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

2025-09-29

AI Summary

Prior low-diameter decomposition (LDD) algorithms for directed graphs suffer from a fundamental theoretical limitation: they only guarantee decompositions for super-logarithmic diameters (ω(log n)), failing to provide meaningful guarantees for sub-logarithmic diameters—e.g., D = Ω(log log n). Method: We introduce the first *separating* LDD definition tailored for small diameters in directed graphs, strengthening probabilistic guarantees and edge-wise independence while generalizing key properties of undirected-graph LDDs. Building on structural refinements of the algorithms by [Bri+25; Li25], we design a near-linear-time Õ(m) sampling algorithm. Contribution/Results: Our algorithm is the first to compute separating LDDs for directed graphs with controllable diameter D = Ω(log log n), offering strong theoretical guarantees—including separation, bounded diameter, and near-optimal cut probability. This breaks the long-standing diameter lower-bound barrier, significantly broadening the applicability of LDDs to directed-graph algorithms (e.g., flow, distance estimation, clustering) and establishing a new foundational tool for structural analysis of directed graphs.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Social Networks and Social Media: Social mining and social search on the Web

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 $D = Ω(loglog n)$ in contrast to the state of the art that only applies to super-logarithmic diameters $D = ω(log n)$. 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 [Bri+25; Li25] can be used to sample decompositions with our new guarantees in near-linear time $ ilde{O}(m)$.

Problem

• Strengthening directed low-diameter decompositions with sub-logarithmic diameter guarantees
• Introducing separated low-diameter decompositions for directed graphs
• Transferring undirected decomposition properties to directed graph setting

Innovation

• Introduces separated low-diameter decompositions for directed graphs
• Achieves sub-logarithmic diameter guarantees with probabilistic improvements
• Provides near-linear time algorithmic modifications for sampling decompositions