Work-Efficient Parallel Counting via Sampling

Hongyang Liu; Yitong Yin; Yiyao Zhang

arxiv: 2408.09719 · v3 · submitted 2024-08-19 · 💻 cs.DS · cs.CC· cs.DC

Work-Efficient Parallel Counting via Sampling

Hongyang Liu , Yitong Yin , Yiyao Zhang This is my paper

Pith reviewed 2026-05-23 22:23 UTC · model grok-4.3

classification 💻 cs.DS cs.CCcs.DC

keywords parallel algorithmscountingsamplingpartition functionsimulated annealinghardcore modelIsing modelGibbs distribution

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The pith

An algorithm reduces counting to sampling with logarithmic depth and near-optimal total work.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a parallel algorithm for approximating the partition function of a Gibbs distribution that reduces counting to sampling. It achieves near-optimal total work while ensuring the computation has only logarithmic depth, allowing efficient use of parallelism. This closes the gap between earlier parallel algorithms that incurred extra cost and sequential ones that could not run in parallel. The result directly yields work-efficient parallel counting procedures for the hardcore and Ising models inside the uniqueness regime.

Core claim

We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.

What carries the argument

A parallelized simulated annealing schedule that converts a sampling oracle into a counting procedure while keeping total work near-optimal and depth logarithmic.

If this is right

Work-efficient parallel approximation of the partition function for the hardcore model in the uniqueness regime.
Work-efficient parallel approximation of the partition function for the Ising model in the uniqueness regime.
The same reduction applies to any Gibbs distribution whose sampling oracle can be simulated efficiently.
The method unifies prior adaptive sequential techniques with non-adaptive parallel ones under a single work bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The logarithmic depth suggests the algorithm can be scheduled on standard parallel models such as PRAM or distributed systems with low synchronization cost.
If the sampling oracle itself admits further parallelization, the overall counting procedure could achieve even lower wall-clock time.
The technique may extend to approximate counting tasks outside statistical physics once suitable sampling oracles are available.

Load-bearing premise

The underlying sampling oracles stay efficient and the simulated annealing schedule admits a parallel implementation whose total work remains near-optimal.

What would settle it

A concrete construction or lower bound showing that any reduction from counting to sampling for the hardcore model in the uniqueness regime must incur either super-logarithmic depth or work exceeding the sequential optimum by more than a logarithmic factor.

read the original abstract

A canonical approach to approximating the partition function of a Gibbs distribution via sampling is simulated annealing. This method has led to efficient reductions from counting to sampling, including: $\bullet$ classic non-adaptive (parallel) algorithms with sub-optimal cost (Dyer-Frieze-Kannan '89; Bez\'akov\'a-\v{S}tefankovi\v{c}-Vazirani-Vigoda '08); $\bullet$ adaptive (sequential) algorithms with near-optimal cost (\v{S}tefankovi\v{c}-Vempala-Vigoda '09; Huber '15; Kolmogorov '18; Harris-Kolmogorov '24). We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a counting-to-sampling reduction that gets both logarithmic depth and near-optimal work, at least on paper.

read the letter

The headline result is a reduction from approximate counting to sampling that runs in logarithmic parallel depth while keeping total work near the sequential adaptive bound. That combination is what the abstract positions against the two earlier families: non-adaptive parallel methods with extra work and adaptive sequential methods with good work but no parallelism. If the construction holds, it directly gives work-efficient parallel algorithms for the hardcore and Ising models inside the uniqueness regime. The paper does a clean job stating the prior separation and claiming the new algorithm closes it via a parallelized simulated annealing schedule. The applications to those two models are the concrete payoff. The soft spot is exactly where the stress-test note flags it: the adaptive schedule must be made parallel without inflating the number of oracle calls or the per-call work by more than a constant. The oracles here are MCMC chains whose mixing times are known only in the sequential setting, so the paper needs an explicit argument that the telescoping product estimator still has the right variance with a parallel schedule. Without seeing the derivation or the work bound written out, it is hard to tell whether that step is tight or whether hidden logarithmic factors appear. The citation pattern looks standard for the area and does not rely on self-referential fitting. This paper is for people who care about parallel algorithms for partition functions or who want to implement counting routines on many cores. It is worth a serious referee to check the work analysis in the parallel case; the claim is narrow enough that a careful review can settle it.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a new reduction from counting (approximating the partition function of a Gibbs distribution) to sampling that simultaneously achieves near-optimal total work and O(log n) depth. This improves upon prior non-adaptive parallel algorithms (sub-optimal cost) and adaptive sequential algorithms (near-optimal cost but sequential). As a consequence, it yields work-efficient parallel counting algorithms for the hardcore and Ising models in the uniqueness regime via parallelizable simulated annealing schedules.

Significance. If the central reduction holds, the result would be significant for parallel algorithms in statistical physics and approximate counting, as it resolves the tension between work-optimality (previously requiring sequential adaptivity) and efficient parallelism. It would enable practical parallel implementations for important models without sacrificing asymptotic efficiency.

major comments (1)

[Extension to hardcore and Ising models (likely §4 or §5)] The load-bearing claim for the hardcore and Ising models (uniqueness regime) is that the adaptive simulated annealing schedule can be parallelized while keeping total work near-optimal. The manuscript must explicitly bound the number of MCMC oracle calls (and per-call work) under the parallel schedule to show that variance bounds on the telescoping product estimator are preserved without more than constant-factor inflation relative to the sequential case (e.g., Stefankovic-Vempala-Vigoda). Without such a calculation, the near-optimality assertion for these models is not yet supported.

minor comments (1)

[Introduction] Notation for the parallel depth and work bounds should be introduced with explicit big-O expressions early in the introduction or preliminaries for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for explicit bounds in the application to the hardcore and Ising models. We address the major comment below.

read point-by-point responses

Referee: [Extension to hardcore and Ising models (likely §4 or §5)] The load-bearing claim for the hardcore and Ising models (uniqueness regime) is that the adaptive simulated annealing schedule can be parallelized while keeping total work near-optimal. The manuscript must explicitly bound the number of MCMC oracle calls (and per-call work) under the parallel schedule to show that variance bounds on the telescoping product estimator are preserved without more than constant-factor inflation relative to the sequential case (e.g., Stefankovic-Vempala-Vigoda). Without such a calculation, the near-optimality assertion for these models is not yet supported.

Authors: We agree that an explicit calculation is required to fully support the near-optimality claim. In the revised manuscript we will add a dedicated subsection (in §4 or §5) that bounds the total number of MCMC oracle calls under the parallelized simulated annealing schedule. We will show that the schedule can be executed in O(log n) parallel rounds while the aggregate number of calls across all rounds remains within a constant factor of the sequential adaptive schedule of Stefankovic-Vempala-Vigoda. Because each call uses the same per-call work as in the sequential case and the variance analysis of the telescoping product depends only on the total number of samples (not their distribution across rounds), the variance bounds are preserved up to a constant factor. This establishes that the overall work remains near-optimal. revision: yes

Circularity Check

0 steps flagged

New algorithmic reduction from counting to sampling presented without reduction to fitted inputs or self-citations

full rationale

The paper describes a novel algorithm achieving both near-optimal total work and O(log n) depth for reductions from counting to sampling, extending prior sequential adaptive methods (e.g., Stefankovic-Vempala-Vigoda) and non-adaptive parallel ones. No equations, fitted parameters, or self-definitional steps appear in the abstract. The construction is presented as an independent algorithmic advance rather than a renaming or re-derivation of prior results by construction. External citations are to non-overlapping prior work and do not form a load-bearing self-citation chain. The derivation chain is self-contained as a new parallel schedule construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions about efficient sampling oracles under the uniqueness regime; no free parameters or invented entities are visible from the abstract.

axioms (1)

domain assumption Sampling oracles are efficient for the models considered when the uniqueness condition holds.
Invoked when stating consequences for hardcore and Ising models.

pith-pipeline@v0.9.0 · 5693 in / 1091 out tokens · 18728 ms · 2026-05-23T22:23:23.647363+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2
math.PR 2026-05 unverdicted novelty 7.0

A polynomial-time algorithm samples the SK model Gibbs measure with o(1) TVD error for β < 1/2 by combining potential Hessian ascent, stochastic localization, Jarzynski equality, and Glauber dynamics.
Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2
math.PR 2026-05 unverdicted novelty 7.0

Polynomial-time algorithm samples the Sherrington-Kirkpatrick Gibbs measure at beta < 1/2 with o(1) TVD error by combining potential Hessian ascent, stochastic localization, covariance estimates, and Jarzynski equalit...

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Accelerating Simulated Annealing for the Permanent and Combinatorial Co unting Problems

[Bez+08] Ivona Bezáková, Daniel Stefankovic, Vijay V . V azi rani, and Eric Vigoda. “Accelerating Simulated Annealing for the Permanent and Combinatorial Co unting Problems”. In: SIAM J. Comput. 37.5 (2008), pp. 1429–1454. [DFK91] Martin Dyer, Alan Frieze, and Ravi Kannan. “A random pol ynomial-time algorithm for approximating the volume of convex bodies”...

work page 2008
[2]

Approximation algorithms for the norma lizing constant of Gibbs distri- butions

15 [Hub15] Mark Huber. “ Approximation algorithms for the norma lizing constant of Gibbs distri- butions”. In: The Annals of Applied Probability (2015), pp. 974–985. [JS93] Mark Jerrum and Alistair Sinclair. “Polynomial-Time Approximation Algorithms for the Ising Model”. In: SIAM Journal on Computing 22.5 (Oct. 1993). Publisher: Society for Industrial and...

work page 2015

[1] [1]

Accelerating Simulated Annealing for the Permanent and Combinatorial Co unting Problems

[Bez+08] Ivona Bezáková, Daniel Stefankovic, Vijay V . V azi rani, and Eric Vigoda. “Accelerating Simulated Annealing for the Permanent and Combinatorial Co unting Problems”. In: SIAM J. Comput. 37.5 (2008), pp. 1429–1454. [DFK91] Martin Dyer, Alan Frieze, and Ravi Kannan. “A random pol ynomial-time algorithm for approximating the volume of convex bodies”...

work page 2008

[2] [2]

Approximation algorithms for the norma lizing constant of Gibbs distri- butions

15 [Hub15] Mark Huber. “ Approximation algorithms for the norma lizing constant of Gibbs distri- butions”. In: The Annals of Applied Probability (2015), pp. 974–985. [JS93] Mark Jerrum and Alistair Sinclair. “Polynomial-Time Approximation Algorithms for the Ising Model”. In: SIAM Journal on Computing 22.5 (Oct. 1993). Publisher: Society for Industrial and...

work page 2015