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Simpler and Faster Directed Low-Diameter Decompositions

Jason Li

TL;DR

This work presents a simpler and faster algorithm for directed low-diameter decompositions that matches the best known loss factor $ell(n)=O(log n\ log log n)$ and improves running time to $O((m+n\ log\log n)\ log^2 n)$ for graphs with integral, polynomially-bounded weights. The approach replaces geometric ball-growing with CKR-style uniform-radius balls processed in random order and introduces heavy-vertex elimination to reduce recursion depth, enabling a robust, near-linear-time decomposition. By combining these techniques with a careful probabilistic analysis of edge cuts and a recursive Seymour-like framework, the paper achieves the same asymptotic loss as prior work while significantly improving practical efficiency. The results advance directed LDDs as a tool for near-linear time graph algorithms and related applications in directed settings, including negative-weight shortest-path-related techniques and multi-cut problems.

Abstract

We present a simpler and faster algorithm for low-diameter decompositions on directed graphs, matching the $O(\log n\log\log n)$ loss factor from Bringmann, Fischer, Haeupler, and Latypov (ICALP 2025) and improving the running time to $O((m+n\log\log n)\log^2n)$.

Simpler and Faster Directed Low-Diameter Decompositions

TL;DR

This work presents a simpler and faster algorithm for directed low-diameter decompositions that matches the best known loss factor and improves running time to for graphs with integral, polynomially-bounded weights. The approach replaces geometric ball-growing with CKR-style uniform-radius balls processed in random order and introduces heavy-vertex elimination to reduce recursion depth, enabling a robust, near-linear-time decomposition. By combining these techniques with a careful probabilistic analysis of edge cuts and a recursive Seymour-like framework, the paper achieves the same asymptotic loss as prior work while significantly improving practical efficiency. The results advance directed LDDs as a tool for near-linear time graph algorithms and related applications in directed settings, including negative-weight shortest-path-related techniques and multi-cut problems.

Abstract

We present a simpler and faster algorithm for low-diameter decompositions on directed graphs, matching the loss factor from Bringmann, Fischer, Haeupler, and Latypov (ICALP 2025) and improving the running time to .
Paper Structure (10 sections, 10 theorems, 9 equations)

This paper contains 10 sections, 10 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our Techniques
  4. CKR instead of geometric ball-growing.
  5. Elimination of heavy vertices.
  6. Main Algorithm
  7. Eliminating Heavy Vertices
  8. Case 1.
  9. Case 2.
  10. Combining both cases.

Key Result

Theorem 1

Given a graph with integral and polynomially-bounded edge weights, there is a directed LDD algorithm that achieves loss factor $\ell(n)=O(\log n\log\log n)$, runs in $O((m+n\log\log n)\log^2n)$ time, and succeeds with high probability.

Theorems & Definitions (20)

  • Theorem 1
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 10 more