Flow operates in O(V^2 E) time and O(V + E) space within the Network Flow family. Often compared with Push-Relabel and Gomory-Hu Tree.
Sources (2)
What each reference supports for this entry.
Network Flows: Theory, Algorithms, and Applications
BookReference for Capacity Scaling Max Flow's network-flow, cut, residual-network, or matching formulation.
Introduction to Algorithms (CLRS)
BookTextbook support for Capacity Scaling Max Flow's algorithm family, complexity framing, and standard implementation concepts.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Alternative toPush-Relabel
Graph Traversal · Core
Capacity Scaling Max Flow and Push-Relabel are practical alternatives within maximum flow, with different assumptions and tradeoffs.
Compare with thisAlternative toStoer-Wagner Min Cut
Graph Traversal · Core
Capacity Scaling Max Flow and Stoer-Wagner Min Cut are practical alternatives within maximum flow, with different assumptions and tradeoffs.
Compare with thisRelationships
View full relationship pageAlternative toPush-Relabel
Capacity Scaling Max Flow and Push-Relabel are practical alternatives within maximum flow, with different assumptions and tradeoffs.
Alternative toGomory-Hu Tree
Capacity Scaling Max Flow and Gomory-Hu Tree are practical alternatives within maximum flow, with different assumptions and tradeoffs.
Alternative toStoer-Wagner Min Cut
Capacity Scaling Max Flow and Stoer-Wagner Min Cut are practical alternatives within maximum flow, with different assumptions and tradeoffs.
Tags
GraphTraversalNetwork Flow