# Local K-Cut Paper Implementation

## Overview

This document describes the paper-compliant implementation of the Local K-Cut algorithm from the December 2024 paper "Deterministic and Exact Fully-dynamic Minimum Cut of Superpolylogarithmic Size" (arxiv:2512.13105).

## Location

- **Implementation**: `/home/user/ruvector/crates/ruvector-mincut/src/localkcut/paper_impl.rs`
- **Tests**: Embedded in paper_impl.rs (16 comprehensive tests)
- **Integration Tests**: `/home/user/ruvector/crates/ruvector-mincut/tests/localkcut_paper_integration.rs`

## API Specification

### Core Types

#### LocalKCutQuery
```rust
pub struct LocalKCutQuery {
    /// Seed vertices defining the search region
    pub seed_vertices: Vec<VertexId>,

    /// Maximum acceptable cut value
    pub budget_k: u64,

    /// Maximum search radius (BFS depth)
    pub radius: usize,
}
```

#### LocalKCutResult
```rust
pub enum LocalKCutResult {
    /// Found a cut with value ≤ budget_k
    Found {
        witness: WitnessHandle,
        cut_value: u64,
    },

    /// No cut ≤ budget_k found in the local region
    NoneInLocality,
}
```

#### LocalKCutOracle (Trait)
```rust
pub trait LocalKCutOracle: Send + Sync {
    fn search(&self, graph: &DynamicGraph, query: LocalKCutQuery) -> LocalKCutResult;
}
```

### Implementation

#### DeterministicLocalKCut
```rust
pub struct DeterministicLocalKCut {
    max_radius: usize,
    family_generator: DeterministicFamilyGenerator,
}

impl DeterministicLocalKCut {
    pub fn new(max_radius: usize) -> Self;

    pub fn with_family_generator(
        max_radius: usize,
        family_generator: DeterministicFamilyGenerator,
    ) -> Self;
}

impl LocalKCutOracle for DeterministicLocalKCut {
    fn search(&self, graph: &DynamicGraph, query: LocalKCutQuery) -> LocalKCutResult;
}
```

## Algorithm Details

### Deterministic Properties

1. **No Randomness**: The algorithm is completely deterministic
   - BFS exploration order determined by sorted vertex IDs
   - Seed selection based on deterministic ordering
   - Same input always produces same output

2. **Bounded Range**: Searches for cuts with value ≤ budget_k
   - Early termination when budget exceeded
   - Returns smallest cut found or NoneInLocality

3. **Local Exploration**: BFS-based approach
   - Starts from seed vertices
   - Expands outward layer by layer
   - Tracks boundary size at each layer
   - Returns witness when cut found

### Time Complexity

- **Per Query**: O(radius × (|V| + |E|)) worst case
- **Typical**: Much faster due to early termination
- **Deterministic**: No probabilistic guarantees needed

## Usage Examples

### Basic Usage
```rust
use ruvector_mincut::{
    DynamicGraph, LocalKCutQuery, PaperLocalKCutResult,
    LocalKCutOracle, DeterministicLocalKCut,
};
use std::sync::Arc;

// Create graph
let graph = Arc::new(DynamicGraph::new());
graph.insert_edge(1, 2, 1.0).unwrap();
graph.insert_edge(2, 3, 1.0).unwrap();
graph.insert_edge(3, 4, 1.0).unwrap();

// Create oracle
let oracle = DeterministicLocalKCut::new(5);

// Create query
let query = LocalKCutQuery {
    seed_vertices: vec![1],
    budget_k: 2,
    radius: 3,
};

// Execute search
match oracle.search(&graph, query) {
    PaperLocalKCutResult::Found { cut_value, witness } => {
        println!("Found cut with value: {}", cut_value);
        println!("Witness cardinality: {}", witness.cardinality());
    }
    PaperLocalKCutResult::NoneInLocality => {
        println!("No cut found in locality");
    }
}
```

### With Custom Family Generator
```rust
let generator = DeterministicFamilyGenerator::new(5);
let oracle = DeterministicLocalKCut::with_family_generator(10, generator);

let query = LocalKCutQuery {
    seed_vertices: vec![1, 2, 3],
    budget_k: 5,
    radius: 8,
};

let result = oracle.search(&graph, query);
```

### Accessing Witness Details
```rust
if let PaperLocalKCutResult::Found { witness, cut_value } = result {
    // Check witness properties
    assert_eq!(witness.boundary_size(), cut_value);
    assert!(witness.contains(witness.seed()));

    // Materialize full partition (expensive)
    let (u, v_minus_u) = witness.materialize_partition();
    println!("Cut separates {} from {} vertices", u.len(), v_minus_u.len());
}
```

## Test Coverage

### Unit Tests (16 tests)

1. **API Tests**
   - `test_local_kcut_query_creation` - Query structure creation
   - `test_deterministic_family_generator` - Family generator determinism
   - `test_deterministic_local_kcut_creation` - Oracle instantiation

2. **Algorithm Tests**
   - `test_simple_path_cut` - Basic path graph
   - `test_triangle_no_cut` - Graph with no small cuts
   - `test_dumbbell_bridge_cut` - Finding bridge in dumbbell graph
   - `test_determinism` - Verify deterministic behavior

3. **Edge Case Tests**
   - `test_empty_seeds` - Empty seed list
   - `test_invalid_seed` - Non-existent vertex
   - `test_zero_radius` - No expansion
   - `test_large_radius` - Radius capping

4. **Correctness Tests**
   - `test_boundary_calculation` - Boundary size computation
   - `test_witness_creation` - Witness handle creation
   - `test_multiple_seeds` - Multiple starting points
   - `test_budget_enforcement` - Budget constraints
   - `test_witness_properties` - Witness invariants

### Integration Tests (10 tests)

See `/home/user/ruvector/crates/ruvector-mincut/tests/localkcut_paper_integration.rs`

## Module Structure

```
ruvector-mincut/
├── src/
│   └── localkcut/
│       ├── mod.rs (updated with exports)
│       └── paper_impl.rs (new implementation)
├── tests/
│   └── localkcut_paper_integration.rs (new)
└── docs/
    └── localkcut-paper-implementation.md (this file)
```

## Exports

The paper implementation is exported at the crate root:

```rust
pub use localkcut::{
    // Paper API types
    LocalKCutQuery,
    LocalKCutResult as PaperLocalKCutResult,  // Aliased to avoid conflict
    LocalKCutOracle,

    // Implementation
    DeterministicLocalKCut,
    DeterministicFamilyGenerator,

    // Original implementation (still available)
    LocalKCut,
    LocalCutResult,
    EdgeColor,
    ColorMask,
    ForestPacking,
};
```

## Key Features

### 1. Strict API Compliance
- Matches paper specification exactly
- Clear enum variants for results
- Witness-based certification

### 2. Deterministic Algorithm
- No randomness whatsoever
- Sorted vertex traversal
- Reproducible results

### 3. Efficient Implementation
- Early termination on budget violation
- BFS layer-by-layer expansion
- Minimal memory allocation

### 4. Comprehensive Testing
- 16 unit tests covering all functionality
- 10 integration tests
- Edge case coverage
- Determinism verification

## Differences from Original Implementation

| Feature | Original LocalKCut | Paper Implementation |
|---------|-------------------|---------------------|
| API | `find_cut(v)` returns `Option<LocalCutResult>` | `search(graph, query)` returns `LocalKCutResult` enum |
| Input | Single vertex + k parameter | Query struct with seeds, budget, radius |
| Output | Result struct with various fields | Enum: Found or NoneInLocality |
| Witness | Optional in result | Always included when found |
| Determinism | Uses edge coloring | BFS with sorted traversal |
| Complexity | 4^d enumerations | Single deterministic BFS |

## Performance Characteristics

- **Best Case**: O(radius) when cut found immediately
- **Average Case**: O(radius × avg_degree × |found_vertices|)
- **Worst Case**: O(radius × (|V| + |E|))
- **Memory**: O(|V|) for visited set

## Future Enhancements

1. **Parallel Search**: Multiple seeds in parallel
2. **Incremental Updates**: Cache results across queries
3. **Adaptive Radius**: Auto-tune radius based on graph density
4. **Witness Compression**: Reduce witness storage
5. **Batch Queries**: Process multiple queries efficiently

## References

- Paper: "Deterministic and Exact Fully-dynamic Minimum Cut of Superpolylogarithmic Size"
- ArXiv: https://arxiv.org/abs/2512.13105
- Implementation: `/home/user/ruvector/crates/ruvector-mincut/src/localkcut/paper_impl.rs`

## Verification

All 16 unit tests pass:
```bash
cargo test -p ruvector-mincut --lib localkcut::paper_impl::tests
```

Expected output:
```
running 16 tests
test localkcut::paper_impl::tests::test_boundary_calculation ... ok
test localkcut::paper_impl::tests::test_budget_enforcement ... ok
test localkcut::paper_impl::tests::test_determinism ... ok
test localkcut::paper_impl::tests::test_deterministic_family_generator ... ok
test localkcut::paper_impl::tests::test_deterministic_local_kcut_creation ... ok
test localkcut::paper_impl::tests::test_dumbbell_bridge_cut ... ok
test localkcut::paper_impl::tests::test_empty_seeds ... ok
test localkcut::paper_impl::tests::test_invalid_seed ... ok
test localkcut::paper_impl::tests::test_large_radius ... ok
test localkcut::paper_impl::tests::test_local_kcut_query_creation ... ok
test localkcut::paper_impl::tests::test_multiple_seeds ... ok
test localkcut::paper_impl::tests::test_simple_path_cut ... ok
test localkcut::paper_impl::tests::test_triangle_no_cut ... ok
test localkcut::paper_impl::tests::test_witness_creation ... ok
test localkcut::paper_impl::tests::test_witness_properties ... ok
test localkcut::paper_impl::tests::test_zero_radius ... ok

test result: ok. 16 passed; 0 failed
```