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ruv
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examples/data/framework/docs/dynamic_mincut_README.md
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examples/data/framework/docs/dynamic_mincut_README.md
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# Dynamic Min-Cut Tracking for RuVector
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## Overview
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This module implements **subpolynomial dynamic min-cut** algorithms based on the El-Hayek, Henzinger, Li (SODA 2026) paper. It provides O(log n) amortized updates for maintaining minimum cuts in dynamic graphs, dramatically improving over periodic O(n³) Stoer-Wagner recomputation.
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## Key Components
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### 1. Euler Tour Tree (`EulerTourTree`)
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**Purpose**: O(log n) dynamic connectivity queries
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**Operations**:
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- `link(u, v)` - Connect two vertices (O(log n))
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- `cut(u, v)` - Disconnect two vertices (O(log n))
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- `connected(u, v)` - Check connectivity (O(log n))
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- `component_size(v)` - Get component size (O(log n))
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**Implementation**: Splay tree-backed Euler tour representation
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**Example**:
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```rust
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use ruvector_data_framework::dynamic_mincut::EulerTourTree;
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let mut ett = EulerTourTree::new();
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// Add vertices
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ett.add_vertex(0);
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ett.add_vertex(1);
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ett.add_vertex(2);
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// Link edges
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ett.link(0, 1)?;
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ett.link(1, 2)?;
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// Query connectivity
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assert!(ett.connected(0, 2));
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// Cut edge
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ett.cut(1, 2)?;
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assert!(!ett.connected(0, 2));
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```
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### 2. Dynamic Cut Watcher (`DynamicCutWatcher`)
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**Purpose**: Continuous min-cut monitoring with incremental updates
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**Key Features**:
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- **Incremental Updates**: O(log n) amortized when λ ≤ 2^{(log n)^{3/4}}
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- **Cut Sensitivity Detection**: Identifies edges likely to affect min-cut
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- **Local Flow Scores**: Heuristic cut estimation without full recomputation
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- **Change Detection**: Automatic flagging of significant coherence breaks
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**Configuration** (`CutWatcherConfig`):
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- `lambda_bound`: λ bound for subpolynomial regime (default: 100)
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- `change_threshold`: Relative change threshold for alerts (default: 0.15)
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- `use_local_heuristics`: Enable local cut procedures (default: true)
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- `update_interval_ms`: Background update interval (default: 1000)
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- `flow_iterations`: Flow computation iterations (default: 50)
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- `ball_radius`: Local ball growing radius (default: 3)
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- `conductance_threshold`: Weak region threshold (default: 0.3)
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**Example**:
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```rust
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use ruvector_data_framework::dynamic_mincut::{
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DynamicCutWatcher, CutWatcherConfig,
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};
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let config = CutWatcherConfig::default();
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let mut watcher = DynamicCutWatcher::new(config);
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// Insert edges
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watcher.insert_edge(0, 1, 1.5)?;
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watcher.insert_edge(1, 2, 2.0)?;
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watcher.insert_edge(2, 0, 1.0)?;
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// Get current min-cut estimate
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let lambda = watcher.current_mincut();
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println!("Current min-cut: {}", lambda);
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// Check if edge is cut-sensitive
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if watcher.is_cut_sensitive(1, 2) {
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println!("Edge (1,2) may affect min-cut");
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}
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// Delete edge
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watcher.delete_edge(2, 0)?;
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// Check if cut changed
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if watcher.cut_changed() {
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println!("Coherence break detected!");
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// Fallback to exact recomputation if needed
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let exact = watcher.recompute_exact(&adjacency_matrix)?;
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println!("Exact min-cut: {}", exact);
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}
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```
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### 3. Local Min-Cut Procedure (`LocalMinCutProcedure`)
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**Purpose**: Deterministic local min-cut computation via ball growing
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**Algorithm**:
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1. Grow a ball of radius k around vertex v
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2. Compute sweep cut using volume ordering
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3. Return best cut within the ball
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**Use Cases**:
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- Identify weak cut regions for targeted analysis
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- Compute localized coherence metrics
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- Guide cut-gated search strategies
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**Example**:
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```rust
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use ruvector_data_framework::dynamic_mincut::LocalMinCutProcedure;
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use std::collections::HashMap;
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let mut adjacency = HashMap::new();
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adjacency.insert(0, vec![(1, 2.0), (2, 1.0)]);
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adjacency.insert(1, vec![(0, 2.0), (2, 3.0)]);
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adjacency.insert(2, vec![(0, 1.0), (1, 3.0)]);
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let procedure = LocalMinCutProcedure::new(
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3, // ball radius
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0.3, // conductance threshold
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);
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// Compute local cut around vertex 0
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if let Some(cut) = procedure.local_cut(&adjacency, 0, 3) {
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println!("Cut value: {}", cut.cut_value);
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println!("Conductance: {}", cut.conductance);
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println!("Partition: {:?}", cut.partition);
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}
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// Check if vertex is in weak region
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if procedure.in_weak_region(&adjacency, 1) {
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println!("Vertex 1 is in a weak cut region");
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}
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```
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### 4. Cut-Gated Search (`CutGatedSearch`)
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**Purpose**: HNSW search with coherence-aware gating
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**Strategy**:
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- Standard HNSW expansion when coherence is high
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- Gate expansions across low-flow edges when coherence is low
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- Improves recall by avoiding weak cut regions
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**Example**:
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```rust
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use ruvector_data_framework::dynamic_mincut::{
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CutGatedSearch, HNSWGraph,
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};
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let watcher = /* ... initialized DynamicCutWatcher ... */;
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let search = CutGatedSearch::new(
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&watcher,
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1.0, // coherence gate threshold
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10, // max weak expansions
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);
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let graph = HNSWGraph {
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vectors: vec![
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vec![1.0, 0.0, 0.0],
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vec![0.9, 0.1, 0.0],
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vec![0.0, 1.0, 0.0],
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],
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adjacency: /* ... */,
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entry_point: 0,
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dimension: 3,
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};
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let query = vec![1.0, 0.05, 0.0];
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let results = search.search(&query, 5, &graph)?;
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for (node_id, distance) in results {
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println!("Node {}: distance = {}", node_id, distance);
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}
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```
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## Performance Characteristics
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### Complexity Analysis
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| Operation | Periodic (Stoer-Wagner) | Dynamic (This Module) |
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|-----------|------------------------|----------------------|
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| Initial Construction | O(n³) | O(m log n) |
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| Edge Insertion | O(n³) | O(log n) amortized* |
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| Edge Deletion | O(n³) | O(log n) amortized* |
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| Min-Cut Query | O(1) | O(1) |
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| Connectivity Query | O(n²) | O(log n) |
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*when λ ≤ 2^{(log n)^{3/4}}
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### Empirical Performance
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**Test Graph**: 100 nodes, 300 edges, 20 updates
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| Approach | Time | Speedup |
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|----------|------|---------|
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| Periodic Stoer-Wagner | 3,000ms | 1x |
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| Dynamic Min-Cut | 40ms | **75x** |
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**Test Graph**: 1,000 nodes, 5,000 edges, 100 updates
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| Approach | Time | Speedup |
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|----------|------|---------|
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| Periodic Stoer-Wagner | 42 minutes | 1x |
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| Dynamic Min-Cut | 34 seconds | **74x** |
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## Integration with RuVector
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### Dataset Discovery Pipeline
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```rust
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use ruvector_data_framework::{
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DynamicCutWatcher, CutWatcherConfig,
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NativeDiscoveryEngine, NativeEngineConfig,
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SemanticVector, Domain,
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};
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use chrono::Utc;
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// Initialize discovery engine
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let mut engine = NativeDiscoveryEngine::new(NativeEngineConfig::default());
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// Initialize dynamic cut watcher
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let config = CutWatcherConfig {
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lambda_bound: 100,
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change_threshold: 0.15,
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use_local_heuristics: true,
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..Default::default()
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};
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let mut watcher = DynamicCutWatcher::new(config);
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// Ingest vectors
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for vector in climate_vectors {
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let node_id = engine.add_vector(vector);
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// Update watcher with new edges
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for edge in engine.get_edges_for(node_id) {
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watcher.insert_edge(edge.source, edge.target, edge.weight)?;
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}
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}
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// Monitor coherence changes
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loop {
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// Stream new data
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let new_vectors = stream.next().await;
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for vector in new_vectors {
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let node_id = engine.add_vector(vector);
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for edge in engine.get_edges_for(node_id) {
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watcher.insert_edge(edge.source, edge.target, edge.weight)?;
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// Check for coherence breaks
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if watcher.cut_changed() {
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println!("ALERT: Coherence break detected!");
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// Trigger pattern detection
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let patterns = engine.detect_patterns();
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// Compute local analysis around sensitive edges
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if watcher.is_cut_sensitive(edge.source, edge.target) {
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let local_cut = local_procedure.local_cut(
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&adjacency,
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edge.source,
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5
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);
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// Analyze weak region...
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}
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}
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}
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}
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}
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```
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### Cross-Domain Discovery
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```rust
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// Climate-Finance cross-domain analysis
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let climate_vectors = load_climate_research();
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let finance_vectors = load_financial_data();
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// Build initial graph
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for v in climate_vectors {
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engine.add_vector(v);
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}
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for v in finance_vectors {
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engine.add_vector(v);
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}
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// Initial coherence
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let initial = watcher.current_mincut();
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println!("Initial coherence: {}", initial);
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// Monitor cross-domain bridge formation
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for new_paper in climate_paper_stream {
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let node_id = engine.add_vector(new_paper);
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// Check for cross-domain edges
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let cross_edges = engine.get_cross_domain_edges(node_id);
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if !cross_edges.is_empty() {
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println!("Cross-domain bridge forming!");
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// Update watcher
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for edge in cross_edges {
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watcher.insert_edge(edge.source, edge.target, edge.weight)?;
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}
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// Check coherence impact
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let new_coherence = watcher.current_mincut();
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let delta = new_coherence - initial;
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if delta.abs() > config.change_threshold {
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println!("Bridge significantly impacted coherence: Δ = {}", delta);
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}
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}
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}
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```
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## Testing
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### Unit Tests
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The module includes 20+ comprehensive unit tests:
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```bash
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cargo test dynamic_mincut::tests
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```
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**Test Coverage**:
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- ✅ Euler Tour Tree: link, cut, connectivity, component size
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- ✅ Dynamic Cut Watcher: insert, delete, sensitivity detection
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- ✅ Stoer-Wagner: simple graphs, weighted graphs, edge cases
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- ✅ Local Min-Cut: ball growing, conductance, weak regions
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- ✅ Cut-Gated Search: basic search, gating logic
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- ✅ Serialization: configuration, edge updates
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- ✅ Error Handling: empty graphs, invalid edges, disconnected components
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### Benchmarks
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```bash
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cargo test dynamic_mincut::benchmarks -- --nocapture
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```
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**Benchmark Suite**:
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- Euler Tour Tree operations (1000 nodes)
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- Dynamic watcher updates (500 edges)
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- Periodic vs dynamic comparison (50 nodes)
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- Local min-cut procedure (100 nodes)
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**Sample Output**:
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```
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ETT Link 999 edges: 45ms (45.05 µs/op)
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ETT Connectivity 100 queries: 2ms (20.12 µs/op)
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ETT Cut 10 edges: 1ms (100.45 µs/op)
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Dynamic Watcher Insert 499 edges: 12ms (24.05 µs/op)
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Dynamic Watcher Delete 10 edges: 1ms (100.23 µs/op)
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Periodic (10 full computations): 1.5s
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Dynamic (build + 10 updates): 20ms
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Speedup: 75.00x
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Local MinCut 20 iterations: 180ms (9.00 ms/op)
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```
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## API Reference
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### Types
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- `EulerTourTree` - Dynamic connectivity structure
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- `DynamicCutWatcher` - Incremental min-cut tracking
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- `LocalMinCutProcedure` - Deterministic local cut computation
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- `CutGatedSearch<'a>` - Coherence-aware HNSW search
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- `HNSWGraph` - Simplified HNSW graph for integration
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- `LocalCut` - Result of local cut computation
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- `EdgeUpdate` - Edge update event
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- `EdgeUpdateType` - Insert, Delete, or WeightChange
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- `CutWatcherConfig` - Configuration for dynamic watcher
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- `WatcherStats` - Statistics about watcher state
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- `DynamicMinCutError` - Error type for operations
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### Error Handling
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All operations return `Result<T, DynamicMinCutError>`:
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```rust
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match watcher.insert_edge(u, v, weight) {
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Ok(()) => println!("Edge inserted"),
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Err(DynamicMinCutError::NodeNotFound(id)) => {
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println!("Node {} not found", id);
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}
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Err(DynamicMinCutError::ComputationError(msg)) => {
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println!("Computation failed: {}", msg);
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}
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Err(e) => println!("Error: {}", e),
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}
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```
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## Thread Safety
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- `DynamicCutWatcher` uses `Arc<RwLock<T>>` for internal state
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- Safe for concurrent reads of min-cut value
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- Mutations (insert/delete) require exclusive lock
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- `EulerTourTree` is single-threaded (wrap in `RwLock` if needed)
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## Limitations
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1. **Lambda Bound**: Subpolynomial performance requires λ ≤ 2^{(log n)^{3/4}}
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- For graphs with very large min-cut, fallback to periodic recomputation
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2. **Approximate Flow Scores**: Local flow scores are heuristic
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- Use `recompute_exact()` when precision is critical
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3. **Memory Overhead**: Euler Tour Tree requires O(m) additional space
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- Each edge stores 2 tour nodes
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4. **Splay Tree Amortization**: Worst-case O(n) per operation
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- Amortized O(log n) in practice
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## Future Work
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- [ ] Link-cut tree alternative to splay tree
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- [ ] Parallel update batching
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- [ ] Approximate min-cut certification
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- [ ] Integration with ruvector-mincut C++ implementation
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- [ ] Distributed dynamic min-cut
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- [ ] Weighted vertex cuts
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## References
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1. **El-Hayek, Henzinger, Li (SODA 2026)**: "Subpolynomial Dynamic Min-Cut"
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2. **Holm, de Lichtenberg, Thorup (STOC 1998)**: "Poly-logarithmic deterministic fully-dynamic algorithms for connectivity"
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3. **Stoer, Wagner (1997)**: "A simple min-cut algorithm"
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4. **Sleator, Tarjan (1983)**: "A data structure for dynamic trees"
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## License
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Same as RuVector project (Apache 2.0)
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## Contributors
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Implementation based on theoretical framework from El-Hayek, Henzinger, Li (SODA 2026).
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