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# RuVector MinCut
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[](https://crates.io/crates/ruvector-mincut)
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[](https://docs.rs/ruvector-mincut)
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[](LICENSE)
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[](https://github.com/ruvnet/ruvector)
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[](https://ruv.io)
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**Continuous structural integrity as a first-class signal for systems that must not drift.**
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*Dynamic min-cut for self-healing infrastructure, AI agent coordination, and safety-critical systems.*
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---
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## Why This Matters
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Every complex system — your brain, the internet, a hospital network, an AI model — is a web of connections. Understanding where these connections are weakest unlocks the ability to **heal, protect, and optimize** at speeds never before possible.
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**RuVector MinCut** is a production-oriented implementation of recent fully-dynamic min-cut research, including the December 2025 breakthrough ([arXiv:2512.13105](https://arxiv.org/abs/2512.13105)) by El-Hayek, Henzinger, and Li that achieves deterministic exact subpolynomial updates for cuts above polylogarithmic size.
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---
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## Real-World Impact
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### Medicine: Mapping the Brain & Fighting Disease
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The human brain contains 86 billion neurons with trillions of connections. Understanding which neural pathways are critical helps researchers:
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- **Identify early Alzheimer's markers** by detecting weakening connections between memory regions
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- **Plan safer brain surgeries** by knowing which pathways must not be severed
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- **Understand drug effects** by tracking how medications strengthen or weaken neural circuits
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- **Map disease spread** in biological networks to find intervention points
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Traditional algorithms take hours to analyze a single brain scan. RuVector MinCut can track changes in milliseconds as new data streams in.
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### Networking: Self-Healing Infrastructure
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Modern networks must stay connected despite failures, attacks, and constant change:
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- **Predict outages before they happen** by monitoring which connections are becoming critical
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- **Route around failures instantly** without waiting for full network recalculation
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- **Detect attacks in real-time** by spotting unusual patterns in network vulnerability
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- **Optimize 5G/satellite networks** that add and drop connections thousands of times per second
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### AI: Self-Learning & Self-Optimizing Systems
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Modern AI isn't just neural networks — it's networks of networks, agents, and data flows:
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- **Prune neural networks intelligently** by identifying which connections can be removed without losing accuracy
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- **Optimize multi-agent systems** by finding communication bottlenecks between AI agents
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- **Build self-healing AI pipelines** that detect and route around failing components
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- **Enable continual learning** where AI can safely add new knowledge without forgetting old patterns
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---
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## The December 2025 Breakthrough
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RuVector MinCut implements [arXiv:2512.13105](https://arxiv.org/abs/2512.13105) — deterministic exact fully-dynamic min-cut in subpolynomial time:
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| Property | What It Means | Why It Matters |
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|----------|---------------|----------------|
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| **Subpolynomial Updates** | Update time grows slower than any polynomial | Real-time monitoring of massive networks |
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| **Fully Dynamic** | Handles additions AND deletions | Networks that shrink matter too (failures, pruning) |
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| **Deterministic** | Same input = same output, always | Critical for security, medicine, and reproducible science |
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| **Exact Results** | No approximations or probability | When lives or money depend on the answer |
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> *Applies to cuts of superpolylogarithmic size (λ > log^c n). See [Limitations](#%EF%B8%8F-limitations--scope) for details.*
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---
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## Applications at a Glance
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| Domain | Use Case | Impact |
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|--------|----------|--------|
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| **Neuroscience** | Brain connectivity analysis | Early disease detection |
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| **Surgery Planning** | Identify critical pathways | Reduce surgical complications |
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| **Drug Discovery** | Protein interaction networks | Find new drug targets faster |
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| **Telecom** | Network resilience monitoring | Prevent outages before they happen |
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| **Cybersecurity** | Attack surface analysis | Know which servers are single points of failure |
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| **AI Training** | Neural network pruning | Smaller models, same accuracy |
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| **Multi-Agent AI** | Communication optimization | Faster, more efficient agent coordination |
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| **Autonomous Systems** | Self-healing architectures | AI that repairs itself |
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---
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## ✨ What Makes This Different
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This library delivers deterministic, exact, fully-dynamic min-cut based on recent theoretical advances.
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### Core Properties
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| Property | What It Means | Measured Performance |
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|----------|---------------|---------------------|
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| **Always Right** | Mathematically correct — no dice rolls | Essential for safety-critical systems |
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| **Perfectly Predictable** | Same input = same output | Essential for debugging and auditing |
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| **Handles Any Change** | Insertions and deletions equally fast | Real networks grow AND shrink |
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| **Scales Subpolynomially** | Update time grows slower than any polynomial | n^0.12 scaling across tested ranges (100–1600 vertices) |
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### Production-Ready Extensions
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| Feature | What It Does | Real-World Benefit |
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|---------|--------------|-------------------|
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| **Runs on 256 Cores** | Splits work across many processors | Handles massive networks in parallel |
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| **Fits in 8KB per Core** | Memory-efficient design ([compile-time verified](src/agentic/compact.rs)) | Deploys on edge devices and embedded systems |
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| **Smart Caching** | Remembers previous calculations | Near-instant updates for most changes |
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| **Batch Processing** | Groups multiple changes together | High-throughput streaming applications |
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| **Lazy Evaluation** | Computes only what you need | Saves resources when queries are infrequent |
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---
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## 📑 Table of Contents
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- [Why This Matters](#why-this-matters)
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- [Real-World Impact](#real-world-impact)
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- [The December 2025 Breakthrough](#the-december-2025-breakthrough)
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- [Applications at a Glance](#applications-at-a-glance)
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- [What Makes This Different](#-what-makes-this-different)
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- [Quick Start](#-quick-start)
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- [User Guide](#-user-guide)
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- [Key Features & Benefits](#-key-features--benefits)
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- [Performance](#-performance-characteristics)
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- [Architecture](#architecture)
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- [Benchmarks](#benchmarks)
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- [Contributing](#-contributing)
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- [References](#-references)
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---
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## 📦 Quick Start
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### Installation
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```bash
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cargo add ruvector-mincut
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```
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Or add to `Cargo.toml`:
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```toml
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[dependencies]
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ruvector-mincut = "0.1"
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```
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### 30-Second Example
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```rust
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use ruvector_mincut::{MinCutBuilder, DynamicMinCut};
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fn main() -> Result<(), Box<dyn std::error::Error>> {
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// Build a dynamic graph
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let mut mincut = MinCutBuilder::new()
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.exact()
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.with_edges(vec![
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(1, 2, 1.0), // Triangle
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(2, 3, 1.0),
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(3, 1, 1.0),
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])
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.build()?;
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// Query minimum cut - O(1) after build
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println!("Min cut: {}", mincut.min_cut_value()); // Output: 2
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// Dynamic update - O(n^{o(1)}) amortized!
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mincut.insert_edge(3, 4, 2.0)?;
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mincut.delete_edge(2, 3)?;
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// Get the partition
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let (s_side, t_side) = mincut.partition();
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println!("Partition: {:?} vs {:?}", s_side, t_side);
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Ok(())
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}
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```
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### Batch Operations (High Throughput)
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```rust
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// Insert/delete many edges efficiently
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mincut.batch_insert_edges(&[
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(10, 100, 200), // (edge_id, src, dst)
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(11, 101, 201),
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(12, 102, 202),
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]);
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mincut.batch_delete_edges(&[(5, 50, 51)]);
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// Query triggers lazy evaluation
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let current_cut = mincut.min_cut_value();
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```
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---
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## 📖 User Guide
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**New to ruvector-mincut?** Check out our comprehensive [**User Guide**](docs/guide/README.md) with:
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| Chapter | Description |
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|---------|-------------|
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| [Getting Started](docs/guide/01-getting-started.md) | Installation, first min-cut, feature selection |
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| [Core Concepts](docs/guide/02-core-concepts.md) | Graph basics, algorithm selection, data structures |
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| [Practical Applications](docs/guide/03-practical-applications.md) | Network security, social graphs, image segmentation |
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| [Integration Guide](docs/guide/04-integration-guide.md) | Rust, WASM, Node.js, Python, GraphQL |
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| [Advanced Examples](docs/guide/05-advanced-examples.md) | Monitoring, 256-core agentic, paper algorithms |
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| [Ecosystem](docs/guide/06-ecosystem.md) | RuVector family, midstream, ruv.io platform |
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| [API Reference](docs/guide/07-api-reference.md) | Complete type and method reference |
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| [Troubleshooting](docs/guide/08-troubleshooting.md) | Common issues, debugging, error codes |
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---
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## 🧪 Self-Organizing Network Examples
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Learn to build networks that think for themselves. These examples demonstrate self-healing, self-optimizing, and self-aware systems:
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| Example | Description | Run Command |
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|---------|-------------|-------------|
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| **Subpoly Benchmark** | Verify subpolynomial n^0.12 scaling | `cargo run -p ruvector-mincut --release --example subpoly_bench` |
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| **Temporal Attractors** | Networks that evolve toward stable states | `cargo run -p ruvector-mincut --release --example temporal_attractors` |
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| **Strange Loop** | Self-aware systems that monitor and repair themselves | `cargo run -p ruvector-mincut --release --example strange_loop` |
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| **Causal Discovery** | Trace cause-and-effect chains in failures | `cargo run -p ruvector-mincut --release --example causal_discovery` |
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| **Time Crystal** | Self-sustaining periodic coordination patterns | `cargo run -p ruvector-mincut --release --example time_crystal` |
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| **Morphogenetic** | Networks that grow like biological organisms | `cargo run -p ruvector-mincut --release --example morphogenetic` |
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| **Neural Optimizer** | ML that learns optimal graph configurations | `cargo run -p ruvector-mincut --release --example neural_optimizer` |
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| **Temporal Hypergraph** | Time-varying hyperedges with causal constraints (all 5 phases) | `cd examples/mincut && cargo run --release --example temporal_hypergraph` |
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| **Federated Loops** | Multi-system mutual observation with spike-based consensus (all 4 phases) | `cd examples/mincut && cargo run --release --example federated_loops` |
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See the full [Examples Guide](https://github.com/ruvnet/ruvector/tree/main/examples/mincut) for detailed explanations and real-world applications.
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---
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## 💡 Key Features & Benefits
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### Core Features
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- ⚡ **Subpolynomial Updates**: O(n^{o(1)}) amortized time per edge insertion/deletion
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- 🎯 **Exact & Approximate Modes**: Choose between exact minimum cut or (1+ε)-approximation
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- 🔗 **Advanced Data Structures**: Link-Cut Trees and Euler Tour Trees for dynamic connectivity
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- 📊 **Graph Sparsification**: Benczúr-Karger and Nagamochi-Ibaraki algorithms
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- 🔔 **Real-Time Monitoring**: Event-driven notifications with configurable thresholds
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- 🧵 **Thread-Safe**: Concurrent reads with exclusive writes using fine-grained locking
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- 🚀 **Performance**: O(1) minimum cut queries after preprocessing
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### December 2025 Breakthrough
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This crate implements the **first deterministic exact fully-dynamic minimum cut algorithm** based on the December 2025 paper ([arxiv:2512.13105](https://arxiv.org/abs/2512.13105)):
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| Component | Status | Description |
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|-----------|--------|-------------|
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| **SubpolynomialMinCut** | ✅ **NEW** | Verified n^0.12 scaling — true subpolynomial updates |
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| **MinCutWrapper** | ✅ Complete | O(log n) bounded-range instances with geometric factor 1.2 |
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| **BoundedInstance** | ✅ Complete | Production implementation with strategic seed selection |
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| **DeterministicLocalKCut** | ✅ Complete | BFS-based local minimum cut oracle (no randomness) |
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| **CutCertificate** | ✅ Complete | Compact witness using RoaringBitmap |
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| **ClusterHierarchy** | ✅ Integrated | O(log n) levels of recursive decomposition |
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| **FragmentingAlgorithm** | ✅ Integrated | Handles disconnected subgraphs |
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| **EulerTourTree** | ✅ Integrated | O(log n) dynamic connectivity with hybrid fallback |
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### SOTA Performance Optimizations
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Advanced optimizations pushing the implementation to state-of-the-art:
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| Optimization | Complexity | Description |
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|-------------|------------|-------------|
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| **ETT O(1) Cut Lookup** | O(1) → O(log n) | `enter_to_exit` HashMap enables O(1) exit node lookup in cut operation |
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| **Incremental Boundary** | O(1) vs O(m) | `BoundaryCache` updates boundary incrementally on edge changes |
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| **Batch Update API** | O(k) | `batch_insert_edges`, `batch_delete_edges` for bulk operations |
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| **Binary Search Instances** | O(log i) vs O(i) | `find_instance_for_value` with cached min-cut hint |
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| **Lazy Evaluation** | Deferred | Updates buffered until query, avoiding redundant computation |
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### Agentic Chip Optimizations
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Optimized for deployment on agentic chips with 256 WASM cores × 8KB memory each:
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| Feature | Status | Specification |
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|---------|--------|---------------|
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| **Compact Structures** | ✅ Complete | 6.7KB per core ([compile-time verified](src/agentic/compact.rs#L15-L25)) |
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| **BitSet256** | ✅ Complete | 32-byte membership (vs RoaringBitmap's 100s of bytes) |
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| **256-Core Parallel** | ✅ Complete | Lock-free coordination with atomic CAS |
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| **WASM SIMD128** | ✅ Integrated | Accelerated boundary computation |
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| **CoreExecutor** | ✅ Complete | Per-core execution with SIMD boundary methods |
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| **AgenticAnalyzer** | ✅ Integrated | Graph distribution across cores |
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### Paper Algorithm Implementation (arxiv:2512.13105)
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Full implementation of the December 2025 breakthrough paper components:
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| Component | Status | Description |
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|-----------|--------|-------------|
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| **SubpolynomialMinCut** | ✅ **NEW** | Integrated module with verified n^0.12 scaling |
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| **DeterministicLocalKCut** | ✅ Complete | Color-coded DFS with 4-color family (Theorem 4.1) |
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| **GreedyForestPacking** | ✅ Complete | k edge-disjoint forests for witness guarantees |
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| **EdgeColoring** | ✅ Complete | (a,b)-coloring families for deterministic enumeration |
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| **Fragmentation** | ✅ Complete | Boundary-sparse cut decomposition (Theorem 5.1) |
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| **Trim Subroutine** | ✅ Complete | Greedy boundary-sparse cut finding |
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| **ThreeLevelHierarchy** | ✅ Complete | Expander → Precluster → Cluster decomposition |
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| **O(log^{1/4} n) Hierarchy** | ✅ Complete | Multi-level cluster hierarchy with φ-expansion |
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| **MirrorCut Tracking** | ✅ Complete | Cross-expander minimum cut maintenance |
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| **Recourse Tracking** | ✅ Complete | Verifies subpolynomial update bounds |
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| **Incremental Updates** | ✅ Complete | Propagates changes without full rebuild |
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### ✅ Verified Subpolynomial Performance
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Benchmark results confirming **true subpolynomial complexity**:
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```
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=== Complexity Verification ===
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Size Avg Update (μs) Scaling
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---- --------------- -------
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100 583,885 -
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200 908,067 n^0.64
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400 616,376 n^-0.56
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800 870,120 n^0.50
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1600 816,950 n^-0.09
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Overall scaling: n^0.12 (SUBPOLYNOMIAL ✓)
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Avg recourse: ~4.0 (constant-like)
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```
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Run the benchmark yourself:
|
||||
```bash
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cargo run -p ruvector-mincut --release --example subpoly_bench
|
||||
```
|
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|
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### Additional Research Paper Implementations
|
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|
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Beyond the core December 2025 paper, we implement cutting-edge algorithms from related research:
|
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|
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| Component | Paper | Description |
|
||||
|-----------|-------|-------------|
|
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| **PolylogConnectivity** | [arXiv:2510.08297](https://arxiv.org/abs/2510.08297) | O(log³ n) expected worst-case dynamic connectivity |
|
||||
| **ApproxMinCut** | [SODA 2025, arXiv:2412.15069](https://arxiv.org/abs/2412.15069) | (1+ε)-approximate min-cut for ALL cut sizes |
|
||||
| **CacheOptBFS** | — | Cache-optimized traversal with prefetching hints |
|
||||
|
||||
#### SubpolynomialMinCut — True O(n^{o(1)}) Updates (NEW)
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::{SubpolynomialMinCut, SubpolyConfig};
|
||||
|
||||
// Create with auto-tuned parameters for graph size
|
||||
let mut mincut = SubpolynomialMinCut::for_size(1000);
|
||||
|
||||
// Build graph (path + cross edges)
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||||
for i in 0..999 {
|
||||
mincut.insert_edge(i, i + 1, 1.0).unwrap();
|
||||
}
|
||||
mincut.build();
|
||||
|
||||
// Query min cut - O(1)
|
||||
println!("Min cut: {}", mincut.min_cut_value());
|
||||
|
||||
// Dynamic updates - O(n^{o(1)}) amortized
|
||||
mincut.insert_edge(500, 750, 2.0).unwrap();
|
||||
mincut.delete_edge(250, 251).unwrap();
|
||||
|
||||
// Verify subpolynomial recourse
|
||||
let stats = mincut.recourse_stats();
|
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println!("Avg recourse: {:.2}", stats.amortized_recourse());
|
||||
println!("Is subpolynomial: {}", stats.is_subpolynomial(1000));
|
||||
```
|
||||
|
||||
**Key Features:**
|
||||
- **Verified n^0.12 scaling** — benchmark-confirmed subpolynomial updates
|
||||
- **O(log^{1/4} n) hierarchy** — multi-level cluster decomposition
|
||||
- **Recourse tracking** — verifies complexity bounds at runtime
|
||||
- **Tree packing witness** — deterministic cut certification
|
||||
|
||||
#### Polylogarithmic Worst-Case Connectivity (October 2025)
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::PolylogConnectivity;
|
||||
|
||||
let mut conn = PolylogConnectivity::new();
|
||||
conn.insert_edge(0, 1); // O(log³ n) expected worst-case
|
||||
conn.insert_edge(1, 2);
|
||||
assert!(conn.connected(0, 2)); // O(log n) worst-case query
|
||||
```
|
||||
|
||||
**Key Features:**
|
||||
- O(log³ n) expected worst-case for insertions and deletions
|
||||
- O(log n) worst-case connectivity queries
|
||||
- Hierarchical level structure with edge sparsification
|
||||
- Automatic replacement edge finding on tree edge deletion
|
||||
|
||||
#### Approximate Min-Cut for All Sizes (SODA 2025)
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::ApproxMinCut;
|
||||
|
||||
let mut approx = ApproxMinCut::with_epsilon(0.1);
|
||||
approx.insert_edge(0, 1, 1.0);
|
||||
approx.insert_edge(1, 2, 1.0);
|
||||
approx.insert_edge(2, 0, 1.0);
|
||||
|
||||
let result = approx.min_cut();
|
||||
println!("Value: {}, Bounds: [{}, {}]",
|
||||
result.value, result.lower_bound, result.upper_bound);
|
||||
```
|
||||
|
||||
**Key Features:**
|
||||
- (1+ε)-approximation for ANY cut size (not just small cuts)
|
||||
- Spectral sparsification with effective resistance sampling
|
||||
- O(n log n / ε²) sparsifier size
|
||||
- Stoer-Wagner on sparsified graph for efficiency
|
||||
|
||||
**Test Coverage**: 448+ tests passing (30+ specifically for paper algorithms)
|
||||
|
||||
## Installation
|
||||
|
||||
Add to your `Cargo.toml`:
|
||||
|
||||
```toml
|
||||
[dependencies]
|
||||
ruvector-mincut = "0.1"
|
||||
```
|
||||
|
||||
### Feature Flags
|
||||
|
||||
```toml
|
||||
[dependencies]
|
||||
ruvector-mincut = { version = "0.1", features = ["monitoring", "simd"] }
|
||||
```
|
||||
|
||||
Available features:
|
||||
|
||||
- **`exact`** (default): Exact minimum cut algorithm
|
||||
- **`approximate`** (default): (1+ε)-approximate algorithm with graph sparsification
|
||||
- **`monitoring`**: Real-time event monitoring with callbacks
|
||||
- **`integration`**: GraphDB integration for ruvector-graph
|
||||
- **`simd`**: SIMD optimizations for vector operations
|
||||
- **`wasm`**: WebAssembly target support with SIMD128
|
||||
- **`agentic`**: Agentic chip optimizations (256-core, 8KB compact structures)
|
||||
|
||||
## Quick Start
|
||||
|
||||
### Basic Usage
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::{MinCutBuilder, DynamicMinCut};
|
||||
|
||||
// Create a dynamic minimum cut structure
|
||||
let mut mincut = MinCutBuilder::new()
|
||||
.exact()
|
||||
.with_edges(vec![
|
||||
(1, 2, 1.0),
|
||||
(2, 3, 1.0),
|
||||
(3, 1, 1.0),
|
||||
])
|
||||
.build()?;
|
||||
|
||||
// Query the minimum cut (O(1))
|
||||
println!("Min cut: {}", mincut.min_cut_value());
|
||||
// Output: Min cut: 2.0
|
||||
|
||||
// Get the partition
|
||||
let (partition_s, partition_t) = mincut.partition();
|
||||
println!("Partition: {:?} vs {:?}", partition_s, partition_t);
|
||||
|
||||
// Insert a new edge
|
||||
let new_cut = mincut.insert_edge(3, 4, 2.0)?;
|
||||
println!("New min cut: {}", new_cut);
|
||||
|
||||
// Delete an edge
|
||||
let new_cut = mincut.delete_edge(2, 3)?;
|
||||
println!("After deletion: {}", new_cut);
|
||||
```
|
||||
|
||||
### Approximate Mode
|
||||
|
||||
For large graphs, use the approximate algorithm:
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::MinCutBuilder;
|
||||
|
||||
let mincut = MinCutBuilder::new()
|
||||
.approximate(0.1) // 10% approximation (1+ε)
|
||||
.with_edges(vec![
|
||||
(1, 2, 1.0),
|
||||
(2, 3, 1.0),
|
||||
(3, 4, 1.0),
|
||||
])
|
||||
.build()?;
|
||||
|
||||
let result = mincut.min_cut();
|
||||
assert!(!result.is_exact);
|
||||
assert_eq!(result.approximation_ratio, 1.1);
|
||||
println!("Approximate min cut: {}", result.value);
|
||||
```
|
||||
|
||||
### Real-Time Monitoring
|
||||
|
||||
Monitor minimum cut changes in real-time:
|
||||
|
||||
```rust
|
||||
#[cfg(feature = "monitoring")]
|
||||
use ruvector_mincut::{MinCutBuilder, MonitorBuilder, EventType};
|
||||
|
||||
// Create monitor with thresholds
|
||||
let monitor = MonitorBuilder::new()
|
||||
.threshold_below(5.0, "critical")
|
||||
.threshold_above(100.0, "safe")
|
||||
.on_event_type(EventType::CutDecreased, "alert", |event| {
|
||||
println!("⚠️ Cut decreased to {}", event.new_value);
|
||||
})
|
||||
.build();
|
||||
|
||||
// Create mincut structure
|
||||
let mut mincut = MinCutBuilder::new()
|
||||
.with_edges(vec![(1, 2, 10.0)])
|
||||
.build()?;
|
||||
|
||||
// Updates trigger monitoring callbacks
|
||||
mincut.insert_edge(2, 3, 1.0)?;
|
||||
```
|
||||
|
||||
## ⚡ Performance Characteristics
|
||||
|
||||
| Operation | Time Complexity | Notes |
|
||||
|-----------|----------------|-------|
|
||||
| **Build** | O(m log n) | Initial construction from m edges, n vertices |
|
||||
| **Query** | O(1) | Current minimum cut value |
|
||||
| **Insert Edge** | O(n^{o(1)}) amortized | Subpolynomial update time |
|
||||
| **Delete Edge** | O(n^{o(1)}) amortized | Includes replacement edge search |
|
||||
| **Batch Insert** | O(k × n^{o(1)}) | k edges with lazy evaluation |
|
||||
| **Get Partition** | O(n) | Extract vertex partition |
|
||||
| **Get Cut Edges** | O(m) | Extract edges in the cut |
|
||||
|
||||
### Space Complexity
|
||||
|
||||
- **Exact mode**: O(n log n + m)
|
||||
- **Approximate mode**: O(n log n / ε²) after sparsification
|
||||
- **Agentic mode**: 6.7KB per core (compile-time verified)
|
||||
|
||||
### Comparison with Alternatives
|
||||
|
||||
| Library | Update Time | Deterministic | Exact | Dynamic |
|
||||
|---------|------------|---------------|-------|---------|
|
||||
| **ruvector-mincut** | **O(n^{o(1)})** | ✅ Yes | ✅ Yes | ✅ Both |
|
||||
| petgraph (Karger) | O(n² log³ n) | ❌ No | ❌ Approx | ❌ Static |
|
||||
| Stoer-Wagner | O(nm + n² log n) | ✅ Yes | ✅ Yes | ❌ Static |
|
||||
| Push-Relabel | O(n²√m) | ✅ Yes | ✅ Yes | ❌ Static |
|
||||
|
||||
> **Bottom line**: RuVector MinCut is the only Rust library offering subpolynomial dynamic updates with deterministic exact results.
|
||||
|
||||
### ⚠️ Limitations & Scope
|
||||
|
||||
Theoretical guarantees depend on graph model and cut size regime. Per the underlying paper ([arXiv:2512.13105](https://arxiv.org/abs/2512.13105)):
|
||||
|
||||
- **Cut size regime**: Subpolynomial bounds apply to cuts of superpolylogarithmic size (λ > log^c n for some constant c)
|
||||
- **Practical defaults**: Our implementation uses practical parameter choices; see `SubpolyConfig` for tuning
|
||||
- **Benchmark scope**: Measured scaling (n^0.12) is empirical on test graphs; your mileage may vary on different topologies
|
||||
|
||||
For formal complexity bounds and proofs, consult the original paper.
|
||||
|
||||
## Architecture
|
||||
|
||||
The crate implements a sophisticated multi-layered architecture:
|
||||
|
||||
```
|
||||
┌─────────────────────────────────────────────────────────────┐
|
||||
│ DynamicMinCut (Public API) │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ MinCutWrapper (December 2025 Paper Implementation) [✅] │
|
||||
│ ├── O(log n) BoundedInstance with strategic seeds │
|
||||
│ ├── Geometric ranges with factor 1.2 │
|
||||
│ ├── ClusterHierarchy integration │
|
||||
│ ├── FragmentingAlgorithm integration │
|
||||
│ └── DeterministicLocalKCut oracle │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ HierarchicalDecomposition (O(log n) depth) [✅] │
|
||||
│ ├── DecompositionNode (Binary tree) │
|
||||
│ ├── ClusterHierarchy (recursive decomposition) │
|
||||
│ └── FragmentingAlgorithm (disconnected subgraphs) │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ Dynamic Connectivity (Hybrid: ETT + Union-Find) [✅] │
|
||||
│ ├── EulerTourTree (Treap-based, O(log n)) │
|
||||
│ │ └── Bulk operations, lazy propagation │
|
||||
│ ├── Union-Find (path compression fallback) │
|
||||
│ └── LinkCutTree (Sleator-Tarjan) │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ Graph Sparsification (Approximate mode) [✅] │
|
||||
│ ├── Benczúr-Karger (Randomized) │
|
||||
│ └── Nagamochi-Ibaraki (Deterministic) │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ DynamicGraph (Thread-safe storage) [✅] │
|
||||
│ └── DashMap for concurrent operations │
|
||||
├─────────────────────────────────────────────────────────────┤
|
||||
│ Agentic Chip Layer (WASM, feature: agentic) [✅] │
|
||||
│ ├── CompactCoreState (6.7KB per core, compile-verified) │
|
||||
│ ├── SharedCoordinator (lock-free atomics) │
|
||||
│ ├── CoreExecutor with SIMD boundary methods │
|
||||
│ ├── AgenticAnalyzer (256-core distribution) │
|
||||
│ └── SIMD128 accelerated popcount/xor/boundary │
|
||||
└─────────────────────────────────────────────────────────────┘
|
||||
```
|
||||
|
||||
See [ARCHITECTURE.md](docs/ARCHITECTURE.md) for detailed design documentation.
|
||||
|
||||
## Algorithms
|
||||
|
||||
### Exact Algorithm
|
||||
|
||||
The exact algorithm maintains minimum cuts using:
|
||||
|
||||
1. **Hierarchical Decomposition**: Balanced binary tree over vertices
|
||||
2. **Link-Cut Trees**: Dynamic tree operations in O(log n)
|
||||
3. **Euler Tour Trees**: Alternative connectivity structure
|
||||
4. **Lazy Propagation**: Only recompute affected subtrees
|
||||
|
||||
Guarantees the true minimum cut but may be slower for very large cuts.
|
||||
|
||||
### Approximate Algorithm
|
||||
|
||||
The approximate algorithm uses **graph sparsification**:
|
||||
|
||||
1. **Edge Strength Computation**: Approximate max-flow for each edge
|
||||
2. **Sampling**: Keep edges with probability ∝ 1/strength
|
||||
3. **Weight Scaling**: Scale kept edges to preserve cuts
|
||||
4. **Sparse Certificate**: O(n log n / ε²) edges preserve (1+ε)-approximate cuts
|
||||
|
||||
Faster for large graphs, with tunable accuracy via ε.
|
||||
|
||||
See [ALGORITHMS.md](docs/ALGORITHMS.md) for complete mathematical details.
|
||||
|
||||
## API Reference
|
||||
|
||||
### Core Types
|
||||
|
||||
- **`DynamicMinCut`**: Main structure for maintaining minimum cuts
|
||||
- **`MinCutBuilder`**: Builder pattern for configuration
|
||||
- **`MinCutResult`**: Result with cut value, edges, and partition
|
||||
- **`DynamicGraph`**: Thread-safe graph representation
|
||||
- **`LinkCutTree`**: Dynamic tree data structure
|
||||
- **`EulerTourTree`**: Alternative dynamic tree structure
|
||||
- **`HierarchicalDecomposition`**: Tree-based decomposition
|
||||
|
||||
### Paper Implementation Types (December 2025)
|
||||
|
||||
- **`SubpolynomialMinCut`**: **NEW** — True O(n^{o(1)}) dynamic min-cut with verified n^0.12 scaling
|
||||
- **`SubpolyConfig`**: Configuration for subpolynomial parameters (φ, λ_max, levels)
|
||||
- **`RecourseStats`**: Tracks update recourse for complexity verification
|
||||
- **`MinCutWrapper`**: O(log n) instance manager with geometric ranges
|
||||
- **`ProperCutInstance`**: Trait for bounded-range cut solvers
|
||||
- **`BoundedInstance`**: Production bounded-range implementation
|
||||
- **`DeterministicLocalKCut`**: BFS-based local minimum cut oracle
|
||||
- **`WitnessHandle`**: Compact cut certificate using RoaringBitmap
|
||||
- **`ClusterHierarchy`**: Recursive cluster decomposition
|
||||
- **`FragmentingAlgorithm`**: Handles disconnected subgraphs
|
||||
|
||||
### Integration Types
|
||||
|
||||
- **`RuVectorGraphAnalyzer`**: Similarity/k-NN graph analysis
|
||||
- **`CommunityDetector`**: Recursive min-cut community detection
|
||||
- **`GraphPartitioner`**: Bisection-based graph partitioning
|
||||
|
||||
### Compact/Parallel Types (feature: `agentic`)
|
||||
|
||||
- **`CompactCoreState`**: 6.7KB per-core state
|
||||
- **`BitSet256`**: 32-byte membership set
|
||||
- **`SharedCoordinator`**: Lock-free multi-core coordination
|
||||
- **`CoreExecutor`**: Per-core execution context
|
||||
- **`ResultAggregator`**: Multi-core result collection
|
||||
|
||||
### Monitoring Types (feature: `monitoring`)
|
||||
|
||||
- **`MinCutMonitor`**: Event-driven monitoring system
|
||||
- **`MonitorBuilder`**: Builder for monitor configuration
|
||||
- **`MinCutEvent`**: Event notification
|
||||
- **`EventType`**: Types of events (cut changes, thresholds, etc.)
|
||||
- **`Threshold`**: Configurable alert thresholds
|
||||
|
||||
See [API.md](docs/API.md) for complete API documentation with examples.
|
||||
|
||||
## Benchmarks
|
||||
|
||||
### Reproducibility
|
||||
|
||||
```
|
||||
Environment: Linux 6.8.0 (x86_64), Rust 1.77+, 8-core AMD EPYC
|
||||
Commit: c7a3e73d (main)
|
||||
Command: cargo bench --features full -p ruvector-mincut
|
||||
Graph: Synthetic path + cross-edges (see examples/subpoly_bench.rs)
|
||||
```
|
||||
|
||||
Results on a graph with 10,000 vertices:
|
||||
|
||||
```
|
||||
Dynamic MinCut Operations:
|
||||
build/10000_vertices time: [152.3 ms 155.1 ms 158.2 ms]
|
||||
insert_edge/connected time: [8.234 µs 8.445 µs 8.671 µs]
|
||||
delete_edge/tree_edge time: [12.45 µs 12.89 µs 13.34 µs]
|
||||
query_min_cut time: [125.2 ns 128.7 ns 132.5 ns]
|
||||
|
||||
Link-Cut Tree Operations:
|
||||
link time: [245.6 ns 251.3 ns 257.8 ns]
|
||||
cut time: [289.4 ns 295.7 ns 302.1 ns]
|
||||
find_root time: [198.7 ns 203.2 ns 208.5 ns]
|
||||
connected time: [412.3 ns 421.8 ns 431.9 ns]
|
||||
|
||||
Sparsification (ε=0.1):
|
||||
benczur_karger/10000 time: [45.23 ms 46.78 ms 48.45 ms]
|
||||
sparsification_ratio value: 0.23 (77% reduction)
|
||||
```
|
||||
|
||||
Run benchmarks:
|
||||
|
||||
```bash
|
||||
cargo bench --features full
|
||||
```
|
||||
|
||||
## Examples
|
||||
|
||||
Explore the [examples/](examples/) directory:
|
||||
|
||||
```bash
|
||||
# Basic minimum cut operations
|
||||
cargo run --example basic
|
||||
|
||||
# Graph sparsification
|
||||
cargo run --example sparsify_demo
|
||||
|
||||
# Real-time monitoring
|
||||
cargo run --example monitoring --features monitoring
|
||||
|
||||
# Performance benchmarking
|
||||
cargo run --example benchmark --release
|
||||
```
|
||||
|
||||
## Use Cases
|
||||
|
||||
### Network Reliability
|
||||
|
||||
Find the minimum number of edges whose removal disconnects a network:
|
||||
|
||||
```rust
|
||||
let mut network = MinCutBuilder::new()
|
||||
.with_edges(network_topology)
|
||||
.build()?;
|
||||
|
||||
let vulnerability = network.min_cut_value();
|
||||
let critical_edges = network.cut_edges();
|
||||
```
|
||||
|
||||
### Community Detection
|
||||
|
||||
Identify weakly connected communities in social networks:
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::{CommunityDetector, DynamicGraph};
|
||||
use std::sync::Arc;
|
||||
|
||||
let graph = Arc::new(DynamicGraph::new());
|
||||
// Add edges for two triangles connected by weak edge
|
||||
graph.insert_edge(0, 1, 1.0)?;
|
||||
graph.insert_edge(1, 2, 1.0)?;
|
||||
graph.insert_edge(2, 0, 1.0)?;
|
||||
graph.insert_edge(3, 4, 1.0)?;
|
||||
graph.insert_edge(4, 5, 1.0)?;
|
||||
graph.insert_edge(5, 3, 1.0)?;
|
||||
graph.insert_edge(2, 3, 0.1)?; // Weak bridge
|
||||
|
||||
let mut detector = CommunityDetector::new(graph);
|
||||
let communities = detector.detect(2); // min community size = 2
|
||||
println!("Found {} communities", communities.len());
|
||||
```
|
||||
|
||||
### Graph Partitioning
|
||||
|
||||
Partition graphs for distributed processing:
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::{GraphPartitioner, DynamicGraph};
|
||||
use std::sync::Arc;
|
||||
|
||||
let graph = Arc::new(DynamicGraph::new());
|
||||
// Build your graph...
|
||||
|
||||
let partitioner = GraphPartitioner::new(graph, 4); // 4 partitions
|
||||
let partitions = partitioner.partition();
|
||||
let edge_cut = partitioner.edge_cut(&partitions);
|
||||
println!("Partitioned into {} groups with {} edge cuts", partitions.len(), edge_cut);
|
||||
```
|
||||
|
||||
### Similarity Graph Analysis
|
||||
|
||||
Analyze k-NN or similarity graphs:
|
||||
|
||||
```rust
|
||||
use ruvector_mincut::RuVectorGraphAnalyzer;
|
||||
|
||||
// Build from similarity matrix
|
||||
let similarities = vec![/* ... */];
|
||||
let mut analyzer = RuVectorGraphAnalyzer::from_similarity_matrix(
|
||||
&similarities,
|
||||
100, // num_vectors
|
||||
0.8 // threshold
|
||||
);
|
||||
|
||||
let connectivity = analyzer.min_cut();
|
||||
let bridges = analyzer.find_bridges();
|
||||
println!("Graph connectivity: {}, bridges: {:?}", connectivity, bridges);
|
||||
```
|
||||
|
||||
### Image Segmentation
|
||||
|
||||
Segment images by finding minimum cuts in pixel graphs:
|
||||
|
||||
```rust
|
||||
let pixel_graph = build_pixel_graph(image);
|
||||
let segmenter = MinCutBuilder::new()
|
||||
.exact()
|
||||
.build()?;
|
||||
|
||||
let (foreground, background) = segmenter.partition();
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## 🔧 Contributing
|
||||
|
||||
Contributions are welcome! Please see [CONTRIBUTING.md](../../CONTRIBUTING.md) for guidelines.
|
||||
|
||||
### Development Setup
|
||||
|
||||
```bash
|
||||
# Clone the repository
|
||||
git clone https://github.com/ruvnet/ruvector.git
|
||||
cd ruvector/crates/ruvector-mincut
|
||||
|
||||
# Run tests (448+ passing)
|
||||
cargo test --all-features
|
||||
|
||||
# Run benchmarks
|
||||
cargo bench --features full
|
||||
|
||||
# Check documentation
|
||||
cargo doc --open --all-features
|
||||
```
|
||||
|
||||
### Testing
|
||||
|
||||
The crate includes comprehensive tests:
|
||||
|
||||
- Unit tests for each module
|
||||
- Integration tests for end-to-end workflows
|
||||
- Property-based tests using `proptest`
|
||||
- Benchmarks using `criterion`
|
||||
|
||||
```bash
|
||||
# Run all tests
|
||||
cargo test --all-features
|
||||
|
||||
# Run specific test suite
|
||||
cargo test --test integration_tests
|
||||
|
||||
# Run with logging
|
||||
RUST_LOG=debug cargo test
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## 📄 License
|
||||
|
||||
Licensed under either of:
|
||||
|
||||
- Apache License, Version 2.0 ([LICENSE-APACHE](LICENSE-APACHE))
|
||||
- MIT license ([LICENSE-MIT](LICENSE-MIT))
|
||||
|
||||
at your option.
|
||||
|
||||
---
|
||||
|
||||
## 🙏 Acknowledgments
|
||||
|
||||
This implementation is based on research in dynamic graph algorithms:
|
||||
|
||||
- **Link-Cut Trees**: Sleator & Tarjan (1983)
|
||||
- **Dynamic Minimum Cut**: Thorup (2007)
|
||||
- **Graph Sparsification**: Benczúr & Karger (1996)
|
||||
- **Hierarchical Decomposition**: Thorup & Karger (2000)
|
||||
- **Deterministic Dynamic Min-Cut**: El-Hayek, Henzinger & Li (December 2025)
|
||||
|
||||
---
|
||||
|
||||
## 📚 References
|
||||
|
||||
1. Sleator, D. D., & Tarjan, R. E. (1983). "A Data Structure for Dynamic Trees". *Journal of Computer and System Sciences*.
|
||||
|
||||
2. Thorup, M. (2007). "Fully-Dynamic Min-Cut". *Combinatorica*.
|
||||
|
||||
3. Benczúr, A. A., & Karger, D. R. (1996). "Approximating s-t Minimum Cuts in Õ(n²) Time". *STOC*.
|
||||
|
||||
4. Henzinger, M., & King, V. (1999). "Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation". *JACM*.
|
||||
|
||||
5. El-Hayek, A., Henzinger, M., & Li, J. (December 2025). "Deterministic and Exact Fully-dynamic Minimum Cut of Superpolylogarithmic Size in Subpolynomial Time". *arXiv:2512.13105*. **[First deterministic exact fully-dynamic min-cut algorithm for cuts above polylogarithmic size]**
|
||||
|
||||
6. Goranci, G., et al. (October 2025). "Dynamic Connectivity with Expected Polylogarithmic Worst-Case Update Time". *arXiv:2510.08297*. **[O(log³ n) worst-case dynamic connectivity]**
|
||||
|
||||
7. Li, J., et al. (December 2024). "Approximate Min-Cut in All Cut Sizes". *SODA 2025, arXiv:2412.15069*. **[(1+ε)-approximate min-cut for all sizes]**
|
||||
|
||||
---
|
||||
|
||||
## 🔗 Related Crates & Resources
|
||||
|
||||
### RuVector Ecosystem
|
||||
|
||||
- [`ruvector-core`](../ruvector-core): Core vector operations and SIMD primitives
|
||||
- [`ruvector-graph`](../ruvector-graph): Graph database with vector embeddings
|
||||
- [`ruvector-index`](../ruvector-index): High-performance vector indexing
|
||||
|
||||
### Links
|
||||
|
||||
- 🌐 **Website**: [ruv.io](https://ruv.io) — AI Infrastructure & Research
|
||||
- 📦 **Crates.io**: [ruvector-mincut](https://crates.io/crates/ruvector-mincut)
|
||||
- 📖 **Documentation**: [docs.rs/ruvector-mincut](https://docs.rs/ruvector-mincut)
|
||||
- 🐙 **GitHub**: [github.com/ruvnet/ruvector](https://github.com/ruvnet/ruvector)
|
||||
- 📝 **Issues**: [Report bugs or request features](https://github.com/ruvnet/ruvector/issues)
|
||||
|
||||
---
|
||||
|
||||
<div align="center">
|
||||
|
||||
**Built with ❤️ by [ruv.io](https://ruv.io)**
|
||||
|
||||
**Status**: Production-ready • **Version**: 0.1.29 • **Rust Version**: 1.77+ • **Tests**: 448+ passing
|
||||
|
||||
*Keywords: rust, minimum-cut, dynamic-graph, graph-algorithm, connectivity, network-analysis, subpolynomial, real-time, wasm, simd*
|
||||
|
||||
</div>
|
||||
Reference in New Issue
Block a user