890 lines
25 KiB
Markdown
890 lines
25 KiB
Markdown
# Algorithm Implementation Plan - Sublinear Time Solver
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## Overview
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This document outlines the implementation strategy for three complementary algorithmic techniques in the sublinear-time solver: Neumann Series Expansion, Forward & Backward Push Methods, and Hybrid Random-Walk Estimation.
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## 1. Neumann Series Implementation
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### 1.1 Core Algorithm Structure
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```pseudocode
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function neumannSeries(M: Matrix, b: Vector, tolerance: f64) -> Vector:
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// Pre-process: Ensure ||M|| < 1 for convergence
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scaling_factor = 1.0 / spectralRadius(M)
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M_scaled = M * scaling_factor
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b_scaled = b * scaling_factor
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result = b_scaled.clone()
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power_term = b_scaled.clone()
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residual_norm = inf
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iteration = 0
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while residual_norm > tolerance and iteration < MAX_ITERATIONS:
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power_term = M_scaled * power_term
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result += power_term
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// Compute residual: ||b - (I - M)x||
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residual = b_scaled - (power_term - M_scaled * result)
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residual_norm = residual.l2_norm()
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iteration += 1
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// Early termination check
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if power_term.l2_norm() < tolerance * 1e-3:
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break
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return result
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```
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### 1.2 Matrix Scaling Strategies
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**Spectral Radius Estimation:**
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- Power iteration method for dominant eigenvalue
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- Gershgorin circle theorem for bounds
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- Adaptive scaling based on matrix structure
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**Implementation Details:**
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```rust
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pub struct ScalingStrategy {
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method: ScalingMethod,
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max_iterations: usize,
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tolerance: f64,
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}
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enum ScalingMethod {
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SpectralRadius,
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GershgorinBounds,
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FrobeniusNorm,
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Adaptive,
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}
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impl ScalingStrategy {
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pub fn compute_scaling_factor(&self, matrix: &SparseMatrix) -> f64 {
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match self.method {
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ScalingMethod::SpectralRadius => self.power_iteration(matrix),
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ScalingMethod::GershgorinBounds => self.gershgorin_estimate(matrix),
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ScalingMethod::FrobeniusNorm => 1.0 / matrix.frobenius_norm(),
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ScalingMethod::Adaptive => self.adaptive_scaling(matrix),
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}
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}
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}
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```
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### 1.3 Series Truncation Logic
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**Convergence Criteria:**
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1. **Residual-based:** `||r_k|| < tolerance`
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2. **Term magnitude:** `||M^k b|| < epsilon * ||b||`
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3. **Relative improvement:** `||x_{k+1} - x_k|| / ||x_k|| < delta`
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**Adaptive Truncation:**
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```pseudocode
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function adaptiveTruncation(term_sequence: Iterator<Vector>) -> usize:
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terms = []
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for (i, term) in term_sequence.enumerate():
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terms.push(term)
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if i >= 3: // Need minimum terms for analysis
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// Check for geometric decay
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ratio = terms[i].norm() / terms[i-1].norm()
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if ratio > 0.95: // Poor convergence
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return i
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// Richardson extrapolation for acceleration
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if i % 3 == 0:
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extrapolated = richardsonExtrapolation(terms.last_n(3))
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if extrapolated.converged():
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return i
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return MAX_TERMS
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```
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### 1.4 Vectorized Operations Design
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**SIMD Optimization:**
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```rust
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use std::simd::{f64x4, Simd};
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pub fn vectorized_axpy(alpha: f64, x: &[f64], y: &mut [f64]) {
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let alpha_vec = Simd::splat(alpha);
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for (x_chunk, y_chunk) in x.chunks_exact(4).zip(y.chunks_exact_mut(4)) {
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let x_vec = f64x4::from_slice(x_chunk);
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let y_vec = f64x4::from_slice(y_chunk);
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let result = alpha_vec * x_vec + y_vec;
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y_chunk.copy_from_slice(result.as_array());
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}
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}
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```
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### 1.5 Error Bound Calculations
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**A Posteriori Error Estimates:**
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```pseudocode
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function errorBounds(x_approx: Vector, M: Matrix, b: Vector) -> ErrorBounds:
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residual = b - (I - M) * x_approx
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residual_norm = residual.l2_norm()
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// Condition number estimate
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condition_estimate = estimateConditionNumber(I - M)
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error_bound = ErrorBounds {
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absolute: residual_norm / (1 - spectralRadius(M)),
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relative: residual_norm / (condition_estimate * b.l2_norm()),
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backward: residual_norm / b.l2_norm(),
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}
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return error_bound
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```
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## 2. Push Methods (Forward/Backward)
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### 2.1 Graph Representation for Push
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**Sparse Matrix Structure:**
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```rust
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pub struct PushGraph {
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adjacency: CompressedSparseRow<f64>,
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reverse_adjacency: CompressedSparseRow<f64>, // For backward push
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degrees: Vec<f64>,
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reverse_degrees: Vec<f64>,
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}
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impl PushGraph {
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pub fn from_matrix(matrix: &SparseMatrix) -> Self {
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let adjacency = matrix.to_csr();
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let reverse_adjacency = adjacency.transpose();
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Self {
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degrees: adjacency.row_sums(),
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reverse_degrees: reverse_adjacency.row_sums(),
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adjacency,
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reverse_adjacency,
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}
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}
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}
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```
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### 2.2 Residual Vector Management
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**Forward Push Algorithm:**
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```pseudocode
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function forwardPush(graph: PushGraph, source: NodeId, alpha: f64, epsilon: f64) -> (Vector, Vector):
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n = graph.num_nodes()
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estimate = Vector::zeros(n)
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residual = Vector::zeros(n)
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residual[source] = 1.0
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work_queue = PriorityQueue::new()
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work_queue.push(source, residual[source])
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while let Some((node, _)) = work_queue.pop():
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if residual[node] < epsilon * graph.degrees[node]:
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continue
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push_amount = alpha * residual[node]
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estimate[node] += push_amount
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residual[node] -= push_amount
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remaining = (1.0 - alpha) * residual[node]
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residual[node] = 0.0
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// Distribute to neighbors
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for (neighbor, weight) in graph.adjacency.row(node):
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delta = remaining * weight / graph.degrees[node]
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residual[neighbor] += delta
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if residual[neighbor] >= epsilon * graph.degrees[neighbor]:
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work_queue.push(neighbor, residual[neighbor])
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return (estimate, residual)
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```
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### 2.3 Work Queue Optimization
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**Priority-Based Processing:**
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```rust
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pub struct WorkQueue {
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heap: BinaryHeap<WorkItem>,
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in_queue: BitSet,
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threshold: f64,
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}
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#[derive(PartialEq, PartialOrd)]
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struct WorkItem {
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priority: OrderedFloat<f64>,
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node_id: usize,
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}
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impl WorkQueue {
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pub fn push_if_threshold(&mut self, node: usize, residual: f64, degree: f64) {
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let priority = residual / degree;
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if priority >= self.threshold && !self.in_queue.contains(node) {
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self.heap.push(WorkItem {
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priority: OrderedFloat(priority),
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node_id: node,
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});
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self.in_queue.insert(node);
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}
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}
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pub fn adaptive_threshold(&mut self, queue_size: usize) {
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// Increase threshold if queue too large
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if queue_size > MAX_QUEUE_SIZE {
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self.threshold *= 1.1;
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} else if queue_size < MIN_QUEUE_SIZE {
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self.threshold *= 0.9;
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}
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}
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}
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```
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### 2.4 Single-Index vs Full-Solution Modes
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**Mode Selection Strategy:**
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```rust
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pub enum PushMode {
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SingleSource { source: usize, target: Option<usize> },
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MultiSource { sources: Vec<usize> },
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FullSolution,
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}
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impl PushSolver {
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pub fn solve(&self, mode: PushMode) -> SolutionResult {
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match mode {
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PushMode::SingleSource { source, target } => {
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let (estimate, residual) = self.forward_push(source);
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if let Some(t) = target {
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SolutionResult::SingleValue(estimate[t])
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} else {
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SolutionResult::SparseVector(estimate)
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}
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},
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PushMode::FullSolution => {
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self.solve_all_sources()
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},
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PushMode::MultiSource { sources } => {
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self.solve_multiple_sources(&sources)
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}
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}
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}
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}
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```
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### 2.5 Visited Node Tracking
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**Efficient Set Operations:**
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```rust
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pub struct VisitedTracker {
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visited: BitSet,
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visit_order: Vec<usize>,
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timestamps: Vec<u32>,
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current_time: u32,
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}
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impl VisitedTracker {
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pub fn mark_visited(&mut self, node: usize) -> bool {
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if !self.visited.contains(node) {
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self.visited.insert(node);
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self.visit_order.push(node);
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self.timestamps[node] = self.current_time;
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true
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} else {
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false
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}
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}
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pub fn reset_for_new_query(&mut self) {
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self.current_time += 1;
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if self.current_time == u32::MAX {
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self.full_reset();
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}
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}
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}
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```
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## 3. Hybrid Random-Walk
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### 3.1 Random Walk Simulation Engine
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**Core Random Walk Implementation:**
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```pseudocode
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function randomWalk(graph: Graph, start: NodeId, max_steps: usize, restart_prob: f64) -> WalkResult:
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current = start
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steps = 0
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path = [start]
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rng = RandomGenerator::new()
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while steps < max_steps:
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if rng.uniform() < restart_prob:
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return WalkResult::Restart(steps, path)
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neighbors = graph.neighbors(current)
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if neighbors.is_empty():
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return WalkResult::Sink(steps, path)
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// Weighted random selection
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weights = graph.edge_weights(current)
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next_node = weightedRandomChoice(neighbors, weights, rng)
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current = next_node
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path.push(current)
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steps += 1
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return WalkResult::MaxSteps(steps, path)
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```
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### 3.2 Sampling Strategies
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**Adaptive Sampling:**
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```rust
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pub struct AdaptiveSampler {
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base_samples: usize,
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variance_threshold: f64,
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max_samples: usize,
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confidence_level: f64,
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}
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impl AdaptiveSampler {
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pub fn sample_until_converged<F>(&self, mut sampler: F) -> SamplingResult
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where
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F: FnMut() -> f64,
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{
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let mut samples = Vec::with_capacity(self.base_samples);
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let mut sum = 0.0;
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let mut sum_squares = 0.0;
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// Initial batch
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for _ in 0..self.base_samples {
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let sample = sampler();
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samples.push(sample);
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sum += sample;
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sum_squares += sample * sample;
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}
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loop {
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let n = samples.len() as f64;
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let mean = sum / n;
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let variance = (sum_squares - sum * sum / n) / (n - 1.0);
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let std_error = (variance / n).sqrt();
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// Check convergence using confidence interval
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let margin = self.confidence_level * std_error;
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if margin / mean.abs() < self.variance_threshold || samples.len() >= self.max_samples {
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break;
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}
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// Add more samples
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let batch_size = (samples.len() / 4).max(10);
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for _ in 0..batch_size {
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let sample = sampler();
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samples.push(sample);
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sum += sample;
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sum_squares += sample * sample;
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}
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}
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SamplingResult {
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estimate: sum / samples.len() as f64,
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variance: sum_squares / samples.len() as f64 - (sum / samples.len() as f64).powi(2),
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num_samples: samples.len(),
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}
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}
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}
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```
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### 3.3 Push-Walk Coordination
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**Hybrid Strategy:**
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```pseudocode
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function hybridSolver(graph: Graph, source: NodeId, target: NodeId, epsilon: f64) -> f64:
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// Phase 1: Forward push to reduce problem size
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(push_estimate, residual) = forwardPush(graph, source, alpha=0.2, epsilon)
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// Phase 2: Random walks from high-residual nodes
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high_residual_nodes = residual.nonzero_indices_above(epsilon)
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walk_contribution = 0.0
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for node in high_residual_nodes:
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// Estimate transition probability from node to target
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prob_estimate = estimateTransitionProbability(graph, node, target, residual[node])
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walk_contribution += prob_estimate
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// Phase 3: Backward push from target (if beneficial)
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if shouldUseBackwardPush(graph, target, high_residual_nodes):
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(backward_estimate, _) = backwardPush(graph, target, alpha=0.2, epsilon)
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// Combine estimates using residual weights
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combined = combineBidirectionalEstimates(push_estimate, walk_contribution, backward_estimate, residual)
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return combined
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return push_estimate[target] + walk_contribution
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```
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### 3.4 Bidirectional Exploration
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**Meet-in-the-Middle Strategy:**
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```rust
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pub struct BidirectionalWalker {
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forward_frontier: HashMap<usize, f64>,
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backward_frontier: HashMap<usize, f64>,
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meeting_probability: f64,
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}
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impl BidirectionalWalker {
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pub fn explore(&mut self, graph: &Graph, source: usize, target: usize) -> f64 {
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let forward_steps = self.forward_walk(graph, source);
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let backward_steps = self.backward_walk(graph, target);
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// Find intersection points
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let mut total_probability = 0.0;
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for (&node, &forward_prob) in &self.forward_frontier {
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if let Some(&backward_prob) = self.backward_frontier.get(&node) {
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total_probability += forward_prob * backward_prob;
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}
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}
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total_probability
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}
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fn should_meet(&self, forward_depth: usize, backward_depth: usize) -> bool {
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// Use graph diameter estimate to decide when frontiers should meet
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forward_depth + backward_depth >= self.estimated_diameter()
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}
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}
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```
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### 3.5 Stochastic Error Estimates
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**Confidence Intervals:**
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```pseudocode
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function computeConfidenceInterval(samples: Vec<f64>, confidence: f64) -> (f64, f64):
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n = samples.len()
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mean = samples.mean()
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variance = samples.variance()
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std_error = sqrt(variance / n)
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// Use t-distribution for small samples, normal for large
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if n < 30:
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t_value = tDistributionQuantile(confidence, n - 1)
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margin = t_value * std_error
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else:
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z_value = normalQuantile(confidence)
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margin = z_value * std_error
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return (mean - margin, mean + margin)
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```
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## 4. Algorithm Selection Logic
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### 4.1 Condition Number Estimation
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**Fast Condition Number Bounds:**
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```rust
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pub fn estimate_condition_number(matrix: &SparseMatrix) -> f64 {
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// Use power iteration for largest eigenvalue
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let lambda_max = power_iteration(matrix, 50);
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// Use inverse power iteration for smallest eigenvalue
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let lambda_min = inverse_power_iteration(matrix, 50);
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lambda_max / lambda_min.abs()
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}
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pub fn power_iteration(matrix: &SparseMatrix, max_iter: usize) -> f64 {
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let n = matrix.nrows();
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let mut v = vec![1.0 / (n as f64).sqrt(); n];
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let mut lambda = 0.0;
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for _ in 0..max_iter {
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let w = matrix * &v;
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lambda = v.dot(&w);
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let norm = w.l2_norm();
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v = w / norm;
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}
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lambda
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}
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```
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### 4.2 Method Auto-Selection Heuristics
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**Decision Tree:**
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```rust
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pub enum SolverMethod {
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NeumannSeries,
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ForwardPush,
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BackwardPush,
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HybridRandomWalk,
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DirectSolver,
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}
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pub struct MethodSelector {
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matrix_analyzer: MatrixAnalyzer,
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performance_history: PerformanceTracker,
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}
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impl MethodSelector {
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pub fn select_method(&self, problem: &LinearSystemProblem) -> SolverMethod {
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let analysis = self.matrix_analyzer.analyze(&problem.matrix);
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// Decision criteria
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if analysis.condition_number < 10.0 {
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return SolverMethod::DirectSolver;
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}
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if analysis.sparsity > 0.99 && problem.query_type == QueryType::SingleEntry {
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return SolverMethod::ForwardPush;
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}
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if analysis.spectral_radius < 0.5 {
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return SolverMethod::NeumannSeries;
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}
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if problem.precision_requirement < 1e-6 {
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return SolverMethod::HybridRandomWalk;
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}
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// Default to adaptive hybrid
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SolverMethod::HybridRandomWalk
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}
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}
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```
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### 4.3 Adaptive Switching During Solve
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**Dynamic Method Switching:**
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```pseudocode
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function adaptiveSolve(problem: LinearSystemProblem) -> Solution:
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current_method = selectInitialMethod(problem)
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solution_state = SolutionState::new()
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while not solution_state.converged():
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progress = executeMethod(current_method, problem, solution_state)
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|
|
if progress.stagnated():
|
|
// Switch to different method
|
|
candidates = alternativeMethods(current_method, problem)
|
|
current_method = selectBestCandidate(candidates, solution_state)
|
|
|
|
if progress.error_increased():
|
|
// Fallback to more stable method
|
|
current_method = conservativeFallback(problem)
|
|
|
|
updateSolutionState(solution_state, progress)
|
|
|
|
return solution_state.extract_solution()
|
|
```
|
|
|
|
### 4.4 Performance Profiling Hooks
|
|
|
|
**Profiling Framework:**
|
|
```rust
|
|
pub struct PerformanceProfiler {
|
|
timers: HashMap<String, Instant>,
|
|
counters: HashMap<String, u64>,
|
|
memory_tracker: MemoryTracker,
|
|
}
|
|
|
|
impl PerformanceProfiler {
|
|
pub fn profile<F, R>(&mut self, name: &str, f: F) -> R
|
|
where
|
|
F: FnOnce() -> R,
|
|
{
|
|
let start = Instant::now();
|
|
let start_memory = self.memory_tracker.current_usage();
|
|
|
|
let result = f();
|
|
|
|
let duration = start.elapsed();
|
|
let memory_delta = self.memory_tracker.current_usage() - start_memory;
|
|
|
|
self.record_timing(name, duration);
|
|
self.record_memory(name, memory_delta);
|
|
|
|
result
|
|
}
|
|
}
|
|
```
|
|
|
|
## 5. Numerical Stability
|
|
|
|
### 5.1 Precision Management (f32/f64)
|
|
|
|
**Mixed Precision Strategy:**
|
|
```rust
|
|
pub enum PrecisionMode {
|
|
Single, // f32
|
|
Double, // f64
|
|
Mixed, // f32 for bulk operations, f64 for critical computations
|
|
Adaptive, // Switch based on condition number
|
|
}
|
|
|
|
pub struct PrecisionManager {
|
|
mode: PrecisionMode,
|
|
promotion_threshold: f64,
|
|
demotion_threshold: f64,
|
|
}
|
|
|
|
impl PrecisionManager {
|
|
pub fn should_promote(&self, condition_number: f64) -> bool {
|
|
condition_number > self.promotion_threshold
|
|
}
|
|
|
|
pub fn execute_with_precision<F, R>(&self, cond_num: f64, f: F) -> R
|
|
where
|
|
F: FnOnce(PrecisionLevel) -> R,
|
|
{
|
|
let precision = if self.should_promote(cond_num) {
|
|
PrecisionLevel::Double
|
|
} else {
|
|
PrecisionLevel::Single
|
|
};
|
|
|
|
f(precision)
|
|
}
|
|
}
|
|
```
|
|
|
|
### 5.2 Overflow/Underflow Handling
|
|
|
|
**Safe Arithmetic Operations:**
|
|
```rust
|
|
pub trait SafeArithmetic {
|
|
fn safe_add(&self, other: &Self) -> Result<Self, ArithmeticError>
|
|
where
|
|
Self: Sized;
|
|
|
|
fn safe_multiply(&self, other: &Self) -> Result<Self, ArithmeticError>
|
|
where
|
|
Self: Sized;
|
|
}
|
|
|
|
impl SafeArithmetic for f64 {
|
|
fn safe_add(&self, other: &f64) -> Result<f64, ArithmeticError> {
|
|
let result = self + other;
|
|
if result.is_infinite() {
|
|
Err(ArithmeticError::Overflow)
|
|
} else if result == 0.0 && (*self != 0.0 || *other != 0.0) {
|
|
Err(ArithmeticError::Underflow)
|
|
} else {
|
|
Ok(result)
|
|
}
|
|
}
|
|
}
|
|
```
|
|
|
|
### 5.3 Ill-Conditioned System Detection
|
|
|
|
**Early Warning System:**
|
|
```pseudocode
|
|
function detectIllConditioning(matrix: Matrix) -> ConditioningReport:
|
|
// Quick checks
|
|
diagonal_dominance = checkDiagonalDominance(matrix)
|
|
if diagonal_dominance < 0.1:
|
|
return ConditioningReport::PoorlyConditioned
|
|
|
|
// Eigenvalue spread estimation
|
|
eigenvalue_ratio = estimateEigenvalueRatio(matrix)
|
|
if eigenvalue_ratio > 1e12:
|
|
return ConditioningReport::IllConditioned
|
|
|
|
// Numerical rank estimation
|
|
rank_deficiency = estimateRankDeficiency(matrix)
|
|
if rank_deficiency > 0:
|
|
return ConditioningReport::RankDeficient
|
|
|
|
return ConditioningReport::WellConditioned
|
|
```
|
|
|
|
### 5.4 Residual Computation Strategies
|
|
|
|
**High-Precision Residual:**
|
|
```rust
|
|
pub fn compute_residual_extended_precision(
|
|
matrix: &SparseMatrix<f64>,
|
|
solution: &Vector<f64>,
|
|
rhs: &Vector<f64>
|
|
) -> Vector<f64> {
|
|
// Use extended precision for critical computation
|
|
let matrix_ext: SparseMatrix<f128> = matrix.cast();
|
|
let solution_ext: Vector<f128> = solution.cast();
|
|
let rhs_ext: Vector<f128> = rhs.cast();
|
|
|
|
let residual_ext = &rhs_ext - &matrix_ext * &solution_ext;
|
|
|
|
// Cast back to working precision
|
|
residual_ext.cast()
|
|
}
|
|
```
|
|
|
|
## 6. Incremental Updates
|
|
|
|
### 6.1 Delta Cost Propagation
|
|
|
|
**Incremental Matrix Updates:**
|
|
```pseudocode
|
|
function incrementalUpdate(solver_state: SolverState, delta_matrix: SparseMatrix) -> SolverState:
|
|
// Sherman-Morrison-Woodbury formula for low-rank updates
|
|
if delta_matrix.rank() <= MAX_RANK_UPDATE:
|
|
return shermanMorrisonUpdate(solver_state, delta_matrix)
|
|
|
|
// Incremental push updates for localized changes
|
|
affected_nodes = delta_matrix.nonzero_pattern()
|
|
if affected_nodes.len() < solver_state.num_nodes() * 0.1:
|
|
return localizedPushUpdate(solver_state, delta_matrix, affected_nodes)
|
|
|
|
// Full recomputation for major changes
|
|
return fullRecompute(solver_state.matrix + delta_matrix)
|
|
```
|
|
|
|
### 6.2 Partial Recomputation Logic
|
|
|
|
**Smart Invalidation:**
|
|
```rust
|
|
pub struct IncrementalSolver {
|
|
cached_solutions: HashMap<QueryKey, CachedSolution>,
|
|
dependency_graph: DependencyGraph,
|
|
invalidation_frontier: BitSet,
|
|
}
|
|
|
|
impl IncrementalSolver {
|
|
pub fn update_matrix(&mut self, changes: &MatrixDelta) {
|
|
let affected_queries = self.dependency_graph.find_dependent_queries(changes);
|
|
|
|
for query in affected_queries {
|
|
self.invalidate_cached_solution(query);
|
|
self.invalidation_frontier.insert(query.node_id);
|
|
}
|
|
|
|
// Propagate invalidation using push-based approach
|
|
self.propagate_invalidation();
|
|
}
|
|
|
|
fn propagate_invalidation(&mut self) {
|
|
while let Some(node) = self.invalidation_frontier.pop_first() {
|
|
let dependents = self.dependency_graph.dependents(node);
|
|
for dependent in dependents {
|
|
if self.should_invalidate(dependent, node) {
|
|
self.invalidation_frontier.insert(dependent);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
```
|
|
|
|
### 6.3 State Caching Mechanisms
|
|
|
|
**Multi-Level Cache:**
|
|
```rust
|
|
pub struct SolutionCache {
|
|
l1_cache: LruCache<QueryKey, Vector<f64>>, // Recent exact solutions
|
|
l2_cache: LruCache<QueryKey, ApproximateSolution>, // Approximate solutions
|
|
l3_cache: PersistentCache<QueryKey, CompressedSolution>, // Compressed historical data
|
|
}
|
|
|
|
impl SolutionCache {
|
|
pub fn get_cached_solution(&self, query: &QueryKey) -> Option<CachedSolution> {
|
|
// Check L1 first
|
|
if let Some(exact) = self.l1_cache.get(query) {
|
|
return Some(CachedSolution::Exact(exact.clone()));
|
|
}
|
|
|
|
// Check L2 for approximation
|
|
if let Some(approx) = self.l2_cache.get(query) {
|
|
if approx.meets_tolerance(query.tolerance) {
|
|
return Some(CachedSolution::Approximate(approx.clone()));
|
|
}
|
|
}
|
|
|
|
// Check L3 for warm start
|
|
if let Some(compressed) = self.l3_cache.get(query) {
|
|
let warm_start = compressed.decompress();
|
|
return Some(CachedSolution::WarmStart(warm_start));
|
|
}
|
|
|
|
None
|
|
}
|
|
}
|
|
```
|
|
|
|
### 6.4 Update Verification
|
|
|
|
**Correctness Checking:**
|
|
```pseudocode
|
|
function verifyIncrementalUpdate(
|
|
original_solution: Vector,
|
|
updated_solution: Vector,
|
|
matrix_delta: SparseMatrix,
|
|
tolerance: f64
|
|
) -> VerificationResult:
|
|
// Check solution validity
|
|
original_residual = computeResidual(original_matrix, original_solution)
|
|
updated_residual = computeResidual(updated_matrix, updated_solution)
|
|
|
|
if updated_residual.norm() > original_residual.norm() * 1.1:
|
|
return VerificationResult::ResidualIncreased
|
|
|
|
// Check incremental consistency
|
|
expected_change = estimateExpectedChange(matrix_delta, original_solution)
|
|
actual_change = updated_solution - original_solution
|
|
|
|
relative_error = (actual_change - expected_change).norm() / expected_change.norm()
|
|
|
|
if relative_error < tolerance:
|
|
return VerificationResult::Verified
|
|
else:
|
|
return VerificationResult::SuspiciousChange(relative_error)
|
|
```
|
|
|
|
## Complexity Analysis
|
|
|
|
### Time Complexity Summary
|
|
|
|
| Algorithm | Single Query | Full Solution | Space |
|
|
|-----------|--------------|---------------|-------|
|
|
| Neumann Series | O(k·nnz) | O(k·n²) | O(n) |
|
|
| Forward Push | O(1/ε) | O(n/ε) | O(n) |
|
|
| Backward Push | O(1/ε) | O(n/ε) | O(n) |
|
|
| Hybrid Random-Walk | O(√n/ε) | O(n√n/ε) | O(√n) |
|
|
|
|
Where:
|
|
- `k` = number of series terms
|
|
- `nnz` = number of non-zeros
|
|
- `ε` = accuracy parameter
|
|
- `n` = matrix dimension
|
|
|
|
### Space-Time Tradeoffs
|
|
|
|
**Memory-Efficient Mode:**
|
|
- Streaming computation for large matrices
|
|
- On-demand residual computation
|
|
- Compressed intermediate storage
|
|
|
|
**Speed-Optimized Mode:**
|
|
- Full matrix precomputation
|
|
- Aggressive caching
|
|
- Parallel execution of multiple queries
|
|
|
|
## Implementation Priority
|
|
|
|
1. **Phase 1:** Core Neumann Series (Week 1-2)
|
|
2. **Phase 2:** Forward Push Method (Week 3-4)
|
|
3. **Phase 3:** Random Walk Engine (Week 5-6)
|
|
4. **Phase 4:** Algorithm Selection & Hybrid Methods (Week 7-8)
|
|
5. **Phase 5:** Numerical Stability & Incremental Updates (Week 9-10)
|
|
6. **Phase 6:** Performance Optimization & Benchmarking (Week 11-12)
|
|
|
|
## Testing Strategy
|
|
|
|
- Unit tests for each algorithm component
|
|
- Integration tests for method selection
|
|
- Property-based testing for numerical stability
|
|
- Benchmarking against reference implementations
|
|
- Stress testing with ill-conditioned matrices
|
|
- Performance regression testing
|
|
|
|
This implementation plan provides a comprehensive roadmap for building a robust, efficient, and numerically stable sublinear-time linear system solver. |