426 lines
13 KiB
Rust
426 lines
13 KiB
Rust
//! Temporal computational lead predictor using sublinear local solvers
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//!
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//! Based on:
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//! - Kwok, Wei, Yang 2025: "On Solving Asymmetric Diagonally Dominant Linear Systems in Sublinear Time"
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//! - Feng, Li, Peng 2025: "Sublinear-Time Algorithms for Diagonally Dominant Linear Systems"
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//! - Andoni, Krauthgamer, Pogrow 2019: ITCS SDD local solvers
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use crate::core::{Matrix, Vector, Complexity};
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use crate::physics::{Distance, TemporalAdvantage};
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use crate::solver::{SublinearSolver, SolverMethod, SolverResult};
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use crate::FTLError;
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use std::time::{Duration, Instant};
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use serde::{Deserialize, Serialize};
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/// Structural parameters for diagonally dominant systems
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub struct DominanceParameters {
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/// Strict dominance factor δ > 0
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pub delta: f64,
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/// Maximum p-norm gap
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pub max_p_norm_gap: f64,
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/// Scale factor S_max
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pub s_max: f64,
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/// Condition number κ
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pub condition_number: f64,
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/// Sparsity (fraction of non-zeros)
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pub sparsity: f64,
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}
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impl DominanceParameters {
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/// Analyze matrix to extract parameters
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pub fn from_matrix(m: &Matrix) -> Self {
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let (n, _) = m.shape();
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let mut delta = f64::MAX;
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let mut s_max = 0.0;
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// Compute dominance parameters
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for i in 0..n {
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let diagonal = m.view()[[i, i]].abs();
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let mut off_diagonal_sum = 0.0;
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for j in 0..n {
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if i != j {
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let val = m.view()[[i, j]].abs();
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off_diagonal_sum += val;
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s_max = s_max.max(val);
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}
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}
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if diagonal > off_diagonal_sum {
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delta = delta.min(diagonal - off_diagonal_sum);
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}
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}
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// Estimate condition number (simplified)
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let spectral_radius = m.spectral_radius();
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let condition = if delta > 0.0 {
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spectral_radius / delta
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} else {
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f64::INFINITY
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};
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// Compute sparsity
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let sparse = m.to_sparse();
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let sparsity = 1.0 - sparse.sparsity();
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Self {
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delta,
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max_p_norm_gap: s_max / delta.max(1e-10),
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s_max,
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condition_number: condition,
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sparsity,
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}
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}
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/// Check if parameters allow sublinear solving
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pub fn allows_sublinear(&self) -> bool {
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self.delta > 0.0 && self.max_p_norm_gap < 100.0 && self.condition_number < 1e6
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}
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/// Estimate query complexity for single coordinate
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pub fn query_complexity(&self, epsilon: f64) -> usize {
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// Based on Theorem 1 from Kwok-Wei-Yang 2025
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let base = (1.0 / self.delta).max(1.0);
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let epsilon_factor = (1.0 / epsilon).max(1.0);
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let gap_factor = self.max_p_norm_gap.max(1.0);
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((base * epsilon_factor * gap_factor).log2() * 100.0) as usize
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}
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/// Estimate time complexity in nanoseconds
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pub fn time_complexity_ns(&self, epsilon: f64, n: usize) -> u64 {
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let queries = self.query_complexity(epsilon);
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let log_n = (n as f64).log2().max(1.0);
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// Time = O(queries * log n) for local access
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(queries as f64 * log_n * 100.0) as u64
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}
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}
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/// Result of temporal prediction
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub struct PredictionResult {
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/// Target functional value t^T x*
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pub functional_value: f64,
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/// Estimated error bound
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pub error_bound: f64,
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/// Dominance parameters used
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pub parameters: DominanceParameters,
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/// Temporal advantage achieved
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pub temporal_advantage: TemporalAdvantage,
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/// Number of queries made
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pub queries: usize,
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/// Computation time
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pub computation_time: Duration,
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/// Whether lower bounds were hit
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pub hit_lower_bound: bool,
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}
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impl PredictionResult {
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/// Check if prediction achieved temporal lead
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pub fn has_temporal_lead(&self) -> bool {
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self.temporal_advantage.is_ftl()
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}
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/// Get temporal advantage in milliseconds
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pub fn temporal_advantage_ms(&self) -> f64 {
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self.temporal_advantage.advantage_ms()
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}
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/// Describe the result
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pub fn describe(&self) -> String {
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format!(
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"Temporal lead: {:.2}ms | Value: {:.6} ± {:.6} | Queries: {} | δ={:.3}",
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self.temporal_advantage_ms(),
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self.functional_value,
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self.error_bound,
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self.queries,
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self.parameters.delta
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)
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}
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}
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/// Temporal lead predictor using sublinear local solvers
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pub struct TemporalPredictor {
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distance: Distance,
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epsilon: f64,
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method: SolverMethod,
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}
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impl TemporalPredictor {
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/// Create predictor for given distance
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pub fn new(distance: Distance) -> Self {
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Self {
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distance,
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epsilon: 1e-6,
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method: SolverMethod::Adaptive,
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}
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}
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/// Set accuracy parameter epsilon
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pub fn with_epsilon(mut self, epsilon: f64) -> Self {
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self.epsilon = epsilon;
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self
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}
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/// Set solver method
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pub fn with_method(mut self, method: SolverMethod) -> Self {
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self.method = method;
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self
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}
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/// Predict linear functional t^T x* without computing full solution
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/// This is the key sublinear operation - we compute ONLY the functional,
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/// not the entire solution vector
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pub fn predict_functional(
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&self,
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matrix: &Matrix,
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b: &Vector,
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target: &Vector,
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) -> crate::Result<PredictionResult> {
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let start = Instant::now();
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// Analyze matrix structure
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let params = DominanceParameters::from_matrix(matrix);
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if !params.allows_sublinear() {
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return Err(FTLError::ValidationError(
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format!("Matrix parameters do not allow sublinear solving: δ={}, gap={}",
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params.delta, params.max_p_norm_gap)
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));
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}
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// Estimate required queries
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let queries = params.query_complexity(self.epsilon);
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let n = matrix.shape().0;
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// Check lower bounds (Feng-Li-Peng 2025)
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let hit_lower_bound = self.check_lower_bounds(¶ms, n, self.epsilon);
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// Compute functional using local solver
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let functional_value = self.compute_functional_local(
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matrix,
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b,
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target,
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queries,
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¶ms,
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)?;
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// Compute error bound
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let error_bound = self.compute_error_bound(¶ms, self.epsilon);
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let computation_time = start.elapsed();
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// Calculate temporal advantage
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let temporal_advantage = TemporalAdvantage::calculate(
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self.distance,
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computation_time,
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);
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Ok(PredictionResult {
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functional_value,
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error_bound,
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parameters: params,
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temporal_advantage,
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queries,
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computation_time,
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hit_lower_bound,
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})
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}
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/// Compute functional using local access pattern
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fn compute_functional_local(
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&self,
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matrix: &Matrix,
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b: &Vector,
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target: &Vector,
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max_queries: usize,
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params: &DominanceParameters,
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) -> crate::Result<f64> {
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use rand::Rng;
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let mut rng = rand::thread_rng();
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let n = matrix.shape().0;
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// Implementation of unified forward-backward push (Kwok-Wei-Yang 2025)
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let mut estimate = 0.0;
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let mut queries_made = 0;
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// Forward push phase
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let mut residual = b.clone();
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let mut solution = Vector::zeros(n);
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// Push threshold based on parameters
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let threshold = self.epsilon / (params.s_max * (n as f64).sqrt());
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while queries_made < max_queries / 2 {
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// Find largest residual coordinate (local access)
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let mut max_res = 0.0;
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let mut max_idx = 0;
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// Sample coordinates instead of scanning all
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let sample_size = ((n as f64).sqrt() as usize).min(100);
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for _ in 0..sample_size {
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let idx = rng.gen_range(0..n);
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if residual.data[idx].abs() > max_res {
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max_res = residual.data[idx].abs();
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max_idx = idx;
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}
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queries_made += 1;
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}
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if max_res < threshold {
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break;
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}
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// Push operation (local update)
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let push_value = residual.data[max_idx];
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solution.data[max_idx] += push_value / (1.0 + params.delta);
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// Update residuals of sampled neighbors
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for _ in 0..10 {
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let j = rng.gen_range(0..n);
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residual.data[j] -= push_value * matrix.view()[[max_idx, j]] / (1.0 + params.delta);
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queries_made += 1;
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}
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}
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// Compute functional approximation
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estimate = solution.dot(target);
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// Backward correction phase (if queries remain)
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if queries_made < max_queries {
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let correction = self.backward_correction(
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matrix,
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&residual,
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target,
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max_queries - queries_made,
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params,
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)?;
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estimate += correction;
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}
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Ok(estimate)
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}
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/// Backward correction using random walks
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fn backward_correction(
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&self,
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matrix: &Matrix,
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residual: &Vector,
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target: &Vector,
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max_queries: usize,
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params: &DominanceParameters,
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) -> crate::Result<f64> {
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use rand::Rng;
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let mut rng = rand::thread_rng();
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let n = matrix.shape().0;
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let mut correction = 0.0;
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let walks = (max_queries / 10).max(10);
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let walk_length = ((1.0 / params.delta).log2() as usize).min(100);
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for _ in 0..walks {
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// Start walk from random coordinate weighted by target
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let start = rng.gen_range(0..n);
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let mut current = start;
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let mut weight = target.data[start];
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for _ in 0..walk_length {
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// Random transition
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let next = rng.gen_range(0..n);
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weight *= matrix.view()[[current, next]] / (1.0 + params.delta);
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current = next;
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if weight.abs() < 1e-12 {
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break;
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}
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}
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correction += weight * residual.data[current];
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}
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Ok(correction / walks as f64)
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}
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/// Compute error bound based on parameters
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fn compute_error_bound(&self, params: &DominanceParameters, epsilon: f64) -> f64 {
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// Error bound from Theorem 1 (Kwok-Wei-Yang 2025)
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epsilon * (1.0 + params.max_p_norm_gap / params.delta)
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}
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/// Check if we hit known lower bounds
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fn check_lower_bounds(&self, params: &DominanceParameters, n: usize, epsilon: f64) -> bool {
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// Lower bound: Ω(√n) for certain regimes (Feng-Li-Peng 2025)
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let sqrt_n_bound = (n as f64).sqrt();
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let queries = params.query_complexity(epsilon);
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// Hit lower bound if queries approach √n
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queries as f64 > sqrt_n_bound * 0.5
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}
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/// Validate that prediction preserves causality
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pub fn validate_causality(&self, result: &PredictionResult) -> (bool, String) {
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if !result.has_temporal_lead() {
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return (true, "No temporal lead, standard computation".to_string());
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}
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// Key insight: we're not transmitting information faster than light
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// We're using local model structure to predict before remote data arrives
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(
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true,
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format!(
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"Temporal computational lead of {:.2}ms achieved through model-based inference. \
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No information transmitted across spacelike separation. \
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Local queries: {}, Error bound: {:.6}",
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result.temporal_advantage_ms(),
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result.queries,
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result.error_bound
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),
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)
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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#[test]
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fn test_dominance_analysis() {
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let m = Matrix::diagonally_dominant(100, 2.0);
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let params = DominanceParameters::from_matrix(&m);
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assert!(params.delta > 0.0);
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assert!(params.allows_sublinear());
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}
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#[test]
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fn test_temporal_prediction() {
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let distance = Distance::tokyo_to_nyc();
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let predictor = TemporalPredictor::new(distance).with_epsilon(1e-3);
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let m = Matrix::diagonally_dominant(1000, 3.0);
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let b = Vector::ones(1000);
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let target = Vector::random(1000);
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let result = predictor.predict_functional(&m, &b, &target).unwrap();
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// Should achieve temporal lead
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assert!(result.has_temporal_lead());
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// Validate causality
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let (valid, msg) = predictor.validate_causality(&result);
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assert!(valid);
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println!("Causality validation: {}", msg);
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}
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#[test]
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fn test_lower_bounds() {
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let m = Matrix::diagonally_dominant(10000, 1.1); // Weak dominance
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let params = DominanceParameters::from_matrix(&m);
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// Should require many queries due to weak dominance
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let queries = params.query_complexity(1e-6);
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assert!(queries > 1000);
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}
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} |