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Simple Linear-Size Additive Emulators

Gary Hoppenworth

TL;DR

This work targets the problem of constructing sparse additive emulators of $n$-vertex graphs with linear size, seeking the smallest possible additive error. It introduces a simple linear-size emulator based on a path-buying framework that leverages a BV16 clustering decomposition and a sampling-based quadratic-expansion argument, with recursive insertion of emulators on large clusters. The resulting emulator achieves an additive error of $+O(n^{1/(3+\sqrt{5})+\epsilon})$ (numerically about $n^{0.191+\epsilon}$) on $O_{\\epsilon}(n)$ edges, improving the previous $+O(n^{0.222})$ bound. This narrows the gap between known upper and lower bounds for linear-size additive emulators and demonstrates the efficacy of path-buying methods beyond traditional dense settings.

Abstract

Given an input graph $G = (V, E)$, an additive emulator $H = (V, E', w)$ is a sparse weighted graph that preserves all distances in $G$ with small additive error. A recent line of inquiry has sought to determine the best additive error achievable in the sparsest setting, when $H$ has a linear number of edges. In particular, the work of [Kogan and Parter, ICALP 2023], following [Pettie, ICALP 2007], constructed linear size emulators with $+O(n^{0.222})$ additive error. It is known that the worst-case additive error must be at least $+Ω(n^{2/29})$ due to [Lu, Vassilevska Williams, Wein, and Xu, SODA 2022]. We present a simple linear-size emulator construction that achieves additive error $+O(n^{0.191})$. Our approach extends the path-buying framework developed by [Baswana, Kavitha, Mehlhorn, and Pettie, SODA 2005] and [Vassilevska Williams and Bodwin, SODA 2016] to the setting of sparse additive emulators.

Simple Linear-Size Additive Emulators

TL;DR

This work targets the problem of constructing sparse additive emulators of -vertex graphs with linear size, seeking the smallest possible additive error. It introduces a simple linear-size emulator based on a path-buying framework that leverages a BV16 clustering decomposition and a sampling-based quadratic-expansion argument, with recursive insertion of emulators on large clusters. The resulting emulator achieves an additive error of (numerically about ) on edges, improving the previous bound. This narrows the gap between known upper and lower bounds for linear-size additive emulators and demonstrates the efficacy of path-buying methods beyond traditional dense settings.

Abstract

Given an input graph , an additive emulator is a sparse weighted graph that preserves all distances in with small additive error. A recent line of inquiry has sought to determine the best additive error achievable in the sparsest setting, when has a linear number of edges. In particular, the work of [Kogan and Parter, ICALP 2023], following [Pettie, ICALP 2007], constructed linear size emulators with additive error. It is known that the worst-case additive error must be at least due to [Lu, Vassilevska Williams, Wein, and Xu, SODA 2022]. We present a simple linear-size emulator construction that achieves additive error . Our approach extends the path-buying framework developed by [Baswana, Kavitha, Mehlhorn, and Pettie, SODA 2005] and [Vassilevska Williams and Bodwin, SODA 2016] to the setting of sparse additive emulators.
Paper Structure (8 sections, 6 theorems, 12 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Preliminaries
  4. Construction
  5. Construction of $H$.
  6. Bounding the error of $H$.
  7. Bounding the size of $H$.
  8. Acknowledgements

Key Result

theorem 1

For any $\epsilon > 0$, every $n$-vertex graph has a $+O(n^{\frac{1}{3 + \sqrt{5}}+ \epsilon})$ additive emulator on $O_{\epsilon}(n)$ edges.

Figures (2)

  • Figure 1: The recursive procedure for the improved emulator upper bounds.
  • Figure 2: The solid black line denotes path $\pi$ in $G$. After edge $(x, y)$ is added to $H$ in the greedy phase, all pairs of vertices $(s', t')$ in $S \times T$ become connected in $H$ for the first time.

Theorems & Definitions (11)

  • Definition 1
  • theorem 1
  • proof : Proof sketch
  • Lemma 1: Lemma 13 of BV16
  • Lemma 2
  • proof
  • proposition 1
  • proof
  • proposition 2
  • proof
  • ...and 1 more