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Lightweight Near-Additive Spanners

Yuval Gitlitz, Ofer Neiman, Richard Spence

TL;DR

This paper focuses on near-additive spanners, where $\alpha=1+\varepsilon$ for arbitrarily small $\varepsilon>0$ and shows the first construction of {\em light} spanners in this setting.

Abstract

An $(α,β)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\leα\cdot d_G(u,v)+β$. The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of $H$ to the weight of a minimum spanning tree). In this paper we focus on near-additive spanners, where $α=1+\varepsilon$ for arbitrarily small $\varepsilon>0$. We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter $k\ge 1$, we obtain an $(1+\varepsilon,O(k/\varepsilon)^k\cdot W(\cdot,\cdot))$-spanner with lightness $\tilde{O}(n^{1/k})$ (where $W(\cdot,\cdot)$ indicates for every pair $u, v \in V$ the heaviest edge in some shortest path between $u,v$). In addition, we can also bound the number of edges in our spanner by $O(kn^{1+3/k})$.

Lightweight Near-Additive Spanners

TL;DR

This paper focuses on near-additive spanners, where for arbitrarily small and shows the first construction of {\em light} spanners in this setting.

Abstract

An -spanner of a weighted graph , is a subgraph such that for every , . The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of to the weight of a minimum spanning tree). In this paper we focus on near-additive spanners, where for arbitrarily small . We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter , we obtain an -spanner with lightness (where indicates for every pair the heaviest edge in some shortest path between ). In addition, we can also bound the number of edges in our spanner by .

Paper Structure

This paper contains 14 sections, 14 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. Related Work
  5. Preliminaries
  6. A lightweight near-additive spanner
  7. Constructing $H_0$.
  8. Adding paths to representatives.
  9. Stretch of the spanner
  10. Lightness of the spanner
  11. Size of the spanner
  12. Proof of Theorem \ref{['thm:spanner-near-additive']}
  13. Proof of \ref{['lemma:distance-long']}
  14. A sparse and lightweight -spanner

Key Result

Theorem 1

Let $G$ be a weighted graph, let $0 < \varepsilon < 1$, and let $k$ be a positive integer. Then $G$ has a $\left(1+\varepsilon,O(\frac{k}{\varepsilon})^{k} \cdot W(\cdot,\cdot)\right)$-spanner of size $O(kn^{1 + 3/k})$ and lightness $\widetilde{O}(\frac{n^{1/k}}{\varepsilon})$.

Figures (2)

  • Figure 1: Illustration of the paths added during the three phases. The dotted and solid circles represent the $\frac{1-\varepsilon}{2}$-bunch, and the 1-bunch of $u \in A_i \setminus A_{i+1}$, respectively. The first phase connects $v$ to its representative $r_j(v)$, the second phase connects $u$ to $r_j(v)$, which is one of the representatives of its $\frac{1-\varepsilon}{2}$-bunch, and the third phase connects $u$ to $p'_{i+1}(u)$, the approximate pivot, via the SLT rooted at $s_{i+1}$.
  • Figure 2: Illustration of Lemma \ref{['lemma:half-bunch-in-bunch']}. The dotted circles represent the $\frac{1-\varepsilon}{2}$-bunches centered at $u_1$, $u_2$, and $u_3$ respectively. Lemma \ref{['lemma:half-bunch-in-bunch']} states that all the $u_i$'s are contained in a 1-bunch of some vertex in $A_i$.

Theorems & Definitions (27)

  • Theorem 1
  • Definition 1: Shallow-light tree (SLT)
  • Theorem 2: khuller1995balancing
  • Definition 2: $\Delta$-net
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 17 more