d803bfe2b1
git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
100 lines
6.1 KiB
Markdown
100 lines
6.1 KiB
Markdown
# Architecture: Bounded-Range Dynamic Minimum Cut
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## Overview
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This crate implements the first deterministic exact fully-dynamic minimum cut
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algorithm with subpolynomial update time, based on arxiv:2512.13105 (December 2024).
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## System Architecture
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```
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┌─────────────────────────────────────────────────────────────────┐
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│ MinCutWrapper │
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│ ┌─────────────────────────────────────────────────────────────┐ │
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│ │ O(log n) Bounded-Range Instances │ │
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│ │ ┌─────────┐ ┌─────────┐ ┌─────────┐ ┌─────────┐ │ │
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│ │ │ [1,1] │ │ [1,1] │ │ [2,2] │ ... │ [λ,1.2λ]│ │ │
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│ │ └────┬────┘ └────┬────┘ └────┬────┘ └────┬────┘ │ │
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│ │ │ │ │ │ │ │
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│ │ ▼ ▼ ▼ ▼ │ │
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│ │ ┌──────────────────────────────────────────────────────┐ │ │
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│ │ │ ProperCutInstance Trait │ │ │
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│ │ │ - apply_inserts(edges) │ │ │
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│ │ │ - apply_deletes(edges) │ │ │
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│ │ │ - query() -> ValueInRange | AboveRange │ │ │
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│ │ └──────────────────────────────────────────────────────┘ │ │
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│ └─────────────────────────────────────────────────────────────┘ │
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│ │ │
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│ ┌───────────────────────────┴───────────────────────────────┐ │
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│ │ DynamicConnectivity │ │
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│ │ (Union-Find with rebuild on delete) │ │
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│ └───────────────────────────────────────────────────────────┘ │
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└─────────────────────────────────────────────────────────────────┘
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┌─────────────────────────────────────────────────────────────────┐
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│ Supporting Components │
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├─────────────────────────────────────────────────────────────────┤
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│ LocalKCutOracle │ DeterministicLocalKCut │
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│ - search(graph, query) │ - BFS exploration │
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│ - deterministic │ - Early termination │
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├───────────────────────────┼─────────────────────────────────────┤
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│ ClusterHierarchy │ FragmentingAlgorithm │
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│ - O(log n) levels │ - Connected components │
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│ - Recursive decomposition│ - Merge/split handling │
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├───────────────────────────┼─────────────────────────────────────┤
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│ CutCertificate │ AuditLogger │
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│ - Witness tracking │ - Provenance logging │
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│ - JSON export │ - Thread-safe │
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└───────────────────────────┴─────────────────────────────────────┘
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```
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## Component Responsibilities
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### MinCutWrapper
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- Manages O(log n) bounded-range instances
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- Geometric ranges with factor 1.2
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- Lazy instantiation
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- Order invariant: inserts before deletes
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### ProperCutInstance
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- Abstract interface for cut maintenance
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- Implementations: StubInstance, BoundedInstance
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### DeterministicLocalKCut
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- BFS-based local minimum cut search
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- Fully deterministic (no randomness)
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- Configurable radius and budget
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### ClusterHierarchy
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- Multi-level vertex clustering
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- Fast boundary updates on edge changes
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### FragmentingAlgorithm
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- Handles graph disconnection
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- Tracks connected components
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## Data Flow
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1. **Update arrives** (insert/delete edge)
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2. **Wrapper buffers** the update with timestamp
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3. **On query**:
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a. Check connectivity (fast path for disconnected)
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b. Process instances in order
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c. Apply buffered updates (inserts then deletes)
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d. Query each instance
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e. Stop at first ValueInRange or end
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4. **Return result** with witness
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## Invariants
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1. **Range invariant**: Instance never sees λ < λ_min during update
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2. **Order invariant**: Inserts applied before deletes
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3. **Certificate invariant**: Every answer can be verified
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4. **Determinism invariant**: Same sequence → same output
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## Complexity
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- **Update**: O(n^{o(1)}) amortized
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- **Query**: O(log n) instances × O(n^{o(1)}) per instance
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- **Space**: O(n + m) per instance
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