# ADR-002 Addendum: SOTA Optimizations for Dynamic Hierarchical j-Tree **Status**: Proposed **Date**: 2026-01-25 **Extends**: ADR-002 (Dynamic Hierarchical j-Tree Decomposition) --- ## Executive Summary This addendum pushes ADR-002 to true state-of-the-art by integrating: 1. **Predictive Dynamics** - SNN predicts updates before they happen 2. **Neural Sparsification** - Learned edge selection via SpecNet 3. **Lazy Hierarchical Evaluation** - Demand-paged j-tree levels 4. **Warm-Start Cut-Matching** - Reuse computation across updates 5. **256-Core Parallel Hierarchy** - Each core owns j-tree levels 6. **Streaming Sketch Fallback** - O(n log n) space for massive graphs **Target**: Sub-microsecond approximate queries, <100μs exact verification --- ## Architecture: Predictive Dynamic j-Tree ``` ┌─────────────────────────────────────────────────────────────────────────────────┐ │ PREDICTIVE DYNAMIC J-TREE ARCHITECTURE │ ├─────────────────────────────────────────────────────────────────────────────────┤ │ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 0: PREDICTION ENGINE ││ │ │ ││ │ │ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ ││ │ │ │ SNN Policy │───►│ TD Learner │───►│ Prefetcher │ ││ │ │ │ (R-STDP) │ │ (Value Net) │ │ (Speculate) │ ││ │ │ └──────────────┘ └──────────────┘ └──────────────┘ ││ │ │ │ │ │ ││ │ │ ▼ ▼ ▼ ││ │ │ Predict which Estimate cut Pre-compute ││ │ │ levels change value change likely queries ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ │ │ ▼ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 1: NEURAL SPARSIFIER ││ │ │ ││ │ │ ┌────────────────────────────────────────────────────────────────────┐ ││ │ │ │ SpecNet Integration (arXiv:2510.27474) │ ││ │ │ │ │ ││ │ │ │ Loss = λ₁·Laplacian_Alignment + λ₂·Feature_Preserve + λ₃·Sparsity │ ││ │ │ │ │ ││ │ │ │ • Joint Graph Evolution layer │ ││ │ │ │ • Spectral Concordance preservation │ ││ │ │ │ • Degree-based fast presparse (DSpar: 5.9x speedup) │ ││ │ │ └────────────────────────────────────────────────────────────────────┘ ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ │ │ ▼ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 2: LAZY HIERARCHICAL J-TREE ││ │ │ ││ │ │ Level L ──┐ ││ │ │ Level L-1 ├── Demand-paged: Only materialize when queried ││ │ │ Level L-2 ├── Dirty marking: Track which levels need recomputation ││ │ │ ... │ Warm-start: Reuse cut-matching state across updates ││ │ │ Level 0 ──┘ ││ │ │ ││ │ │ Memory: O(active_levels × n_level) instead of O(L × n) ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ │ │ ▼ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 3: 256-CORE PARALLEL DISTRIBUTION ││ │ │ ││ │ │ ┌─────────┬─────────┬─────────┬─────────┬─────────┬─────────┐ ││ │ │ │Core 0-31│Core32-63│Core64-95│Core96-127│Core128+ │Core 255│ ││ │ │ │ Level 0 │ Level 1 │ Level 2 │ Level 3 │ ... │ Level L│ ││ │ │ └─────────┴─────────┴─────────┴─────────┴─────────┴─────────┘ ││ │ │ ││ │ │ Work Stealing: Imbalanced levels redistribute to idle cores ││ │ │ Atomic CAS: SharedCoordinator for global min-cut updates ││ │ │ 8KB/core: CompactCoreState fits entire j-tree level ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ │ │ ▼ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 4: STREAMING SKETCH FALLBACK ││ │ │ ││ │ │ When n > 100K vertices: ││ │ │ ┌────────────────────────────────────────────────────────────────────┐ ││ │ │ │ Semi-Streaming Cut Sketch │ ││ │ │ │ • O(n log n) space (two edges per vertex) │ ││ │ │ │ • Reservoir sampling for edge selection │ ││ │ │ │ • (1+ε) approximation maintained incrementally │ ││ │ │ └────────────────────────────────────────────────────────────────────┘ ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ │ │ ▼ │ │ ┌─────────────────────────────────────────────────────────────────────────────┐│ │ │ LAYER 5: EXACT VERIFICATION ││ │ │ ││ │ │ El-Hayek/Henzinger/Li (arXiv:2512.13105) ││ │ │ • Triggered only when approximate cut < threshold ││ │ │ • O(n^{o(1)}) exact verification ││ │ │ • Deterministic, no randomization ││ │ │ ││ │ └─────────────────────────────────────────────────────────────────────────────┘│ │ │ └─────────────────────────────────────────────────────────────────────────────────┘ ``` --- ## Component 1: SNN Prediction Engine Exploits the triple isomorphism already in the codebase: | Graph Theory | Dynamical Systems | Neuromorphic | |--------------|-------------------|--------------| | MinCut value | Lyapunov exponent | Spike synchrony | | Edge contraction | Phase space flow | Synaptic plasticity | | Hierarchy level | Attractor basin | Memory consolidation | ```rust /// Predictive j-tree using SNN dynamics pub struct PredictiveJTree { /// Core j-tree hierarchy hierarchy: JTreeHierarchy, /// SNN policy network for update prediction policy: PolicySNN, /// Value network for cut estimation value_net: ValueNetwork, /// Prefetch cache for speculative computation prefetch: PrefetchCache, /// SONA hooks for continuous adaptation sona_hooks: [usize; 4], // Layers 8, 16, 24, 28 } impl PredictiveJTree { /// Predict which levels will need updates after edge change pub fn predict_affected_levels(&self, edge: (VertexId, VertexId)) -> Vec { // SNN encodes edge as spike pattern let spike_input = self.edge_to_spikes(edge); // Policy network predicts affected regions let activity = self.policy.forward(&spike_input); // Low activity regions are stable, high activity needs update activity.iter() .enumerate() .filter(|(_, &a)| a > ACTIVITY_THRESHOLD) .map(|(level, _)| level) .collect() } /// Speculative update: pre-compute before edge actually changes pub fn speculative_update(&mut self, likely_edge: (VertexId, VertexId), prob: f64) { if prob > SPECULATION_THRESHOLD { let affected = self.predict_affected_levels(likely_edge); // Pre-compute in background cores for level in affected { self.prefetch.schedule(level, likely_edge); } } } /// TD-learning update after observing actual cut change pub fn learn_from_observation(&mut self, predicted_cut: f64, actual_cut: f64) { let td_error = actual_cut - predicted_cut; // R-STDP: Reward-modulated spike-timing-dependent plasticity self.policy.apply_rstdp(td_error); // Update value network self.value_net.td_update(td_error); } } ``` **Performance Target**: Predict 80%+ of affected levels correctly → skip 80% of unnecessary recomputation --- ## Component 2: Neural Sparsifier (SpecNet Integration) Based on arXiv:2510.27474, learn which edges to keep: ```rust /// Neural graph sparsifier with spectral concordance pub struct NeuralSparsifier { /// Graph evolution layer (learned edge selection) evolution_layer: GraphEvolutionLayer, /// Spectral concordance loss weights lambda_laplacian: f64, // λ₁ = 1.0 lambda_feature: f64, // λ₂ = 0.5 lambda_sparsity: f64, // λ₃ = 0.1 /// Degree-based presparse threshold (DSpar optimization) degree_threshold: f64, } impl NeuralSparsifier { /// Fast presparse using degree heuristic (DSpar: 5.9x speedup) pub fn degree_presparse(&self, graph: &DynamicGraph) -> DynamicGraph { let mut sparse = graph.clone(); // Effective resistance ≈ 1/(deg_u × deg_v) // Keep edges with high effective resistance for edge in graph.edges() { let deg_u = graph.degree(edge.source) as f64; let deg_v = graph.degree(edge.target) as f64; let eff_resistance = 1.0 / (deg_u * deg_v); // Sample with probability proportional to effective resistance if eff_resistance < self.degree_threshold { sparse.remove_edge(edge.source, edge.target); } } sparse } /// Spectral concordance loss for training pub fn spectral_concordance_loss( &self, original: &DynamicGraph, sparsified: &DynamicGraph, ) -> f64 { // L₁: Laplacian eigenvalue alignment let laplacian_loss = self.laplacian_alignment(original, sparsified); // L₂: Feature geometry preservation (cut values) let feature_loss = self.cut_preservation_loss(original, sparsified); // L₃: Sparsity inducing trace penalty let sparsity_loss = sparsified.edge_count() as f64 / original.edge_count() as f64; self.lambda_laplacian * laplacian_loss + self.lambda_feature * feature_loss + self.lambda_sparsity * sparsity_loss } /// End-to-end learnable sparsification pub fn learn_sparsify(&mut self, graph: &DynamicGraph) -> SparseGraph { // 1. Fast presparse (DSpar) let presparse = self.degree_presparse(graph); // 2. Neural refinement (SpecNet) let edge_scores = self.evolution_layer.forward(&presparse); // 3. Top-k selection preserving spectral properties let k = (graph.vertex_count() as f64 * (graph.vertex_count() as f64).ln()) as usize; let selected = edge_scores.top_k(k); SparseGraph::from_edges(selected) } } ``` **Performance Target**: 90% edge reduction while maintaining 95%+ cut accuracy --- ## Component 3: Lazy Hierarchical Evaluation Don't compute levels until needed: ```rust /// Lazy j-tree with demand-paged levels pub struct LazyJTreeHierarchy { /// Level states levels: Vec, /// Which levels are materialized materialized: BitSet, /// Dirty flags for incremental update dirty: BitSet, /// Cut-matching state for warm-start warm_state: Vec, } #[derive(Clone)] enum LazyLevel { /// Not yet computed Unmaterialized, /// Computed and valid Materialized(JTree), /// Needs recomputation Dirty(JTree), } impl LazyJTreeHierarchy { /// Query with lazy materialization pub fn approximate_min_cut(&mut self) -> ApproximateCut { // Only materialize levels needed for query let mut current_level = self.levels.len() - 1; while current_level > 0 { self.ensure_materialized(current_level); let cut = self.levels[current_level].as_materialized().min_cut(); // Early termination if cut is good enough if cut.approximation_factor < ACCEPTABLE_APPROX { return cut; } current_level -= 1; } self.levels[0].as_materialized().min_cut() } /// Ensure level is materialized (demand-paging) fn ensure_materialized(&mut self, level: usize) { match &self.levels[level] { LazyLevel::Unmaterialized => { // First-time computation let jtree = self.compute_level(level); self.levels[level] = LazyLevel::Materialized(jtree); self.materialized.insert(level); } LazyLevel::Dirty(old_jtree) => { // Warm-start from previous state (arXiv:2511.02943) let jtree = self.warm_start_recompute(level, old_jtree); self.levels[level] = LazyLevel::Materialized(jtree); self.dirty.remove(level); } LazyLevel::Materialized(_) => { // Already valid, no-op } } } /// Warm-start recomputation avoiding full recursion cost fn warm_start_recompute(&self, level: usize, old: &JTree) -> JTree { // Reuse cut-matching game state from warm_state let state = &self.warm_state[level]; // Only recompute affected regions let mut new_jtree = old.clone(); for node in state.affected_nodes() { new_jtree.recompute_node(node, state); } new_jtree } /// Mark levels dirty after edge update pub fn mark_dirty(&mut self, affected_levels: &[usize]) { for &level in affected_levels { if self.materialized.contains(level) { if let LazyLevel::Materialized(jtree) = &self.levels[level] { self.levels[level] = LazyLevel::Dirty(jtree.clone()); self.dirty.insert(level); } } } } } ``` **Performance Target**: 70% reduction in level computations for typical query patterns --- ## Component 4: 256-Core Parallel Distribution Leverage the existing agentic chip architecture: ```rust /// Parallel j-tree across 256 cores pub struct ParallelJTree { /// Core assignments: which cores handle which levels level_assignments: Vec, /// Shared coordinator for atomic updates coordinator: SharedCoordinator, /// Per-core executors executors: [CoreExecutor; 256], } struct CoreRange { start_core: u8, end_core: u8, level: usize, } impl ParallelJTree { /// Distribute L levels across 256 cores pub fn distribute_levels(num_levels: usize) -> Vec { let cores_per_level = 256 / num_levels; (0..num_levels) .map(|level| { let start = (level * cores_per_level) as u8; let end = ((level + 1) * cores_per_level - 1) as u8; CoreRange { start_core: start, end_core: end, level } }) .collect() } /// Parallel update across all affected levels pub fn parallel_update(&mut self, edge: (VertexId, VertexId)) { // Phase 1: Distribute update to affected cores self.coordinator.phase.store(SharedCoordinator::PHASE_DISTRIBUTE, Ordering::Release); for assignment in &self.level_assignments { for core_id in assignment.start_core..=assignment.end_core { self.executors[core_id as usize].queue_update(edge); } } // Phase 2: Parallel compute self.coordinator.phase.store(SharedCoordinator::PHASE_COMPUTE, Ordering::Release); // Each core processes independently // Work stealing if some cores finish early while !self.coordinator.all_completed() { // Idle cores steal from busy cores self.work_stealing_pass(); } // Phase 3: Collect results self.coordinator.phase.store(SharedCoordinator::PHASE_COLLECT, Ordering::Release); let global_min = self.coordinator.global_min_cut.load(Ordering::Acquire); } /// Work stealing for load balancing fn work_stealing_pass(&mut self) { for core_id in 0..256u8 { if self.executors[core_id as usize].is_idle() { // Find busy core to steal from if let Some(victim) = self.find_busy_core() { let work = self.executors[victim].steal_work(); self.executors[core_id as usize].accept_work(work); } } } } } ``` **Performance Target**: Near-linear speedup up to 256× for independent level updates --- ## Component 5: Streaming Sketch Fallback For graphs with n > 100K vertices: ```rust /// Semi-streaming cut sketch for massive graphs pub struct StreamingCutSketch { /// Two edges per vertex (reservoir sampling) sampled_edges: HashMap; 2]>, /// Total vertices seen vertex_count: usize, /// Reservoir sampling state reservoir: ReservoirSampler, } impl StreamingCutSketch { /// Process edge in streaming fashion: O(1) per edge pub fn process_edge(&mut self, edge: Edge) { // Update reservoir for source vertex self.reservoir.sample(edge.source, edge); // Update reservoir for target vertex self.reservoir.sample(edge.target, edge); } /// Approximate min-cut from sketch: O(n) query pub fn approximate_min_cut(&self) -> ApproximateCut { // Build sparse graph from sampled edges let sparse = self.build_sparse_graph(); // Run exact algorithm on sparse graph // O(n log n) edges → tractable let cut = exact_min_cut(&sparse); ApproximateCut { value: cut.value, approximation_factor: 1.0 + self.epsilon(), partition: cut.partition, } } /// Memory usage: O(n log n) pub fn memory_bytes(&self) -> usize { self.vertex_count * 2 * std::mem::size_of::() } } /// Adaptive system that switches between full j-tree and streaming pub struct AdaptiveJTree { full_jtree: Option, streaming_sketch: Option, threshold: usize, // Switch point (default: 100K vertices) } impl AdaptiveJTree { pub fn new(graph: &DynamicGraph) -> Self { if graph.vertex_count() > 100_000 { Self { full_jtree: None, streaming_sketch: Some(StreamingCutSketch::from_graph(graph)), threshold: 100_000, } } else { Self { full_jtree: Some(LazyJTreeHierarchy::build(graph)), streaming_sketch: None, threshold: 100_000, } } } } ``` **Performance Target**: Handle 1M+ vertex graphs in <1GB memory --- ## Performance Comparison | Metric | ADR-002 Baseline | SOTA Optimized | Improvement | |--------|------------------|----------------|-------------| | **Update Time** | O(n^ε) | O(n^ε) / 256 cores | ~100× | | **Query Time (approx)** | O(log n) | O(1) cached | ~10× | | **Query Time (exact)** | O(n^{o(1)}) | O(n^{o(1)}) lazy | ~5× | | **Memory** | O(n log n) | O(active × n) | ~3× | | **Prediction Accuracy** | N/A | 80%+ | New | | **Edge Reduction** | 1 - ε | 90% neural | ~9× | | **Max Graph Size** | ~100K | 1M+ streaming | ~10× | --- ## Integration with Existing Codebase ### SNN Integration Points ```rust // Use existing SNN components from src/snn/ use crate::snn::{ PolicySNN, // For prediction engine ValueNetwork, // For TD learning NeuralGraphOptimizer, // For neural sparsification compute_synchrony, // For stability detection compute_energy, // For attractor dynamics }; // Connect j-tree to SNN energy landscape impl PredictiveJTree { pub fn snn_energy(&self) -> f64 { let mincut = self.hierarchy.approximate_min_cut().value; let synchrony = compute_synchrony(&self.policy.recent_spikes(), 10.0); compute_energy(mincut, synchrony) } } ``` ### Parallel Architecture Integration ```rust // Use existing parallel components from src/parallel/ use crate::parallel::{ SharedCoordinator, // Atomic coordination CoreExecutor, // Per-core execution CoreDistributor, // Work distribution ResultAggregator, // Result collection NUM_CORES, // 256 cores }; // Extend CoreExecutor for j-tree levels impl CoreExecutor { pub fn process_jtree_level(&mut self, level: &JTree) -> CoreResult { // Process assigned level within 8KB memory budget self.state.process_compact_jtree(level) } } ``` ### SONA Integration ```rust // Connect to SONA hooks for continuous adaptation const SONA_HOOKS: [usize; 4] = [8, 16, 24, 28]; impl PredictiveJTree { pub fn enable_sona(&mut self) { for &hook in &SONA_HOOKS { self.policy.enable_hook(hook); } // Adaptation latency: <0.05ms per hook } } ``` --- ## Implementation Priority | Phase | Component | Effort | Impact | Dependencies | |-------|-----------|--------|--------|--------------| | **P0** | Degree-based presparse | 1 week | High | None | | **P0** | 256-core distribution | 2 weeks | High | parallel/mod.rs | | **P1** | Lazy hierarchy | 2 weeks | High | ADR-002 base | | **P1** | Warm-start cut-matching | 2 weeks | High | Lazy hierarchy | | **P2** | SNN prediction | 3 weeks | Medium | snn/optimizer.rs | | **P2** | Neural sparsifier | 3 weeks | Medium | SNN prediction | | **P3** | Streaming fallback | 2 weeks | Medium | None | | **P3** | SONA integration | 1 week | Medium | SNN prediction | --- ## References ### New Research (2024-2026) 1. **SpecNet**: "Spectral Neural Graph Sparsification" (arXiv:2510.27474) 2. **DSpar**: "Degree-based Sparsification" (OpenReview) 3. **Warm-Start**: "Faster Weak Expander Decomposition" (arXiv:2511.02943) 4. **Parallel Expander**: "Near-Optimal Parallel Expander Decomposition" (SODA 2025) 5. **Semi-Streaming**: "Semi-Streaming Min-Cut" (Dudeja et al.) ### Existing Codebase - `src/snn/mod.rs` - SNN integration (triple isomorphism) - `src/snn/optimizer.rs` - PolicySNN, ValueNetwork, R-STDP - `src/parallel/mod.rs` - 256-core architecture - `src/compact/mod.rs` - 8KB per-core state --- ## Appendix: Complexity Summary | Operation | Baseline | + Prediction | + Neural | + Parallel | + Streaming | |-----------|----------|--------------|----------|------------|-------------| | Insert Edge | O(n^ε) | O(n^ε) × 0.2 | O(n^ε) × 0.1 | O(n^ε / 256) | O(1) | | Delete Edge | O(n^ε) | O(n^ε) × 0.2 | O(n^ε) × 0.1 | O(n^ε / 256) | O(1) | | Approx Query | O(log n) | O(1) cached | O(1) | O(1) | O(n) | | Exact Query | O(n^{o(1)}) | O(n^{o(1)}) × 0.2 | - | - | - | | Memory | O(n log n) | O(n log n) | O(n log n / 10) | O(n log n) | O(n log n) | **Combined**: Average case approaches O(1) for queries, O(n^ε / 256) for updates, with graceful degradation to streaming for massive graphs.