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On Incremental Approximate Shortest Paths in Directed Graphs

Adam Górkiewicz, Adam Karczmarz

TL;DR

The paper tackles incremental all-pairs shortest paths in sparse weighted directed graphs under edge insertions with adaptive adversaries by developing near-linear time data structures for $(1+\epsilon)$-approximate shortest paths. It introduces a source-insertion SSSP data structure, extends it to handle general insertions via a rank-offset framework, and combines these with a phase-based APSP approach that leverages shortcuts and per-source structures to achieve $\tilde{O}(m^{3/2}n^{3/4})$ (deterministic) and $\tilde{O}(m^{4/3}n^{5/6})$ (randomized) total update times in sparse graphs. An offline variant yields near-linear deterministic incremental SSSP and a deterministic all-pairs bounded-leg SP structure for sparse graphs, while randomized resets bound certificate growth with high probability. Together, these results push the boundary on online partially dynamic SSSP/APSP, providing practical near-linear guarantees against adaptive adversaries and shedding light on the potential for offline-all-pairs approaches to inform online structures.

Abstract

In this paper, we show new data structures maintaining approximate shortest paths in sparse directed graphs with polynomially bounded non-negative edge weights under edge insertions. We give more efficient incremental $(1+ε)$-approximate APSP data structures that work against an adaptive adversary: a deterministic one with $\tilde{O}(m^{3/2}n^{3/4})$ total update time and a randomized one with $\tilde{O}(m^{4/3}n^{5/6})$ total update time. For sparse graphs, these both improve polynomially upon the best-known bound against an adaptive adversary. To achieve that, building on the ideas of [Chechik-Zhang, SODA'21] and [Kyng-Meierhans-Probst Gutenberg, SODA'22], we show a near-optimal $(1+ε)$-approximate incremental SSSP data structure for a special case when all edge updates are adjacent to the source, that might be of independent interest. We also describe a very simple and near-optimal \emph{offline} incremental $(1+ε)$-approximate SSSP data structure. While online near-linear partially dynamic SSSP data structures have been elusive so far (except for dense instances), our result excludes using certain types of impossibility arguments to rule them out. Additionally, our offline solution leads to near-optimal and deterministic all-pairs bounded-leg shortest paths data structure for sparse graphs.

On Incremental Approximate Shortest Paths in Directed Graphs

TL;DR

The paper tackles incremental all-pairs shortest paths in sparse weighted directed graphs under edge insertions with adaptive adversaries by developing near-linear time data structures for -approximate shortest paths. It introduces a source-insertion SSSP data structure, extends it to handle general insertions via a rank-offset framework, and combines these with a phase-based APSP approach that leverages shortcuts and per-source structures to achieve (deterministic) and (randomized) total update times in sparse graphs. An offline variant yields near-linear deterministic incremental SSSP and a deterministic all-pairs bounded-leg SP structure for sparse graphs, while randomized resets bound certificate growth with high probability. Together, these results push the boundary on online partially dynamic SSSP/APSP, providing practical near-linear guarantees against adaptive adversaries and shedding light on the potential for offline-all-pairs approaches to inform online structures.

Abstract

In this paper, we show new data structures maintaining approximate shortest paths in sparse directed graphs with polynomially bounded non-negative edge weights under edge insertions. We give more efficient incremental -approximate APSP data structures that work against an adaptive adversary: a deterministic one with total update time and a randomized one with total update time. For sparse graphs, these both improve polynomially upon the best-known bound against an adaptive adversary. To achieve that, building on the ideas of [Chechik-Zhang, SODA'21] and [Kyng-Meierhans-Probst Gutenberg, SODA'22], we show a near-optimal -approximate incremental SSSP data structure for a special case when all edge updates are adjacent to the source, that might be of independent interest. We also describe a very simple and near-optimal \emph{offline} incremental -approximate SSSP data structure. While online near-linear partially dynamic SSSP data structures have been elusive so far (except for dense instances), our result excludes using certain types of impossibility arguments to rule them out. Additionally, our offline solution leads to near-optimal and deterministic all-pairs bounded-leg shortest paths data structure for sparse graphs.

Paper Structure

This paper contains 34 sections, 20 theorems, 48 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Incremental shortest paths
  3. Exact data structures.
  4. Approximate SSSP.
  5. Approximate APSP.
  6. Our contribution
  7. Offline data structures.
  8. Further related work
  9. Organization
  10. Preliminaries
  11. Dijkstra-like propagation
  12. An SSSP data structure for source insertions
  13. Basic setup
  14. Handling source insertions
  15. Running time analysis.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\epsilon\in (0,1)$. Let $G=(V,E)$ be a digraph with edge weights in ${\{0\}\cup [1,W]}$ and a source $s\in V$. There exists a deterministic data structure explicitly maintaining distance estimates $d:V\to\mathbb{R}_{\geq 0}$ satisfying for all $v\in V$ and only supporting insertions (or weight decreases) of source edges$e=sv$, $v\in V$. The total update time of the data structure is $O(m\log

Theorems & Definitions (37)

  • Theorem 1.1: Simplified version of Theorem \ref{['t:source-sssp']}
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • Remark 4.2
  • Definition 4.3: certificates
  • ...and 27 more