Stronger Directed Low-Diameter Decompositions with Sub-Logarithmic Diameter and Separation

Bernhard Haeupler; Richard Hlad\'ik; Shengzhe Wang; Zhijun Zhang

arxiv: 2509.24565 · v2 · submitted 2025-09-29 · 💻 cs.DS

Stronger Directed Low-Diameter Decompositions with Sub-Logarithmic Diameter and Separation

Bernhard Haeupler , Richard Hlad\'ik , Shengzhe Wang , Zhijun Zhang This is my paper

Pith reviewed 2026-05-18 12:55 UTC · model grok-4.3

classification 💻 cs.DS

keywords directed graphslow-diameter decompositionsgraph partitioningseparation propertiessub-logarithmic diameterprobabilistic algorithmsnear-linear timeedge independence

0 comments

The pith

Directed low-diameter decompositions now achieve sub-logarithmic diameters with separation and independence guarantees.

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

The paper introduces separated low-diameter decompositions for directed graphs and shows how small changes to two prior algorithms produce them with stronger probabilistic properties. It establishes that these decompositions work with diameters as small as Omega(log log n) and include new independence between distant edges, moving directed graphs closer to the toolkit long available for undirected ones. The constructions run in near-linear time and tighten existing probability bounds. If the results hold, they transfer widely used decomposition techniques into the directed setting for the first time at these small scales.

Core claim

The central claim is that minor modifications to existing directed low-diameter decomposition algorithms suffice to produce separated decompositions with tightened probability bounds and independence between far-away edges, yielding the first meaningful guarantees for diameter parameters D = Omega(log log n) rather than the prior requirement of omega(log n).

What carries the argument

Separated low-diameter decomposition, a partition of the directed graph into clusters of small diameter that also enforces separation between clusters and statistical independence among distant edges.

If this is right

Transfers key aspects of undirected low-diameter decompositions, including separation and edge independence, into the directed setting.
Delivers the first nontrivial guarantees for decompositions when the diameter parameter is only Omega(log log n).
Provides new independence results between edges that are far apart in the graph.
All constructions remain algorithmic and run in near-linear time in the number of edges.

Where Pith is reading between the lines

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

These decompositions could support improved approximation algorithms for directed versions of clustering, partitioning, or network design problems.
The separation and independence properties may help design more efficient parallel or distributed algorithms on directed graphs.
The approach invites testing whether similar modifications can push diameters even smaller or apply to dynamic or weighted directed graphs.

Load-bearing premise

The modifications to the two prior algorithms preserve their original probabilistic correctness properties.

What would settle it

A concrete counterexample on a family of directed graphs with D set to a constant multiple of log log n where the sampled decompositions violate the claimed separation probability or independence bounds.

read the original abstract

This paper significantly strengthens directed low-diameter decompositions in several ways. We define and give the first results for separated low-diameter decompositions in directed graphs, tighten and generalize probabilistic guarantees, and prove new independence results between (far away) edges. Our results are the first to give meaningful guarantees for decompositions with small diameters $D = \Omega(\log\log n)$ in contrast to the state of the art that only applies to super-logarithmic diameters $D = \omega(\log n)$. These results transfer several important and widely used aspects of undirected low-diameter decompositions to the directed setting. All our results are algorithmic -- small modifications to two existing directed low-diameter decompositions [BFHL25; Li25] can be used to sample decompositions with our new guarantees in near-linear time $\tilde{O}(m)$.

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 adapts two recent directed LDD algorithms to deliver the first separated decompositions and sub-logarithmic diameters, but the probabilistic claims for the smaller diameter rest on unverified modifications.

read the letter

This paper claims the first separated low-diameter decompositions for directed graphs and the first sub-logarithmic diameter bounds. It achieves this by making small modifications to two recent directed LDD algorithms from the cited works. It does well in identifying the gap in prior work, which only handled super-logarithmic diameters, and in generalizing the probabilistic guarantees. The independence results between far away edges are a nice addition that could be useful beyond the decomposition itself. Transferring separation and these other properties from the undirected case is a clear advance for directed graph algorithms. The main soft spot is the reliance on unverified modifications. The headline improvement to D = Ω(log log n) requires that the failure probabilities and separation guarantees from the base algorithms survive the changes. The stress test note points out that reducing the target diameter could affect how the sampling works and whether the probabilistic properties hold. Without an explicit re-analysis or parameter recalculation in the paper, it's unclear whether the original concentration bounds still apply at the smaller diameter. This is a moderate issue rather than a dealbreaker, since the base papers are recent, but it does mean the central claims need careful checking. The citation pattern is appropriate and there are no circular arguments or invented entities here. This paper is aimed at algorithm researchers working on directed graphs and decompositions. Readers who use LDDs in their own directed graph problems would benefit from the stronger guarantees if they check out. It has enough novelty and potential impact to deserve a serious referee. I would recommend sending it for peer review to confirm the modifications work as claimed and to verify the new bounds.

Referee Report

1 major / 1 minor

Summary. The paper claims to strengthen directed low-diameter decompositions by introducing separated low-diameter decompositions in directed graphs, tightening probabilistic guarantees, proving new independence results between far-away edges, and achieving the first meaningful guarantees for sub-logarithmic diameters D = Ω(log log n) (improving on prior work limited to D = ω(log n)). These results are obtained via small modifications to the algorithms of [BFHL25] and [Li25] and run in near-linear time Õ(m).

Significance. If the modifications to the cited algorithms rigorously preserve the failure probabilities, separation, and independence properties, the work would be significant for transferring key features of undirected low-diameter decompositions to the directed setting and enabling their use at smaller diameter scales in directed-graph algorithms.

major comments (1)

[Abstract and §1] Abstract and §1: The central claim that 'small modifications' to the sampling procedures of [BFHL25] and [Li25] yield valid sub-logarithmic diameter D = Ω(log log n) and separation guarantees requires an explicit error analysis or recalculated concentration bounds; the manuscript provides none, so it is impossible to verify that the original probabilistic properties survive the reduction from ω(log n) to Ω(log log n).

minor comments (1)

The citation format for [BFHL25; Li25] should be expanded to full bibliographic entries, and any omitted proof details for the modifications should be placed in an appendix to allow verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the major comment below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1: The central claim that 'small modifications' to the sampling procedures of [BFHL25] and [Li25] yield valid sub-logarithmic diameter D = Ω(log log n) and separation guarantees requires an explicit error analysis or recalculated concentration bounds; the manuscript provides none, so it is impossible to verify that the original probabilistic properties survive the reduction from ω(log n) to Ω(log log n).

Authors: We agree that the current presentation would benefit from an explicit error analysis to make the argument self-contained. The modifications consist of adjusting the sampling probability and the number of iterations in the procedures of [BFHL25] and [Li25] so that the diameter parameter scales as Ω(log log n) while preserving the same Chernoff-style concentration. In the revised version we will add a dedicated subsection (likely in §2) that recalculates the failure probability, separation probability, and edge-independence bounds under this regime, confirming that they remain 1/poly(n) or better. This addition will directly address the verification concern without altering the algorithmic results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results build on external prior algorithms

full rationale

The paper derives its new guarantees for directed low-diameter decompositions with D = Ω(log log n) via small modifications to the sampling procedures of the externally cited algorithms in [BFHL25] and [Li25]. No equations, parameters, or definitions in the provided text reduce the claimed results to themselves by construction, nor is there any fitted-input prediction, self-definitional loop, or uniqueness theorem imported solely from overlapping-author citations. The central claims remain independent of the present paper's own outputs and rest on the assumed correctness of the referenced prior works, which are treated as external benchmarks. This is standard non-circular algorithmic extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work relies on standard probabilistic method arguments and the correctness of two previously published directed LDD algorithms; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

pith-pipeline@v0.9.0 · 5682 in / 1078 out tokens · 28706 ms · 2026-05-18T12:55:24.979519+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our results are the first to give meaningful guarantees for decompositions with small diameters D = Ω(log log n) … small modifications to two existing directed low-diameter decomposition algorithms from [BFHL25] and [Li25]
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Pr(e is not cut) ≥ exp(−de/D · O(log n log log n)) … independence property … separation property

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

Improved Upper Bounds for the Directed Flow-Cut Gap
cs.DS 2026-04 unverdicted novelty 8.0

The directed flow-cut gap is at most n^{1/3 + o(1)}.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

Hardness of approxi- mation in p via short cycle removal: cycle detection, distance oracles, and beyond

[Abb+22] Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. “Hardness of approxi- mation in p via short cycle removal: cycle detection, distance oracles, and beyond”. In: Proceedings of the Symposium on Theory of Computing (STOC). 2022, pp. 1487–1500. [ABN08] Ittai Abraham, Yair Bartal, and Ofer Neiman. “Nearly Tight Low Stretch Span- ning Trees”. In...

work page doi:10.1109/focs.2008.62 2022
[2]

Exploiting spontaneous transmissions for broad- casting and leader election in radio networks

[CD21] Artur Czumaj and Peter Davies. “Exploiting spontaneous transmissions for broad- casting and leader election in radio networks”. In: Journal of the ACM (JACM) 68.2 (2021), pp. 1–22. [Elk+08] Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. “Lower-Stretch Spanning Trees”. In: SIAM J. Comput. 38.2 (2008), pp. 608–628. doi: 10 . 1137 ...

work page doi:10.1145/3274651 2021
[3]

Expander Decomposition in Dynamic Streams

LIPIcs. 2024, 56:1–56:15. doi: 10.4230/LIPICS.SOCG.2024.56. [FKM23] Arnold Filtser, Michael Kapralov, and Mikhail Makarov. “Expander Decomposition in Dynamic Streams”. In: Proceedings of the Conference on Innovations in Theoretical Computer Science (ITCS). Vol

work page doi:10.4230/lipics.socg.2024.56 2024
[4]

In 32nd European Conference on Object-Oriented Programming (ECOOP 2018)

LIPIcs. 2023, 50:1–50:13. doi: 10.4230/LIPICS. ITCS.2023.50. [Fis+25] Nick Fischer, Bernhard Haeupler, Rustam Latypov, Antti Roeyskoe, and Aurelio L Sulser. “A Simple Parallel Algorithm with Near-Linear Work for Negative-Weight Single-Source Shortest Path”. In: Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA) . 2025, pp. 216–225. [Hae+...

work page doi:10.4230/lipics 2023
[5]

A Nearly- m log n Time Solver for SDD Linear Systems

1006/JAGM.1997.0888. [KMP11] Ioannis Koutis, Gary L. Miller, and Richard Peng. “A Nearly- m log n Time Solver for SDD Linear Systems”. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS). 2011, pp. 590–598. doi: 10.1109/FOCS.2011.85. 29 [Kra+04] Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. “Measured De- scent: A N...

work page doi:10.1109/focs.2011.85 1997

[1] [1]

Hardness of approxi- mation in p via short cycle removal: cycle detection, distance oracles, and beyond

[Abb+22] Amir Abboud, Karl Bringmann, Seri Khoury, and Or Zamir. “Hardness of approxi- mation in p via short cycle removal: cycle detection, distance oracles, and beyond”. In: Proceedings of the Symposium on Theory of Computing (STOC). 2022, pp. 1487–1500. [ABN08] Ittai Abraham, Yair Bartal, and Ofer Neiman. “Nearly Tight Low Stretch Span- ning Trees”. In...

work page doi:10.1109/focs.2008.62 2022

[2] [2]

Exploiting spontaneous transmissions for broad- casting and leader election in radio networks

[CD21] Artur Czumaj and Peter Davies. “Exploiting spontaneous transmissions for broad- casting and leader election in radio networks”. In: Journal of the ACM (JACM) 68.2 (2021), pp. 1–22. [Elk+08] Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. “Lower-Stretch Spanning Trees”. In: SIAM J. Comput. 38.2 (2008), pp. 608–628. doi: 10 . 1137 ...

work page doi:10.1145/3274651 2021

[3] [3]

Expander Decomposition in Dynamic Streams

LIPIcs. 2024, 56:1–56:15. doi: 10.4230/LIPICS.SOCG.2024.56. [FKM23] Arnold Filtser, Michael Kapralov, and Mikhail Makarov. “Expander Decomposition in Dynamic Streams”. In: Proceedings of the Conference on Innovations in Theoretical Computer Science (ITCS). Vol

work page doi:10.4230/lipics.socg.2024.56 2024

[4] [4]

In 32nd European Conference on Object-Oriented Programming (ECOOP 2018)

LIPIcs. 2023, 50:1–50:13. doi: 10.4230/LIPICS. ITCS.2023.50. [Fis+25] Nick Fischer, Bernhard Haeupler, Rustam Latypov, Antti Roeyskoe, and Aurelio L Sulser. “A Simple Parallel Algorithm with Near-Linear Work for Negative-Weight Single-Source Shortest Path”. In: Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA) . 2025, pp. 216–225. [Hae+...

work page doi:10.4230/lipics 2023

[5] [5]

A Nearly- m log n Time Solver for SDD Linear Systems

1006/JAGM.1997.0888. [KMP11] Ioannis Koutis, Gary L. Miller, and Richard Peng. “A Nearly- m log n Time Solver for SDD Linear Systems”. In: Proceedings of the Symposium on Foundations of Computer Science (FOCS). 2011, pp. 590–598. doi: 10.1109/FOCS.2011.85. 29 [Kra+04] Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. “Measured De- scent: A N...

work page doi:10.1109/focs.2011.85 1997