arXiv subjects

Michael Elkin

Publications and source records attributed to Michael Elkin.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Parallel $(\Delta+1)$-Edge-Coloring

We study the $(\Delta+1)$-edge-coloring problem in the parallel $\left(\mathrm{PRAM}\right)$ model of computation. The celebrated Vizing's theorem [Viz64] states that every simple graph $G = (V,E)$ can be properly $(\Delta+1)$-edge-colored. In a seminal paper, Karloff and Shmoys [KS87] devised a parallel algorithm with time $O\left(\Delta^5\cdot\log n\cdot\left(\log^3 n+\Delta^2\right)\right)$ and $O(m\cdot\Delta)$ processors. This result was improved by Liang et al. [LSH96] to time $O\left(\Delta^{4.5}\cdot \log^3\Delta\cdot \log n + \Delta^4 \cdot\log^4 n\right)$ and $O\left(n\cdot\Delta^{3} +n^2\right)$ processors. [LSH96] claimed $O\left(\Delta^{3.5} \cdot\log^3\Delta\cdot \log n + \Delta^3\cdot \log^4 n\right)$ time, but we point out a flaw in their analysis, which once corrected, results in the above bound. We devise a faster parallel algorithm for this fundamental problem. Specifically, our algorithm uses $O\left(\Delta^4\cdot \log^4 n\right)$ time and $O(m\cdot \Delta)$ processors. Another variant of our algorithm requires $O\left(\Delta^{4+o(1)}\cdot\log^2 n\right)$ time, and $O\left(m\cdot\Delta\cdot\log n\cdot\log^{\delta}\Delta\right)$ processors, for an arbitrarily small $\delta>0$. We also devise a few other tradeoffs between the time and the number of processors, and devise an improved algorithm for graphs with small arboricity. On the way to these results, we also provide a very fast parallel algorithm for updating $(\Delta+1)$-edge-coloring. Our algorithm for this problem is dramatically faster and simpler than the previous state-of-the-art algorithm (due to [LSH96]) for this problem.

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arXiv2025

Spanning Tree Covers for Path-Separable Graphs: Trading Stretch for Size

Given a graph $G=(V,E)$, a collection $\mathcal{T}$ of spanning trees of $G$ is called a spanning tree cover of stretch $\alpha$ if for every $u,v\in V$ there is a tree $T_{uv}\in\mathcal{T}$, such that \[d_{T_{uv}}(u,v)\leq\alpha\cdot d_G(u,v)~.\] Spanning tree covers were introduced in the pioneering work of Gupta et al. [GKR04], that showed that $p$-path-separable graphs admit stretch-$3$ spanning tree covers with size $O(p\log n)$. Many subsequent papers focused on a relaxed notion of non-spanning tree covers, in which the trees are required to be dominating, but may use edges that do not belong to the graph. In particular, Bartal et al. [BFN22] devised a construction of non-spanning tree covers with stretch $1+\epsilon$ and size $O(p\cdot\frac{\log^2n}{\epsilon^2})$. Recently, for $K_r$-minor-free graphs, Chang et al. [CCL+23,CCL+24] devised a non-spanning tree cover with stretch $1+\epsilon$ and size $2^{\frac{1}{\epsilon}r^{O(r)}}$, and an exact spanning tree cover with size $r^{O(diam(G))}$. However, the problem of devising spanning tree covers with stretch smaller than $3$ and small size for general $p$-path-separable graphs remained open. We show that $p$-path-separable graphs admit spanning tree covers with stretch $1+\epsilon$ and size $O(p\cdot\frac{\log^2n}{\epsilon})$. Moreover, we demonstrate that one can trade stretch for size, and devise spanning tree covers with stretch $O(k\log\log p)$ and size $O(kp^{\frac{1}{k}}\cdot\log^{2}n)$ for strongly $p$-path-separable graphs. We also provide a tradeoff for weakly path-separable graphs. For $K_r$-minor-free graphs, we devise spanning tree covers with stretch $O(k\log\log r)$ and size $O(kr^{2+\frac{1}{k}}\cdot\log^{2}n)$. For such graphs, it is only known that $p=r^{4602}$. Thus, for $r=\Omega(\log n)$ and $k\geq2$, this size is much smaller than that of our tree cover of stretch $1+\epsilon$.

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arXiv2025

Faster Multi-Source Reachability and Approximate Distances via Shortcuts, Hopsets and Matrix Multiplication

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a subset $S \subseteq V$ of $|S| = n^{\sigma}$ (for some $0 \le \sigma \le 1$) designated sources, the $S \times V$ reachability problem is to compute the sets $\mathcal V_s$ of vertices reachable from $s$, for every $s \in S$. Naive centralized algorithms run BFS/DFS from each source in $O(m \cdot n^{\sigma})$ time or compute $G$'s transitive closure in $\hat O(n^{\omega})$ time, where $\omega \le 2.371552\ldots$ is the matrix multiplication exponent. Thus, the best known bound is $\hat O(n^{\min \{ 2 + \sigma, \omega\}})$. Leveraging shortcut constructions by Kogan and Parter [SODA 2022, ICALP 2022], we develop a centralized algorithm with running time $\hat O(n^{1 + \frac{2}{3} \omega(\sigma)})$, where $\omega(\sigma)$ is the rectangular matrix multiplication exponent. Using current estimates on $\omega(\sigma)$, our exponent improves upon $\min \{2 + \sigma, \omega \}$ for $\tilde \sigma \leq \sigma \leq 0.53$, where $1/3 < \tilde \sigma < 0.3336$ is a universal constant. In a classical result, Cohen [Journal of Algorithms, 1996] devised parallel algorithms for $S \times V$ reachability on graphs admitting balanced recursive separators of size $n^{\rho}$ for $\rho < 1$, requiring polylogarithmic time and work $n^{\max \{\omega \rho, 2\rho + \sigma \} + o(1)}$. We significantly improve, extend, and generalize Cohen's result. First, our parallel algorithm for graphs with small recursive separators has lower work complexity than Cohen's in boraod paramater ranges. Second, we generalize our algorithm to graphs of treewidth at most $n^{\rho}$ ($\rho < 1$) and provide a centralized algorithm that outperforms existing bounds for $S \times V$ reachability on such graphs. We also do this for some other graph familes with small separators. Finally, we extend these results to $(1 + \epsilon)$-approximate distance computation.

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arXiv2024

Deterministic Simple $(\Delta+\varepsilon\alpha)$-Edge-Coloring in Near-Linear Time

We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(\Delta+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{\Delta\cdot\log n,\sqrt{n}\right\}\right)$, and very recently to randomized $\tilde{O}\left(m\cdot n^{1/3}\right)$. A randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm can be computed in $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$ time, and for large values of $\Delta$, this task requires randomized $O\left(\frac{m\cdot\log\varepsilon^{-1}}{\varepsilon^2}\right)$ time. It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. A randomized variant of our algorithm has running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$. We also study edge-coloring of graphs with arboricity at most $\alpha$. A randomized computation of $(\Delta+1)$-edge-coloring requires $\tilde{O}\left(\min\{m\cdot\sqrt{n},m\cdot\Delta\}\cdot\frac{\alpha}{\Delta}\right)$ time. Deterministically, this task can be done in $O\left(m\cdot\alpha^7\cdot\log n\right)$ time. However, for large values of $\alpha$, these algorithms require super-linear time. We devise a deterministic $(\Delta+\varepsilon\alpha)$-edge-coloring algorithm with running time $O\left(\frac{m\cdot\log n}{\varepsilon^7}\right)$. A randomized version of our algorithm requires $O\left(\frac{m\cdot\log n}{\varepsilon}\right)$ expected time. Our algorithm is based on a novel two-way degree-splitting, which we devise in this paper. We believe that this technique is of independent interest.

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arXiv2024

Faster Multi-Source Directed Reachability via Shortcuts and Matrix Multiplication

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a set $S \subseteq V$, $|S| = n^{\sigma}$ (for some $0 < \sigma \le 1$) of designated sources, the $S \times V$-direachability problem is to compute for every $s \in S$, the set of all the vertices reachable from $s$ in $G$. Known naive algorithms for this problem either run a BFS/DFS separately from every source, and as a result require $O(m \cdot n^{\sigma})$ time, or compute the transitive closure of $G$ in $\tilde O(n^{\omega})$ time, where $\omega < 2.371552\ldots$ is the matrix multiplication exponent. Hence, the current state-of-the-art bound for the problem on graphs with $m = \Theta(n^{\mu})$ edges in $\tilde O(n^{\min \{\mu + \sigma, \omega \}})$. Our first contribution is an algorithm with running time $\tilde O(n^{1 + \tiny{\frac{2}{3}} \omega(\sigma)})$ for this problem, where $\omega(\sigma)$ is the rectangular matrix multiplication exponent. Using current state-of-the-art estimates on $\omega(\sigma)$, our exponent is better than $\min \{2 + \sigma, \omega \}$ for $\tilde \sigma \le \sigma \le 0.53$, where $1/3 < \tilde \sigma < 0.3336$ is a universal constant. Our second contribution is a sequence of algorithms $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$ for the $S \times V$-direachability problem. We argue that under a certain assumption that we introduce, for every $\tilde \sigma \le \sigma < 1$, there exists a sufficiently large index $k = k(\sigma)$ so that $\mathcal A_k$ improves upon the current state-of-the-art bounds for $S \times V$-direachability with $|S| = n^{\sigma}$, in the densest regime $\mu =2$. We show that to prove this assumption, it is sufficient to devise an algorithm that computes a rectangular max-min matrix product roughly as efficiently as ordinary $(+, \cdot)$ matrix product. Our algorithms heavily exploit recent constructions of directed shortcuts by Kogan and Parter.

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arXiv2023

Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n loglog n)

Given an $n$-vertex undirected graph $G=(V,E,w)$, and a parameter $k\geq1$, a path-reporting distance oracle (or PRDO) is a data structure of size $S(n,k)$, that given a query $(u,v)\in V^2$, returns an $f(k)$-approximate shortest $u-v$ path $P$ in $G$ within time $q(k)+O(|P|)$. Here $S(n,k)$, $f(k)$ and $q(k)$ are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has $S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(\log k)$. The size of this oracle is $\Omega(n\log n)$ for all $k$. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal $2k-1$. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with $S(n,k)=O(n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(1)$. In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size $S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$, stretch $f(k)=O(k)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. We can also have size $O(n^{1+\frac{1}{k}})$, stretch $O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any $\epsilon>0$, any $k=1,2,...$, and any graph $G$ and a collection $\mathcal{P}$ of $p$ vertex pairs, there exists a $(1+\epsilon)$-approximate preserver with $O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ edges, where $\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.

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arXiv2022

Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates

This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in $\textrm{polylog}(n)$ rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only $n+o(n)$ edges in which the pairwise distances are stretched by a factor of at most $O(\log n \;\cdot\; 2^{O(\log^* n)})$. We provide a $\textrm{polylog}(n)$-round deterministic distributed algorithm that computes a spanner with stretch $(2k-1)$ and $O(nk + n^{1 + 1/k} \log k)$ edges in unweighted graphs and with $O(n^{1 + 1/k} k)$ edges in weighted graphs. We present the first $\textrm{polylog}(n)$-round randomized distributed algorithm that computes a sparse connectivity certificate. For an $n$-node graph $G$, a certificate for connectivity $k$ is a spanning subgraph $H$ that is $k$-edge-connected if and only if $G$ is $k$-edge-connected, and this subgraph $H$ is called sparse if it has $O(nk)$ edges. Our algorithm achieves a sparsity of $(1 + o(1))nk$ edges, which is within a $2(1 + o(1))$ factor of the best possible.

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arXiv2022

Deterministic Low-Diameter Decompositions for Weighted Graphs and Distributed and Parallel Applications

This paper presents new deterministic and distributed low-diameter decomposition algorithms for weighted graphs. In particular, we show that if one can efficiently compute approximate distances in a parallel or a distributed setting, one can also efficiently compute low-diameter decompositions. This consequently implies solutions to many fundamental distance based problems using a polylogarithmic number of approximate distance computations. Our low-diameter decomposition generalizes and extends the line of work starting from [Rozhoň, Ghaffari STOC 2020] to weighted graphs in a very model-independent manner. Moreover, our clustering results have additional useful properties, including strong-diameter guarantees, separation properties, restricting cluster centers to specified terminals, and more. Applications include: -- The first near-linear work and polylogarithmic depth randomized and deterministic parallel algorithm for low-stretch spanning trees (LSST) with polylogarithmic stretch. Previously, the best parallel LSST algorithm required $m \cdot n^{o(1)}$ work and $n^{o(1)}$ depth and was inherently randomized. No deterministic LSST algorithm with truly sub-quadratic work and sub-linear depth was known. -- The first near-linear work and polylogarithmic depth deterministic algorithm for computing an $\ell_1$-embedding into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for $\ell_1$-embeddings either require large polynomial work or are inherently sequential. Even when we apply our techniques to the classical problem of computing a ball-carving with strong-diameter $O(\log^2 n)$ in an unweighted graph, our new clustering algorithm still leads to an improvement in round complexity from $O(\log^{10} n)$ rounds [Chang, Ghaffari PODC 21] to $O(\log^{4} n)$.

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arXiv2022

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 $1+ε$, for an arbitrarily small constant $ε> 0$ . In this regime existing centralized algorithms require $Ω(\min\{|E|s,n^ω\})$ time, where $ω< 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work $Ω(\min\{|E|s,n^ω\})$. Our centralized algorithm has running time $O((m+ ns)n^ρ)$, and its PRAM counterpart has polylogarithmic depth and work $O((m + ns)n^ρ)$, for an arbitrarily small constant $ρ> 0$. For a pair $(s,v) \in S\times V$, it provides a path of length $\hat{d}(s,v)$ that satisfies $\hat{d}(s,v) \le (1+ε)d_G(s,v) + β\cdot W(s,v)$, 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 $β= (1/ρ)^{O(1/ρ)}$. We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter $κ= 1,2,\ldots$, this algorithm provides for {\em unweighted} graphs a purely additive approximation of $2(κ-1)$ for {\em all pairs shortest paths} (APASP) in time $\tilde{O}(n^{2+1/κ})$. Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of $2(κ- 1) W(u,v)$, 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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arXiv2022

$(1+ε)$-Approximate Shortest Paths in Dynamic Streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied during the last decade. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of $2κ-1$ versus $n^{1+1/κ}$, for an integer parameter $κ$. (In fact, existing solutions also incur an extra factor of $1+ε$ in the stretch for weighted graphs, and an additional factor of $\log^{O(1)}n$ in the space.) The only existing solution of the second type uses $n^{1/2 - O(1/κ)}$ passes over the stream (for space $O(n^{1+1/κ})$), and applies only to unweighted graphs. In this paper we show that $(1+ε)$-approximate single-source shortest paths can be computed in this setting with $\tilde{O}(n^{1+1/κ})$ space using just \emph{constantly} many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs (assuming $ε$ and $κ$ are constant). Moreover, in fact, the same result applies for multi-source shortest paths, as long as the number of sources is $O(n^{1/κ})$. We achieve these results by devising efficient dynamic streaming constructions of $(1 + ε, β)$-spanners and hopsets. We believe that these constructions are of independent interest.

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arXiv2021

Locally-Iterative Distributed (Delta + 1)-Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models

We consider graph coloring and related problems in the distributed message-passing model. {Locally-iterative algorithms} are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Delta+1)-coloring algorithm requires Omega(Delta log Delta + log^* n) rounds, unless there is "a very special type of coloring that can be very efficiently reduced" \cite{SV93}. In this paper we obtain this special type of coloring. Specifically, we devise a locally-iterative (Delta+1)-coloring algorithm with running time O(Delta + log^* n), i.e., {below} Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings: - We obtain self-stabilizing distributed algorithms for (Delta+1)-vertex-coloring, (2Delta-1)-edge-coloring, maximal independent set and maximal matching with O(Delta+log^* n) time. This significantly improves previously-known results that have O(n) or larger running times \cite{GK10}. - We devise a (2Delta-1)-edge-coloring algorithm in the CONGEST model with O(Delta + log^* n) time and in the Bit-Round model with O(Delta + log n) time. Previously-known algorithms had superlinear dependency on Delta for (2Delta-1)-edge-coloring in these models. - We obtain an arbdefective coloring algorithm with running time O(\sqrt Delta + log^* n). We employ it in order to compute proper colorings that improve the recent state-of-the-art bounds of Barenboim from PODC'15 \cite{B15} and Fraigniaud et al. from FOCS'16 \cite{FHK16} by polylogarithmic factors. - Our algorithms are applicable to the SET-LOCAL model of \cite{HKMS15}.

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arXiv2021

Ultra-Sparse Near-Additive Emulators

Near-additive (aka $(1+ε,β)$-) 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 $β$. Specifically, for any pair of parameters $ε>0$, $ κ=1,2,\dots$, one can have a $(1+ε,β)$-emulator with $O(n^{1+1/κ})$ edges, with $β= \left(\frac{\log κ}ε\right)^{\log κ}$. 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 $n^{1+1/κ}$ edges suffice. In particular, our emulators can be \emph{ultra-sparse}, i.e., we can have an emulator with $n+o(n)$ edges and $β= \left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}(1+o(1))}$. We also devise a distributed deterministic algorithm in the CONGEST model that builds these emulators in low polynomial time (i.e., in $O(n^ρ)$ time, for an arbitrarily small constant parameter $ρ>0$). Finally, we also improve the state-of-the-art distributed deterministic \congest-model construction of $(1+ε,β)$-spanners devised in the PODC'19 paper [ElkinM19]. Specifically, the spanners of [ElkinM19] have $O(β\cdot n^{1+1/κ})$ edges, i.e., at their sparsest they employ $ O\left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}}\cdot n$ edges. In this paper, we devise an efficient distributed deterministic CONGEST-model algorithm that builds such spanners with $O(n^{1+1/κ})$ edges for $κ= O\left(\frac{\log n}{\log ^{(3)}n}\right)$. At their sparsest, these spanners employ only $O(n\cdot {\log {\log n}})$ edges.

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arXiv2021

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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arXiv2021

Centralized, Parallel, and Distributed Multi-Source Shortest Paths via Hopsets and Rectangular Matrix Multiplication

Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+ε)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. We devise a significantly improved algorithm for this problem in the entire range of parameter $r$, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of $r$ in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time $\tilde{O}(|E| \cdot n^{o(1)} + n^{ω(r)})$, where $n^{ω(r)}$ is the time required to multiply an $n^r \times n$ matrix by an $n \times n$ one. Our PRAM algorithm has polylogarithmic time $(\log n)^{O(1/ρ)}$, and its work complexity is $\tilde{O}(|E| \cdot n^ρ+ n^{ω(r)})$, for any arbitrarily small constant $ρ>0$. In particular, for $r \le 0.313\ldots$, our centralized algorithm computes $S \times V$ $(1+ε)$-approximate shortest paths in $n^{2 + o(1)}$ time. Our PRAM polylogarithmic-time algorithm has work complexity $O(|E| \cdot n^ρ+ n^{2+o(1)})$, for any arbitrarily small constant $ρ>0$. Previously existing solutions either require centralized time/parallel work of $O(|E| \cdot |S|)$ or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for $|S| = n^r$ sources, for $r \le 0.655$, while previous state-of-the-art algorithms did so only for $r \le 1/2$. Moreover, it improves previous bounds for all $r > 1/2$. For unweighted graphs, the running time is improved further to $poly(\log\log n)$.

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arXiv2020

Near-Additive Spanners and Near-Exact Hopsets, A Unified View

Given an {\em unweighted} undirected graph $G = (V,E)$, and a pair of parameters $ε> 0$, $β= 1,2,\ldots$, a subgraph $G' =(V,H)$, $H \subseteq E$, of $G$ is a {\em $(1+ε,β)$-spanner} (aka, a {\em near-additive spanner}) of $G$ if for every $u,v \in V$, $$d_{G'}(u,v) \le (1+ε)d_G(u,v) + β~.$$ It was shown in \cite{EP01} that for any $n$-vertex $G$ as above, and any $ε> 0$ and $κ= 1,2,\ldots$, there exists a $(1+ε,β)$-spanner $G'$ with $O_{ε,κ}(n^{1+1/κ})$ edges, with $$β= β_{EP} = \left({{\log κ} \over ε}\right)^{\log κ- 2}~.$$ This bound remains state-of-the-art, and its dependence on $ε$ (for the case of small $κ$) was shown to be tight in \cite{ABP18}. Given a {\em weighted} undirected graph $G = (V,E,ω)$, and a pair of parameters $ε> 0$, $β= 1,2,\ldots$, a graph $G'= (V,H,ω')$ is a {\em $(1+ε,β)$-hopset} (aka, a {\em near-exact hopset}) of $G$ if for every $u,v \in V$, $$d_G(u,v) \le d_{G\cup G'}^{(β)}(u,v) \le (1+ε)d_G(u,v)~,$$ where $ d_{G\cup G'}^{(β)}(u,v)$ stands for a $β$-(hop)-bounded distance between $u$ and $v$ in the union graph $G \cup G'$. It was shown in \cite{EN16} that for any $n$-vertex $G$ and $ε$ and $κ$ as above, there exists a $(1+ε,β)$-hopset with $\tilde{O}(n^{1+1/κ})$ edges, with $β= β_{EP}$. Not only the two results of \cite{EP01} and \cite{EN16} are strikingly similar, but so are also their proof techniques. Moreover, Thorup-Zwick's later construction of near-additive spanners \cite{TZ06} was also shown in \cite{EN19,HP17} to provide hopsets with analogous (to that of \cite{TZ06}) properties. In this survey we explore this intriguing phenomenon, sketch the basic proof techniques used for these results, and highlight open questions.

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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.

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arXiv2019

Lossless Prioritized Embeddings

Given metric spaces $(X,d)$ and $(Y,ρ)$ and an ordering $x_1,x_2,\ldots,x_n$ of $(X,d)$, an embedding $f: X \rightarrow Y$ is said to have a prioritized distortion $α(\cdot)$, if for any pair $x_j,x'$ of distinct points in $X$, the distortion provided by $f$ for this pair is at most $α(j)$. If $Y$ is a normed space, the embedding is said to have prioritized dimension $β(\cdot)$, if $f(x_j)$ may have nonzero entries only in its first $β(j)$ coordinates. The notion of prioritized embedding was introduced by \cite{EFN15}, where a general methodology for constructing such embeddings was developed. Though this methodology enables \cite{EFN15} to come up with many prioritized embeddings, it typically incurs some loss in the distortion. This loss is problematic for isometric embeddings. It is also troublesome for Matousek's embedding of general metrics into $\ell_\infty$, which for a parameter $k = 1,2,\ldots$, provides distortion $2k-1$ and dimension $O(k \log n \cdot n^{1/k})$. In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into $\ell_\infty$ with dimension $O(\log j)$. The second one is a prioritized Matousek's embedding of general metrics into $\ell_\infty$, which provides prioritized distortion $2 \lceil k {{\log j} \over {\log n}} \rceil - 1$ and dimension $O(k \log n \cdot n^{1/k})$, again matching the worst-case guarantee $2k-1$ in the distortion of the classical Matousek's embedding. We also provide a dimension-prioritized variant of Matousek's embedding. Finally, we devise prioritized embeddings of general metrics into (single) ultra-metric and of general graphs into (single) spanning tree with asymptotically optimal distortion.

cs.DS↗

arXiv2019

Distributed Construction of Light Networks

A $t$-{\em spanner} $H$ of a weighted graph $G=(V,E,w)$ is a subgraph that approximates all pairwise distances up to a factor of $t$. The {\em lightness} of $H$ is defined as the ratio between the weight of $H$ to that of the minimum spanning tree. An $(α,β)$-{\em Shallow Light Tree} (SLT) is a tree of lightness $β$, that approximates all distances from a designated root vertex up to a factor of $α$. A long line of works resulted in efficient algorithms that produce (nearly) optimal light spanners and SLTs. Some of the most notable algorithmic applications of light spanners and SLTs are in distributed settings. Surprisingly, so far there are no known efficient distributed algorithms for constructing these objects in general graphs. In this paper we devise efficient distributed algorithms in the CONGEST model for constructing light spanners and SLTs, with near optimal parameters. Specifically, for any $k\ge 1$ and $0<ε<1$, we show a $(2k-1)\cdot(1+ε)$-spanner with lightness $O(k\cdot n^{1/k})$ can be built in $\tilde{O}\left(n^{\frac12+\frac{1}{4k+2}}+D\right)$ rounds (where $n=|V|$ and $D$ is the hop-diameter of $G$). In addition, for any $α>1$ we provide an $(α,1+\frac{O(1)}{α-1})$-SLT in $(\sqrt{n}+D)\cdot n^{o(1)}$ rounds. The running time of our algorithms cannot be substantially improved. We also consider spanners for the family of doubling graphs, and devise a $(\sqrt{n}+D)\cdot n^{o(1)}$ rounds algorithm in the CONGEST model that computes a $(1+ε)$-spanner with lightness $(\log n)/ε^{O(1)}$. As a stepping stone, which is interesting in its own right, we first develop a distributed algorithm for constructing nets (for arbitrary weighted graphs), generalizing previous algorithms that worked only for unweighted graphs.

cs.DS↗