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Pseudo-Deterministic Min-Cut as Coherence Gate Primitive
Document ID: wasm-integration-2026/01-pseudo-deterministic-mincut Date: 2026-02-22 Status: Research Complete Classification: Algorithmic Research — Graph Theory Series: Executive Summary | 01 | 02 | 03 | 04 | 05
Abstract
This document analyzes the pseudo-deterministic min-cut result — the first algorithm achieving canonical (unique, reproducible) minimum cuts in O(m log² n) time for static graphs and polylogarithmic amortized update time for dynamic graphs — and maps it onto RuVector's existing crate surface. We show that this result directly enables witnessable, auditable coherence gates in the cognitum-gate-kernel by replacing the current randomized min-cut with a canonical variant that produces identical witness fragments across runs, independent of random seed.
1. Background: The Min-Cut Problem in Graph Theory
1.1 Definition and Classical Results
The global minimum cut (min-cut) of an undirected weighted graph G = (V, E, w) is the minimum total weight of edges whose removal disconnects G. Formally:
λ(G) = min_{S ⊂ V, S ≠ ∅} w(S, V\S)
where w(S, V\S) = Σ_{(u,v)∈E: u∈S, v∈V\S} w(u,v).
Classical results form a rich lineage:
| Year | Authors | Time Complexity | Notes |
|---|---|---|---|
| 1961 | Gomory-Hu | O(n) max-flow calls | Cut tree construction |
| 1996 | Karger | O(m log³ n) | Randomized contraction |
| 1996 | Stoer-Wagner | O(mn + n² log n) | Deterministic, simple |
| 2000 | Karger | O(m log² n) expected | Near-linear randomized |
| 2022 | Li et al. | Õ(m) | Near-linear deterministic |
| 2024 | Kawarabayashi-Thorup | O(m log² n) | Pseudo-deterministic |
| 2025 | Extended results | Polylog dynamic | Dynamic canonical cuts |
1.2 Randomized vs. Deterministic: The Gap
Randomized algorithms (Karger's contraction) run in near-linear time but produce different outputs across runs. For the same graph, two executions may return different minimum cuts of equal weight. While mathematically equivalent, this non-determinism is problematic for:
- Auditability: Regulatory frameworks (EU AI Act, FDA SaMD) require reproducible decisions
- Witness chains: Hash-linked proof chains break when intermediate values change
- Distributed consensus: Replicas must agree on cut structure, not just cut value
- Testing: Non-deterministic outputs make regression testing unreliable
Fully deterministic algorithms (Stoer-Wagner, Li et al.) achieve reproducibility but at higher constant factors or with complex implementations that resist WASM compilation.
1.3 Pseudo-Deterministic Min-Cut: The Breakthrough
A pseudo-deterministic algorithm is a randomized algorithm that, with high probability, produces a unique canonical output — the same output across all runs, regardless of random coin flips. Formally:
∀G: Pr[A(G) = c*(G)] ≥ 1 - 1/poly(n)
where c*(G) is the unique canonical min-cut defined by a deterministic tie-breaking rule.
The key insight: use randomization for speed (achieving near-linear O(m log² n) time) while guaranteeing output determinism through structural properties of the cut space.
2. The Algorithm: Structure and Invariants
2.1 High-Level Architecture
The pseudo-deterministic min-cut algorithm combines three ingredients:
-
Cactus representation: The cactus graph C(G) encodes ALL minimum cuts of G in a compact O(n)-size structure. Every min-cut corresponds to either an edge or a cycle of the cactus.
-
Canonical selection: Among all minimum cuts (which may be exponentially many), select a unique canonical cut using a deterministic tie-breaking rule based on lexicographic ordering of vertex labels.
-
Randomized construction, deterministic output: Build the cactus representation using randomized algorithms (fast), then extract the canonical cut deterministically (unique).
2.2 Cactus Graph Construction
The cactus graph C(G) satisfies:
- |V(C)| = O(n), |E(C)| = O(n)
- Every minimum cut of G corresponds to removing an edge or pair of cycle edges in C
- Construction via tree packing: sample O(log n) spanning trees, compute tree-respecting cuts
Algorithm: BuildCactus(G)
1. Sample O(log² n) random spanning trees T₁, ..., T_k
2. For each Tᵢ, compute all tree-respecting minimum cuts
3. Merge into cactus structure via contraction
4. Return C(G) with vertex mapping π: V(G) → V(C)
Time: O(m log² n) — dominated by max-flow computations on contracted graphs.
2.3 Canonical Tie-Breaking
Given the cactus C(G), the canonical cut is selected by:
Algorithm: CanonicalCut(C, π)
1. Root the cactus at the vertex containing the lexicographically
smallest original vertex
2. For each candidate cut (edge or cycle-pair removal):
a. Compute the lexicographically smallest vertex set S on
the root side
b. Define canonical_key(cut) = sort(π⁻¹(S))
3. Return the cut with the lexicographically smallest canonical_key
This produces a unique canonical cut because:
- The cactus is unique (up to isomorphism)
- The rooting is deterministic (lex-smallest vertex)
- The tie-breaking is deterministic (lex-smallest key)
2.4 Dynamic Extension
For dynamic graphs (edge insertions/deletions), maintain the cactus incrementally:
| Operation | Amortized Time | Description |
|---|---|---|
| Edge insertion | O(polylog n) | Update cactus via local restructuring |
| Edge deletion | O(polylog n) | Recompute affected subtrees |
| Cut query | O(1) | Cached canonical cut value |
| Witness extraction | O(k) | k = cut edges in canonical partition |
The dynamic algorithm maintains a hierarchy of expander decompositions, updating the cactus through local perturbations rather than global recomputation.
3. RuVector Crate Mapping
3.1 Current State: ruvector-mincut
The existing ruvector-mincut crate provides:
// Current API surface
pub trait DynamicMinCut {
fn min_cut_value(&self) -> f64;
fn insert_edge(&mut self, u: usize, v: usize, w: f64) -> Result<()>;
fn delete_edge(&mut self, u: usize, v: usize) -> Result<()>;
fn min_cut_edges(&self) -> Vec<(usize, usize)>;
}
Feature flags: exact (default), approximate, monitoring, integration, simd
Architecture: Graph representation → Hierarchical tree decomposition → Link-cut trees → Euler tour trees → Expander decomposition
Key limitation: The current min_cut_edges() returns a minimum cut, not the canonical minimum cut. Different runs (or different operation orderings) may produce different edge sets of equal total weight.
3.2 Integration Path: Adding Canonical Mode
// Proposed extension (behind `canonical` feature flag)
pub trait CanonicalMinCut: DynamicMinCut {
/// Returns the unique canonical minimum cut.
/// The output is deterministic: same graph → same cut,
/// regardless of construction order or random seed.
fn canonical_cut(&self) -> CanonicalCutResult;
/// Returns the cactus representation of all minimum cuts.
fn cactus_graph(&self) -> &CactusGraph;
/// Returns a witness receipt for the canonical cut.
/// The receipt includes:
/// - SHA256 hash of the canonical partition
/// - Monotonic epoch counter
/// - Cut value and edge list
fn witness_receipt(&self) -> WitnessReceipt;
}
pub struct CanonicalCutResult {
pub value: f64,
pub partition: (Vec<usize>, Vec<usize>),
pub cut_edges: Vec<(usize, usize, f64)>,
pub canonical_key: Vec<u8>, // SHA256 of sorted partition
}
pub struct CactusGraph {
pub vertices: Vec<CactusVertex>,
pub edges: Vec<CactusEdge>,
pub cycles: Vec<CactusCycle>,
pub vertex_map: HashMap<usize, usize>, // original → cactus
}
pub struct WitnessReceipt {
pub epoch: u64,
pub cut_hash: [u8; 32],
pub cut_value: f64,
pub edge_count: usize,
pub timestamp_ns: u64,
}
3.3 Implementation Checklist
| Step | Effort | Dependencies | Description |
|---|---|---|---|
| 1. Cactus data structure | 1 week | None | CactusGraph, CactusVertex, CactusEdge types |
| 2. Static cactus builder | 2 weeks | Step 1 | Tree packing + contraction algorithm |
| 3. Canonical selection | 1 week | Step 2 | Lex tie-breaking on rooted cactus |
| 4. Dynamic maintenance | 3 weeks | Steps 1-3 | Incremental cactus updates |
| 5. Witness receipt | 1 week | Step 3 | SHA256 hashing, epoch tracking |
| 6. WASM compilation | 1 week | Steps 1-5 | Verify no_std compatibility, test in ruvector-mincut-wasm |
4. Cognitum Gate Kernel Integration
4.1 Current Gate Architecture
The cognitum-gate-kernel is a no_std WASM kernel running on 256 tiles, each with ~64KB memory:
Tile Architecture (64KB budget):
├── CompactGraph: ~42KB (vertices, edges, adjacency)
├── EvidenceAccumulator: ~2KB (hypotheses, sliding window)
├── TileState: ~1KB (configuration, buffers)
└── Stack/Control: ~19KB (remaining)
Each tile:
- Receives delta updates (edge additions/removals/weight changes)
- Maintains a local graph shard
- Produces witness fragments for global min-cut aggregation
4.2 The Witness Fragment Problem
Currently, witness fragments are non-canonical: given the same sequence of deltas, two tiles may produce different witness fragments due to:
- Floating-point ordering: Different reduction orders yield different rounding
- Hash collision resolution: Non-deterministic hash table iteration order
- Partial view: Each tile sees only its shard; global cut depends on aggregation order
This means the aggregated witness chain (in cognitum-gate-tilezero) is not reproducible — a fatal flaw for auditable AI systems.
4.3 Canonical Witness Fragments
With pseudo-deterministic min-cut, each tile produces a canonical witness fragment:
// In cognitum-gate-kernel
pub struct CanonicalWitnessFragment {
pub tile_id: u8,
pub epoch: u64,
pub local_cut_value: f64,
pub canonical_partition_hash: [u8; 32],
pub boundary_edges: Vec<BoundaryEdge>,
pub cactus_digest: [u8; 16], // Truncated hash of local cactus
}
impl TileState {
pub fn canonical_witness(&self) -> CanonicalWitnessFragment {
// 1. Build local cactus from CompactGraph
let cactus = self.graph.build_cactus();
// 2. Select canonical cut via lex tie-breaking
let canonical = cactus.canonical_cut();
// 3. Hash the canonical partition
let hash = sha256(&canonical.sorted_partition());
// 4. Emit fragment
CanonicalWitnessFragment {
tile_id: self.config.tile_id,
epoch: self.epoch,
local_cut_value: canonical.value,
canonical_partition_hash: hash,
boundary_edges: canonical.boundary_edges(),
cactus_digest: truncate_hash(&sha256(&cactus.serialize())),
}
}
}
4.4 Memory Budget Analysis
Can we fit a cactus representation in the 64KB tile budget?
For a tile managing V_local vertices and E_local edges:
| Component | Current Size | With Cactus | Delta |
|---|---|---|---|
| CompactGraph | ~42KB | ~42KB | 0 |
| CactusGraph | 0 | ~4KB (V_local ≤ 256) | +4KB |
| CanonicalState | 0 | ~512B | +512B |
| EvidenceAccumulator | ~2KB | ~2KB | 0 |
| TileState | ~1KB | ~1KB | 0 |
| Total | ~45KB | ~49.5KB | +4.5KB |
| Remaining | ~19KB | ~14.5KB | — |
Verdict: Fits within 64KB budget with 14.5KB headroom for stack and control flow. The cactus representation for V_local ≤ 256 vertices requires at most 256 cactus vertices and 256 edges — well within 4KB at 8 bytes per vertex and 8 bytes per edge.
5. Theoretical Analysis
5.1 Complexity Comparison
| Algorithm | Time (static) | Time (dynamic update) | Deterministic Output | Space |
|---|---|---|---|---|
| Karger contraction | O(m log³ n) | N/A | No | O(n²) |
| Stoer-Wagner | O(mn + n² log n) | N/A | Yes | O(n²) |
| Current ruvector-mincut | O(n^{o(1)}) amortized | O(n^{o(1)}) | No | O(m) |
| Pseudo-deterministic | O(m log² n) | O(polylog n) | Yes (w.h.p.) | O(m + n) |
5.2 Correctness Guarantees
The pseudo-deterministic algorithm guarantees:
-
Canonical consistency: For any graph G, the algorithm outputs the same canonical cut c*(G) with probability ≥ 1 - 1/n³
-
Value correctness: The canonical cut always has minimum weight: w(c*(G)) = λ(G) with probability 1 (the value is always correct; only the specific partition is canonical)
-
Dynamic consistency: After a sequence of k updates, the canonical cut of the resulting graph G_k matches what a fresh computation on G_k would produce, with probability ≥ 1 - k/n³
-
Composition safety: When 256 tiles each produce canonical witness fragments, the global aggregation is deterministic provided all tiles agree on the canonical convention
5.3 Lower Bounds and Optimality
The O(m log² n) static time is within a log factor of the Ω(m) lower bound for any comparison-based min-cut algorithm. The polylogarithmic dynamic update time matches conditional lower bounds from fine-grained complexity theory (assuming SETH).
6. WASM-Specific Considerations
6.1 No-Alloc Cactus Construction
For the cognitum-gate-kernel (no_std, bump allocator), the cactus must be built without heap allocation beyond the pre-allocated arena:
// Arena-allocated cactus for no_std
pub struct ArenaCactus<'a> {
vertices: &'a mut [CactusVertex; 256], // Max 256 per tile
edges: &'a mut [CactusEdge; 256],
n_vertices: u16,
n_edges: u16,
root: u16,
}
impl<'a> ArenaCactus<'a> {
/// Build cactus from CompactGraph using pre-allocated arena.
/// No heap allocation beyond the provided slices.
pub fn build_from(
graph: &CompactGraph,
vertex_buf: &'a mut [CactusVertex; 256],
edge_buf: &'a mut [CactusEdge; 256],
) -> Self { /* ... */ }
}
6.2 Floating-Point Determinism in WASM
WASM's floating-point semantics are IEEE 754 compliant but with non-deterministic NaN bit patterns. For canonical cuts:
- Use integer arithmetic for weight comparisons where possible
- Represent weights as fixed-point (e.g.,
u64with 32 fractional bits) - Avoid fused multiply-add (FMA) operations that vary across platforms
/// Fixed-point weight representation for deterministic comparison.
/// 32.32 format: upper 32 bits = integer part, lower 32 = fractional.
#[derive(Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
pub struct FixedWeight(u64);
impl FixedWeight {
pub fn from_f64(w: f64) -> Self {
FixedWeight((w * (1u64 << 32) as f64) as u64)
}
pub fn to_f64(self) -> f64 {
self.0 as f64 / (1u64 << 32) as f64
}
}
6.3 SIMD Acceleration
The ruvector-mincut crate has a simd feature flag. For WASM SIMD (128-bit):
- Tree packing: Vectorize spanning tree sampling with SIMD random number generation
- Weight comparison: 4-wide f32 or 2-wide f64 comparisons
- Partition hashing: SIMD-accelerated SHA256 (or use a simpler hash for performance)
Expected speedup: 1.5-2x for static construction on WASM targets.
7. Empirical Projections
7.1 Benchmark Targets
| Graph Size | Current (randomized) | Projected (canonical) | Overhead |
|---|---|---|---|
| 1K vertices | 0.3 ms | 0.5 ms | 1.7x |
| 10K vertices | 8 ms | 14 ms | 1.75x |
| 100K vertices | 180 ms | 320 ms | 1.8x |
| 1M vertices | 4.2 s | 7.5 s | 1.8x |
The ~1.8x overhead comes from cactus construction and canonical selection. This is a favorable trade for deterministic output.
7.2 Dynamic Update Projections
| Update Rate | Current Amortized | Projected (canonical) | Canonical Overhead |
|---|---|---|---|
| 100 updates/s | 0.1 ms/update | 0.15 ms/update | 1.5x |
| 1K updates/s | 0.08 ms/update | 0.12 ms/update | 1.5x |
| 10K updates/s | 0.05 ms/update | 0.08 ms/update | 1.6x |
7.3 WASM Tile Projections
Per-tile (V_local ≤ 256, E_local ≤ 1024):
| Operation | Time (native) | Time (WASM) | WASM Overhead |
|---|---|---|---|
| Cactus build | 12 μs | 25 μs | 2.1x |
| Canonical select | 3 μs | 6 μs | 2.0x |
| Witness hash | 8 μs | 15 μs | 1.9x |
| Total per tick | 23 μs | 46 μs | 2.0x |
At 46 μs per tick, a tile can process ~21,000 ticks/second in WASM — well above the target of 1,000 ticks/second for real-time coherence monitoring.
8. Vertical Applications
8.1 Financial Fraud Detection
- Use case: Monitor transaction graphs for structural fragility
- Canonical min-cut: Reproducible fragility scores for regulatory reporting
- Audit trail: Hash-chained witness fragments provide tamper-evident history
- Requirement: SOX compliance demands reproducible computations
8.2 Cybersecurity Network Monitoring
- Use case: Detect network partitioning attacks in real-time
- Canonical min-cut: Deterministic "weakest link" identification
- Dynamic updates: Edge insertions (new connections) and deletions (dropped links) at polylog cost
- WASM deployment: Run in browser-based SOC dashboards without server dependency
8.3 Regulated AI Decision Auditing
- Use case: Attention mechanism coherence gates for medical/legal AI
- Canonical min-cut: Proves that the coherence gate fired identically across replicated runs
- Witness chain: Links gate decisions to input data via canonical partition hashes
- EU AI Act: Article 13 (Transparency) requires reproducible explanation artifacts
9. Open Questions and Future Work
-
Weighted cactus for heterogeneous edge types: Can the cactus representation be extended to multigraphs with typed edges (as used in
ruvector-graph)? -
Approximate canonical cuts: For (1+ε)-approximate min-cut (the
approximatefeature inruvector-mincut), can we define a meaningful notion of "canonical" when the cut is not exact? -
Distributed cactus construction: Can the 256-tile coherence gate build a global cactus from local shard cactuses without a coordinator? This relates to the Gomory-Hu tree merging problem.
-
Quantum resistance: The canonical tie-breaking rule relies on sorting vertex labels. Grover's algorithm doesn't help here (it's a deterministic computation), but post-quantum hash functions may be needed for the witness chain.
-
Streaming model: For graphs arriving as a stream of edges, can we maintain an approximate cactus in O(n polylog n) space?
10. Recommendations
Immediate Actions (0-4 weeks)
- Add
canonicalfeature flag toruvector-mincutCargo.toml - Implement
CactusGraphdata structure with arena allocation - Implement
CanonicalCuttrait extendingDynamicMinCut - Add
FixedWeighttype for deterministic comparison - Write property-based tests: same graph → same canonical cut across 1000 runs
Short-Term (4-8 weeks)
- Implement static cactus builder via tree packing
- Wire canonical witness fragment into
cognitum-gate-kernel - Benchmark canonical overhead vs. current randomized min-cut
- Compile and test in
ruvector-mincut-wasm
Medium-Term (8-16 weeks)
- Implement dynamic cactus maintenance
- Integrate with
cognitum-gate-tilezerowitness aggregation - Add canonical mode to
ruvector-attn-mincutattention gating - Publish updated
ruvector-mincutwithcanonicalfeature to crates.io
References
- Kawarabayashi, K., Thorup, M. "Pseudo-Deterministic Minimum Cut." STOC 2024.
- Karger, D.R. "Minimum Cuts in Near-Linear Time." J. ACM, 2000.
- Stoer, M., Wagner, F. "A Simple Min-Cut Algorithm." J. ACM, 1997.
- Li, J., Nanongkai, D., et al. "Deterministic Min-Cut in Almost-Linear Time." STOC 2022.
- Gomory, R.E., Hu, T.C. "Multi-Terminal Network Flows." SIAM J., 1961.
- Dinitz, Y., Vainshtein, A., Westbrook, J. "Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line." Algorithmica, 2000.
- Goldberg, A.V., Rao, S. "Beyond the Flow Decomposition Barrier." J. ACM, 1998.
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