arXiv subjects

Shaked Matar

Publications and source records attributed to Shaked Matar.

4 recordsLinked to original sources

arXiv2024

Massively Parallel Algorithms for Approximate Shortest Paths

We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take $poly(\log{\log{n}})$ rounds in the near-linear memory MPC model. Our results are for unweighted undirected graphs with $n$ vertices and $m$ edges. Our first contribution is a $(1+\epsilon)$-approximation algorithm for Single-Source Shortest Paths (SSSP) that takes $poly(\log{\log{n}})$ rounds in the near-linear MPC model, where the memory per machine is $\tilde{O}(n)$ and the total memory is $\tilde{O}(mn^{\rho})$, where $\rho$ is a small constant. Our second contribution is a distance oracle that allows to approximate the distance between any pair of vertices. The distance oracle is constructed in $poly(\log{\log{n}})$ rounds and allows to query a $(1+\epsilon)(2k-1)$-approximate distance between any pair of vertices $u$ and $v$ in $O(1)$ additional rounds. The algorithm is for the near-linear memory MPC model with total memory of size $\tilde{O}((m+n^{1+\rho})n^{1/k})$, where $\rho$ is a small constant. While our algorithms are for the near-linear MPC model, in fact they only use one machine with $\tilde{O}(n)$ memory, where the rest of machines can have sublinear memory of size $O(n^{\gamma})$ for a small constant $\gamma < 1$. All previous algorithms for approximate shortest paths in the near-linear MPC model either required $\Omega(\log{n})$ rounds or had an $\Omega(\log{n})$ approximation. Our approach is based on fast construction of near-additive emulators, limited-scale hopsets and limited-scale distance sketches that are tailored for the MPC model. While our end-results are for the near-linear MPC model, many of the tools we construct such as hopsets and emulators are constructed in the more restricted sublinear MPC model.

cs.DS↗

arXiv2021

Ultra-Sparse Near-Additive Emulators

Near-additive (aka $(1+\epsilon,\beta)$-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings. Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch $\beta$. Specifically, for any pair of parameters $\epsilon >0$, $ \kappa=1,2,\dots$, one can have a $(1+\epsilon,\beta)$-emulator with $O(n^{1+1/\kappa})$ edges, with $\beta = \left(\frac{\log \kappa}{\epsilon}\right)^{\log \kappa}$. At their sparsest, these emulators employ $c\cdot n$ edges, for some constant $c\geq 2$. We tighten this bound, and show that in fact precisely $n^{1+1/\kappa}$ edges suffice. In particular, our emulators can be \emph{ultra-sparse}, i.e., we can have an emulator with $n+o(n)$ edges and $\beta = \left(\frac{\log {\log n}}{\epsilon }\right)^{{\log {\log n}}(1+o(1))}$. We also devise a distributed deterministic algorithm in the CONGEST model that builds these emulators in low polynomial time (i.e., in $O(n^\rho)$ time, for an arbitrarily small constant parameter $\rho >0$). Finally, we also improve the state-of-the-art distributed deterministic \congest-model construction of $(1+\epsilon,\beta)$-spanners devised in the PODC'19 paper [ElkinM19]. Specifically, the spanners of [ElkinM19] have $O(\beta\cdot n^{1+1/\kappa})$ edges, i.e., at their sparsest they employ $ O\left(\frac{\log {\log n}}{\epsilon }\right)^{{\log {\log n}}}\cdot n$ edges. In this paper, we devise an efficient distributed deterministic CONGEST-model algorithm that builds such spanners with $O(n^{1+1/\kappa})$ edges for $\kappa = O\left(\frac{\log n}{\log ^{(3)}n}\right)$. At their sparsest, these spanners employ only $O(n\cdot {\log {\log n}})$ edges.

cs.DS↗

arXiv2019

Fast Deterministic Constructions of Linear-Size Spanners and Skeletons

In the distributed setting, the only existing constructions of \textit{sparse skeletons}, (i.e., subgraphs with $O(n)$ edges) either use randomization or large messages, or require $Ω(D)$ time, where $D$ is the hop-diameter of the input graph $G$. We devise the first deterministic distributed algorithm in the CONGEST model (i.e., uses small messages) for constructing linear-size skeletons in time $2^{O(\sqrt{{\log n}\cdot{\log{\log n}}})}$. We can also compute a linear-size spanner with stretch $polylog(n)$ in low deterministic polynomial time, i.e., $O(n^ρ)$ for an arbitrarily small constant $ρ>0$, in the CONGEST model. Yet another algorithm that we devise runs in $O({\log n})^{κ-1}$ time, for a parameter $κ=1,2,\dots,$ and constructs an $O({\log n})^{κ-1}$ spanner with $O(n^{1+1/κ})$ edges. All our distributed algorithms are lightweight from the computational perspective, i.e., none of them employs any heavy computations.

cs.DC↗

arXiv2019

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Given parameters $α\geq 1,β\geq 0$, a subgraph $G'=(V,H)$ of an $n$-vertex unweighted undirected graph $G=(V,E)$ is called an $(α,β)$-spanner if for every pair $u,v\in V$ of vertices, $d_{G'}(u,v)\leq αd_{G}(u,v)+β$. If $β=0$ the spanner is called a multiplicative $α$-spanner, and if $α= 1+ε$, for an arbitrarily small $ε>0$, the spanner is said to be a near-additive one. Graph spanners are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Near-additive spanners, introduced in [EP01], preserve large distances much more faithfully than multiplicative spanners. Also, recent lower bounds [AB15] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with $α=1$), and therefore near-additive spanners provide the best approximation of distances that one can hope for. Numerous distributed algorithms for constructing sparse near-additive spanners exist. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [EN17], and also there are efficient deterministic algorithms in the LOCAL model [DGPV09]. The only known deterministic CONGEST-model algorithm for the problem [Elk01] requires superlinear time in $n$. We remedy the situation and devise an efficient deterministic CONGEST-model algorithm for constructing arbitrarily sparse near-additive spanners. The running time of our algorithm is low polynomial, i.e., roughly $O(β\cdot n^ρ)$, where $ρ> 0$ is an arbitrarily small positive constant that affects the additive term $β$. In general, the parameters of our algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm for the problem due to [EN17].

cs.DC↗