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Perfect Simulation of Las Vegas Algorithms via Local Computation

Xinyu Fu, Yonggang Jiang, Yitong Yin

TL;DR

The paper addresses the problem of perfectly simulating Las Vegas LOCAL algorithms within the LOCAL model by enabling local, distributed sampling from hard constraints. It introduces a LOCAL sampling lemma for not-too-scarce LLL instances and builds a framework around Bayes filters, LLL augmentation, and recursive, clustered resampling to preserve the exact output distribution of successful runs. The main contribution is a formal theorem that converts any fixed-round Las Vegas LOCAL algorithm into a zero-error LOCAL algorithm with polylogarithmic overhead, terminating with high probability and reproducing the correct conditioned output. These results extend perfect sampling to Gibbs distributions with strong spatial mixing and yield practical consequences for LOCAL-LCL sampling and distributed LLL-sampling tasks, emphasizing locality does not impede perfect distributed sampling under suitable conditions.

Abstract

The notion of Las Vegas algorithms was introduced by Babai (1979) and can be defined in two ways: * In Babai's original definition, a randomized algorithm is called Las Vegas if it has a finitely bounded running time and certifiable random failure. * Another definition widely accepted today is that Las Vegas algorithms refer to zero-error randomized algorithms with random running times. The equivalence between the two definitions is straightforward. Specifically, for randomized algorithms with certifiable failures, repeatedly running the algorithm until no failure is encountered allows for faithful simulation of the correct output when it executes successfully. We show that a similar perfect simulation can also be achieved in distributed local computation. Specifically, in the LOCAL model, with polylogarithmic overhead in time complexity, any Las Vegas algorithm with finitely bounded running time and locally certifiable failures can be converted to a zero-error Las Vegas algorithm. This transformed algorithm faithfully reproduces the correct output of the original algorithm in successful executions.

Perfect Simulation of Las Vegas Algorithms via Local Computation

TL;DR

The paper addresses the problem of perfectly simulating Las Vegas LOCAL algorithms within the LOCAL model by enabling local, distributed sampling from hard constraints. It introduces a LOCAL sampling lemma for not-too-scarce LLL instances and builds a framework around Bayes filters, LLL augmentation, and recursive, clustered resampling to preserve the exact output distribution of successful runs. The main contribution is a formal theorem that converts any fixed-round Las Vegas LOCAL algorithm into a zero-error LOCAL algorithm with polylogarithmic overhead, terminating with high probability and reproducing the correct conditioned output. These results extend perfect sampling to Gibbs distributions with strong spatial mixing and yield practical consequences for LOCAL-LCL sampling and distributed LLL-sampling tasks, emphasizing locality does not impede perfect distributed sampling under suitable conditions.

Abstract

The notion of Las Vegas algorithms was introduced by Babai (1979) and can be defined in two ways: * In Babai's original definition, a randomized algorithm is called Las Vegas if it has a finitely bounded running time and certifiable random failure. * Another definition widely accepted today is that Las Vegas algorithms refer to zero-error randomized algorithms with random running times. The equivalence between the two definitions is straightforward. Specifically, for randomized algorithms with certifiable failures, repeatedly running the algorithm until no failure is encountered allows for faithful simulation of the correct output when it executes successfully. We show that a similar perfect simulation can also be achieved in distributed local computation. Specifically, in the LOCAL model, with polylogarithmic overhead in time complexity, any Las Vegas algorithm with finitely bounded running time and locally certifiable failures can be converted to a zero-error Las Vegas algorithm. This transformed algorithm faithfully reproduces the correct output of the original algorithm in successful executions.
Paper Structure (50 sections, 30 theorems, 124 equations, 9 algorithms)

This paper contains 50 sections, 30 theorems, 124 equations, 9 algorithms.

Table of Contents

  1. Introduction
  2. The $\mathsf{LOCAL}$ model.
  3. A sampling Lovász local lemma in the $\mathsf{LOCAL}$ model
  4. Perfect simulation via $\mathsf{LOCAL}$ sampling lemma
  5. Preliminary
  6. Graph and LLL notations
  7. Marginal distribution
  8. Decay of correlation in LLL
  9. Las Vegas $\mathsf{SLOCAL}$ algorithm
  10. $\mathsf{SLOCAL}\text{-}\mathsf{LV}$ algorithms
  11. Exposition of the Algorithm: Special Case
  12. Warm-up: Sampling with idealized correlation decay
  13. An attempt to correct resampling.
  14. Construction of the Bayes filter $\mathcal{F}$.
  15. Sampling without correlation decay via LLL augmentation
  16. ...and 35 more sections

Key Result

Theorem 1.1

Any $t(n)$-round Las Vegas $\mathsf{LOCAL}$ algorithm can be converted to a zero-error Las Vegas $\mathsf{LOCAL}$ algorithm, which terminates within $t(n)\cdot\mathrm{polylog}(n)$ rounds with probability $1-n^{-O(1)}$, and returns the output of the $t(n)$-round Las Vegas $\mathsf{LOCAL}$ algorithm c

Theorems & Definitions (81)

  • Theorem 1.1: main theorem, informal
  • Corollary 1.2
  • Remark 1.1
  • Theorem 1.3: $\mathsf{LOCAL}$ sampling lemma
  • Corollary 1.4
  • Theorem 1.5: formal restatement of \ref{['thm:main-informal']}
  • proof
  • Definition 2.1: marginal distribution
  • Definition 2.2: $\epsilon$-correlated sets
  • Proposition 2.1: simulation of $\mathsf{SLOCAL}\text{-}\mathsf{LV}$ in $\mathsf{LOCAL}$
  • ...and 71 more