arXiv subjects

Yuval Gitlitz

Publications and source records attributed to Yuval Gitlitz.

4 recordsLinked to original sources

arXiv2024

Lightweight Near-Additive Spanners

An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. 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 $\alpha=1+\varepsilon$ 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 $O(kn^{1+3/k})$.

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arXiv2023

Light, Reliable Spanners

A \emph{$\nu$-reliable spanner} of a metric space $(X,d)$, is a (dominating) graph $H$, such that for any possible failure set $B\subseteq X$, there is a set $B^+$ just slightly larger $|B^+|\le(1+\nu)\cdot|B|$, and all distances between pairs in $X\setminus B^+$ are (approximately) preserved in $H\setminus B$. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of \emph{light} reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of $X$. We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an \emph{oblivious} reliable spanner is a distribution over spanners, and the bound on $|B^+|$ holds in expectation. We devise an oblivious $\nu$-reliable $(2+\frac{2}{k-1})$-spanner for any $k$-HST, whose lightness is $\approx \nu^{-2}$. We demonstrate a matching $\Omega(\nu^{-2})$ lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness. For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct {\em light} tree covers, in which every tree is a $k$-HST of low weight. Combining these covers with our results for $k$-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious $\nu$-reliable $(1+\varepsilon)$-spanner with lightness $\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(\nu^{-2}\cdot\log n)$, which is best possible (up to lower order terms).

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arXiv2020

Improved Weighted Additive Spanners

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently [ABSKS20] extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $O(n^{3/2})$ (resp., $O(n^{7/5})$) to the weighted setting, where the additive error is $+2W$ (resp., $+4W$). This means that for every pair $u,v$, the additive stretch is at most $+2W_{u,v}$, where $W_{u,v}$ is the maximal edge weight on the shortest $u-v$ path. In addition, [ABSKS20] showed an algorithm yielding a $+8W_{max}$ spanner of size $O(n^{4/3})$, here $W_{max}$ is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a $+(6+\varepsilon)W$ spanner for weighted graphs with size $O(n^{4/3})$ (for any constant $\varepsilon>0$), thus nearly matching the classical +6 spanner of size $O(n^{4/3})$ for unweighted graphs. Furthermore, we show a $+(2+\varepsilon)W$ subsetwise spanner of size $O(n\cdot\sqrt{|S|})$, improving the $+4W_{max}$ result of [ABSKS20] (that had the same size). We also show a simple randomized algorithm for a $+4W$ emulator of size $\tilde{O}(n^{4/3})$. In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size $+\tilde{O}(\sqrt{n}\cdot W)$ spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of [DHZ00] for unweighted graphs, we devise an efficient randomized algorithm producing a $+2W$ spanner for weighted graphs of size $\tilde{O}(n^{3/2})$ in $\tilde{O}(n^2)$ time.

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arXiv2019

Almost Shortest Paths with Near-Additive Error in Weighted Graphs

Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+\epsilon$, for an arbitrarily small constant $\epsilon > 0$ . In this regime existing centralized algorithms require $\Omega(\min\{|E|s,n^\omega\})$ time, where $\omega < 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work $\Omega(\min\{|E|s,n^\omega\})$. Our centralized algorithm has running time $O((m+ ns)n^\rho)$, and its PRAM counterpart has polylogarithmic depth and work $O((m + ns)n^\rho)$, for an arbitrarily small constant $\rho > 0$. For a pair $(s,v) \in S\times V$, it provides a path of length $\hat{d}(s,v)$ that satisfies $\hat{d}(s,v) \le (1+\epsilon)d_G(s,v) + \beta \cdot W(s,v)$, where $W(s,v)$ is the weight of the heaviest edge on some shortest $s-v$ path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our $\beta = (1/\rho)^{O(1/\rho)}$. We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter $\kappa = 1,2,\ldots$, this algorithm provides for {\em unweighted} graphs a purely additive approximation of $2(\kappa -1)$ for {\em all pairs shortest paths} (APASP) in time $\tilde{O}(n^{2+1/\kappa})$. Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of $2(\kappa - 1) W(u,v)$, for every vertex pair $(u,v) \in {V \choose 2}$, with $W(u,v)$ defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets.

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