Near-Optimal Directed Low-Diameter Decompositions

2025-02-08

AI Summary

This paper studies low-diameter decomposition (LDD) for directed graphs, aiming to minimize the edge-cutting probability while approaching the theoretical lower bound Ω(log n / D). We establish the first deep connection between LDD and expander decomposition, enabling a near-optimal construction: edge-cutting probability is reduced to O((log n log log n) / D), the best known to date. We further develop the first randomized algorithm with nearly linear runtime Õ(m), and the first deterministic algorithm whose runtime depends polynomially on the target diameter D (i.e., Õ(m · poly(D)). Our approach integrates expander theory, recursive partitioning via strongly connected components, probabilistic analysis, and derandomization techniques. These results advance the theoretical frontier of directed-graph LDD and provide novel algorithmic tools and foundations for fundamental problems such as directed single-source shortest paths (SSSP).

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Deep Learning Algorithms
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Low Diameter Decompositions (LDDs) are invaluable tools in the design of combinatorial graph algorithms. While historically they have been applied mainly to undirected graphs, in the recent breakthrough for the negative-length Single Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen [FOCS '22] extended the use of LDDs to directed graphs for the first time. Specifically, their LDD deletes each edge with probability at most $O(frac{1}{D} cdot log^2 n)$, while ensuring that each strongly connected component in the remaining graph has a (weak) diameter of at most $D$. In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of $O(frac{1}{D} cdot log n loglog n)$. This improves the previous bound by nearly a logarithmic factor and closely approaches the lower bound of $Omega(frac{1}{D} cdot log n)$. With significantly more technical effort, we also develop two efficient algorithms for computing our LDDs: a deterministic algorithm that runs in time $ ilde O(m cdot poly(D))$ and a randomized algorithm that runs in near-linear time $ ilde O(m)$. We believe that our work provides a solid conceptual and technical foundation for future research relying on directed LDDs, which will undoubtedly follow soon.

Problem

• Extending LDDs to directed graphs
• Improving edge-cutting probability bounds
• Developing efficient LDD computation algorithms

Innovation

• Extended LDDs to directed graphs
• Improved edge-cutting probability
• Developed efficient LDD computation algorithms