arXiv subjects

Merav Parter

Publications and source records attributed to Merav Parter.

At least 37 records · Page 2Linked to original sources

arXiv2022

Having Hope in Hops: New Spanners, Preservers and Lower Bounds for Hopsets

Hopsets and spanners are fundamental graph structures, playing a key role in shortest path computation, distributed communication, and more. A (near-exact) hopset for a given graph $G$ is a (small) subset of weighted edges $H$ that when added to the graph $G$ reduces the number of hops (edges) of near-exact shortest paths. Spanners and distance preservers, on the other hand, ask for removing many edges from the graph while approximately preserving shortest path distances. We provide a general reduction scheme from graph hopsets to the known metric compression schemes of spanners, emulators and distance preservers. Consequently, we get new and improved upper bound constructions for the latter, as well as, new lower bound results for hopsets. Our work makes a significant progress on the tantalizing open problem concerning the formal connection between hopsets and spanners, e.g., as posed by Elkin and Neiman [Bull. EATCS 2020].

cs.DS↗

arXiv2022

Õptimal Vertex Fault-Tolerant Spanners in Õptimal Time: Sequential, Distributed and Parallel

We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal $f$-VFT spanners for any $n$-vertex $m$-edge graph, with near optimal running time in several computational models: - A randomized sequential algorithm with a runtime of $\widetilde{O}(m)$ (i.e., independent in the number of faults $f$). The state-of-the-art time bound is $\widetilde{O}(f^{1-1/k}\cdot n^{2+1/k}+f^2 m)$ by [Bodwin, Dinitz and Robelle, SODA 2021]. - A distributed congest algorithm of $\widetilde{O}(1)$ rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in $\widetilde{O}(f^{2})$ rounds. - A PRAM (CRCW) algorithm with $\widetilde{O}(m)$ work and $\widetilde{O}(1)$ depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using $\widetilde{O}(f^3m)$ work and $\widetilde{O}(f^3)$ depth. An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal $λ$-\emph{vertex} connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of $\widetilde{O}(1)$ depth and $O(λm)$ work, by [Karger and Motwani, STOC'93].

cs.DS↗

arXiv2022

Õptimal Dual Vertex Failure Connectivity Labels

In this paper we present succinct labeling schemes for supporting connectivity queries under vertex faults. For a given $n$-vertex graph $G$, an $f$-VFT (resp., EFT) connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of a vertex pair $u$ and $v$, and the labels of at most $f$ failing vertices (resp., edges) $F$, one can determine if $u$ and $v$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], FT labeling schemes have been devised only for a limited collection of graph families. A recent work [Dory and Parter, PODC 2021] provided EFT labeling schemes for general graphs under edge failures, leaving the vertex failure case fairly open. We provide the first sublinear $f$-VFT labeling schemes for $f \geq 2$ for any $n$-vertex graph. Our key result is $2$-VFT connectivity labels with $O(\log^3 n)$ bits. Our constructions are based on analyzing the structure of dual failure replacement paths on top of the well-known heavy-light tree decomposition technique of [Sleator and Tarjan, STOC 1981]. We also provide $f$-VFT labels with sub-linear length (in $|V|$) for any $f=o(\log\log n)$, that are based on a reduction to the existing EFT labels.

cs.DS↗

arXiv2022

Broadcast CONGEST Algorithms against Adversarial Edges

We consider the corner-stone broadcast task with an adaptive adversary that controls a fixed number of $t$ edges in the input communication graph. In this model, the adversary sees the entire communication in the network and the random coins of the nodes, while maliciously manipulating the messages sent through a set of $t$ edges (unknown to the nodes). Since the influential work of [Pease, Shostak and Lamport, JACM'80], broadcast algorithms against plentiful adversarial models have been studied in both theory and practice for over more than four decades. Despite this extensive research, there is no round efficient broadcast algorithm for general graphs in the CONGEST model of distributed computing. We provide the first round-efficient broadcast algorithms against adaptive edge adversaries. Our two key results for $n$-node graphs of diameter $D$ are as follows: 1. For $t=1$, there is a deterministic algorithm that solves the problem within $\widetilde{O}(D^2)$ rounds, provided that the graph is 3 edge-connected. This round complexity beats the natural barrier of $O(D^3)$ rounds, the existential lower bound on the maximal length of $3$ edge-disjoint paths between a given pair of nodes in $G$. This algorithm can be extended to a $\widetilde{O}(D^{O(t)})$-round algorithm against $t$ adversarial edges in $(2t+1)$ edge-connected graphs. 2. For expander graphs with minimum degree of $Ω(t^2\log n)$, there is an improved broadcast algorithm with $O(t \log ^2 n)$ rounds against $t$ adversarial edges. This algorithm exploits the connectivity and conductance properties of G-subgraphs obtained by employing the Karger's edge sampling technique. Our algorithms mark a new connection between the areas of fault-tolerant network design and reliable distributed communication.

cs.DS↗

arXiv2021

Improved Deterministic $(Δ+1)$-Coloring in Low-Space MPC

We present a deterministic $O(\log \log \log n)$-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of $(Δ+1)$-coloring on $n$-vertex graphs. In this model, every machine has a sublinear local memory of size $n^ϕ$ for any arbitrary constant $ϕ\in (0,1)$. Our algorithm works under the relaxed setting where each machine is allowed to perform exponential (in $n^ϕ$) local computation, while respecting the $n^ϕ$ space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious $(Δ+1)$-coloring LOCAL algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in $T_{local}=poly(\log\log n)$ rounds, which sets the state-of-the-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, most notably pseudorandom generators (PRG) and bounded-independence hash functions. The achieved round complexity of $O(\log\log\log n)$ rounds matches the bound of $\log(T_{local})$, which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the $(Δ+1)$-coloring problem were previously known.

cs.DS↗

arXiv2021

New Diameter-Reducing Shortcuts and Directed Hopsets: Breaking the $\sqrt{n}$ Barrier

For an $n$-vertex digraph $G=(V,E)$, a \emph{shortcut set} is a (small) subset of edges $H$ taken from the transitive closure of $G$ that, when added to $G$ guarantees that the diameter of $G \cup H$ is small. Shortcut sets, introduced by Thorup in 1993, have a wide range of applications in algorithm design, especially in the context of parallel, distributed and dynamic computation on directed graphs. A folklore result in this context shows that every $n$-vertex digraph admits a shortcut set of linear size (i.e., of $O(n)$ edges) that reduces the diameter to $\widetilde{O}(\sqrt{n})$. Despite extensive research over the years, the question of whether one can reduce the diameter to $o(\sqrt{n})$ with $\widetilde{O}(n)$ shortcut edges has been left open. We provide the first improved diameter-sparsity tradeoff for this problem, breaking the $\sqrt{n}$ diameter barrier. Specifically, we show an $O(n^ω)$-time randomized algorithm for computing a linear shortcut set that reduces the diameter of the digraph to $\widetilde{O}(n^{1/3})$. This narrows the gap w.r.t the current diameter lower bound of $Ω(n^{1/6})$ by [Huang and Pettie, SWAT'18]. Moreover, we show that a diameter of $\widetilde{O}(n^{1/2})$ can in fact be achieved with a \emph{sublinear} number of $O(n^{3/4})$ shortcut edges. Formally, letting $S(n,D)$ be the bound on the size of the shortcut set required in order to reduce the diameter of any $n$-vertex digraph to at most $D$, our algorithms yield: \[ S(n,D)=\begin{cases} \widetilde{O}(n^2/D^3), & \text{for~} D\leq n^{1/3},\\ \widetilde{O}((n/D)^{3/2}), & \text{for~} D> n^{1/3}~. \end{cases} \] We also extend our algorithms to provide improved $(β,ε)$ hopsets for $n$-vertex weighted directed graphs.

cs.DS↗

arXiv2021

Low-Congestion Shortcuts in Constant Diameter Graphs

Low congestion shortcuts, introduced by Ghaffari and Haeupler (SODA 2016), provide a unified framework for global optimization problems in the congest model of distributed computing. Roughly speaking, for a given graph $G$ and a collection of vertex-disjoint connected subsets $S_1,\ldots, S_\ell \subseteq V(G)$, $(c,d)$ low-congestion shortcuts augment each subgraph $G[S_i]$ with a subgraph $H_i \subseteq G$ such that: (i) each edge appears on at most $c$ subgraphs (congestion bound), and (ii) the diameter of each subgraph $G[S_i] \cup H_i$ is bounded by $d$ (dilation bound). It is desirable to compute shortcuts of small congestion and dilation as these quantities capture the round complexity of many global optimization problems in the congest model. For $n$-vertex graphs with constant diameter $D=O(1)$, Elkin (STOC 2004) presented an (implicit) shortcuts lower bound with $c+d=\widetildeΩ(n^{(D-2)/(2D-2)})$. A nearly matching upper bound, however, was only recently obtained for $D \in \{3,4\}$ by Kitamura et al. (DISC 2019). In this work, we resolve the long-standing complexity gap of shortcuts in constant diameter graphs, originally posed by Lotker et al. (PODC 2001). We present new shortcut constructions which match, up to poly-logarithmic terms, the lower bounds of Das-Sarma et al. As a result, we provide improved and existentially optimal algorithms for several network optimization tasks in constant diameter graphs, including MST, $(1+ε)$-approximate minimum cuts and more.

cs.DS↗

arXiv2021

Component Stability in Low-Space Massively Parallel Computation

We study the power and limitations of component-stable algorithms in the low-space model of Massively Parallel Computation (MPC). Recently Ghaffari, Kuhn and Uitto (FOCS 2019) introduced the class of component-stable low-space MPC algorithms, which are, informally, defined as algorithms for which the outputs reported by the nodes in different connected components are required to be independent. This very natural notion was introduced to capture most (if not all) of the known efficient MPC algorithms to date, and it was the first general class of MPC algorithms for which one can show non-trivial conditional lower bounds. In this paper we enhance the framework of component-stable algorithms and investigate its effect on the complexity of randomized and deterministic low-space MPC. Our key contributions include: 1) We revise and formalize the lifting approach of Ghaffari, Kuhn and Uitto. This requires a very delicate amendment of the notion of component stability, which allows us to fill in gaps in the earlier arguments. 2) We also extend the framework to obtain conditional lower bounds for deterministic algorithms and fine-grained lower bounds that depend on the maximum degree $Δ$. 3) We demonstrate a collection of natural graph problems for which non-component-stable algorithms break the conditional lower bound obtained for component-stable algorithms. This implies that, for both deterministic and randomized algorithms, component-stable algorithms are conditionally weaker than the non-component-stable ones. Altogether our results imply that component-stability might limit the computational power of the low-space MPC model, at least in certain contexts, paving the way for improved upper bounds that escape the conditional lower bound setting of Ghaffari, Kuhn, and Uitto.

cs.DS↗

arXiv2021

Fault-Tolerant Labeling and Compact Routing Schemes

The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of the vertices $s$ and $t$, and at most $f$ failing edges $F$, one can determine if $s$ and $t$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an $n$-vertex graph and a fault bound $f$ are: -- There is a randomized FT connectivity labeling scheme with a label length of $O(f+\log n)$ bits, hence optimal for $f=O(\log n)$. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG '11]. -- There is a randomized FT connectivity labeling scheme with a label length of $O(\log^3 n)$ bits (independent of the number of faults $f$). This scheme is based on the notion of linear sketches of [Ahn et al., SODA '12]. -- For $k\geq 1$, there is a randomized routing scheme that routes a message from $s$ to $t$ in the presence of a set $F$ of faulty edges, with stretch $O(|F|^2 k)$ and routing tables of size $\tilde{O}(f^3 n^{1/k})$. This significantly improves over the state-of-the-art bounds by [Chechik, ICALP '11], providing the first scheme with sub-linear FT labeling and routing schemes for general graphs.

cs.DS↗

arXiv2020

Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers

Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distributed networks, not much is known on the distributed computational aspects of these structures. In this paper, we present distributed algorithms for constructing fault tolerant distance preservers and $+2$ additive spanners that are resilient to at most \emph{two edge} faults. Prior to our work, the only non-trivial constructions known were for the \emph{single} fault and \emph{single source} setting by [Ghaffari and Parter SPAA'16]. Our key technical contribution is a distributed algorithm for computing distance preservers w.r.t. a subset $S$ of source vertices, resilient to two edge faults. The output structure contains a BFS tree $BFS(s,G \setminus \{e_1,e_2\})$ for every $s \in S$ and every $e_1,e_2 \in G$. The distributed construction of this structure is based on a delicate balance between the edge congestion (formed by running multiple BFS trees simultaneously) and the sparsity of the output subgraph. No sublinear-round algorithms for constructing these structures have been known before.

cs.DC↗

arXiv2020

Spiking Neural Networks Through the Lens of Streaming Algorithms

We initiate the study of biological neural networks from the perspective of streaming algorithms. Like computers, human brains suffer from memory limitations which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the well-known streaming model, which can be traced back to Munro and Paterson `78 and has had significant impact in theory and beyond. In the classical streaming setting, one must compute some function $f$ of a stream of updates $\mathcal{S} = \{u_1,\ldots,u_m\}$, given restricted single-pass access to the stream. The primary complexity measure is the space used by the algorithm. We take the first steps towards understanding the connection between streaming and neural algorithms. On the upper bound side, we design neural algorithms based on known streaming algorithms for fundamental tasks, including distinct elements, approximate median, heavy hitters, and more. The number of neurons in our neural solutions almost matches the space bounds of the corresponding streaming algorithms. As a general algorithmic primitive, we show how to implement the important streaming technique of linear sketching efficient in spiking neural networks. On the lower bound side, we give a generic reduction, showing that any space-efficient spiking neural network can be simulated by a space-efficiently streaming algorithm. This reduction lets us translate streaming-space lower bounds into nearly matching neural-space lower bounds, establishing a close connection between these two models.

cs.DC↗

arXiv2020

Simple, Deterministic, Constant-Round Coloring in the Congested Clique

We settle the complexity of the $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round $(Δ+1)$-list coloring algorithm due to Chang et al. (PODC'19), and significantly improves upon the state-of-the-art $O(\log Δ)$-round deterministic $(Δ+1)$-coloring bound of Parter (ICALP'18). A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithms for this problem are based on the quite involved local coloring algorithm of Chang et al. (STOC'18), our algorithm can be described in just a few lines. At a high level, it applies a careful derandomization of a recursive procedure which partitions the nodes and their respective palettes into separate bins. We show that after $O(1)$ recursion steps, the remaining uncolored subgraph within each bin has linear size, and thus can be solved locally by collecting it to a single node. This algorithm can also be implemented in the Massively Parallel Computation (MPC) model provided that each machine has linear (in $n$, the number of nodes in the input graph) space. We also show an extension of our algorithm to the MPC regime in which machines have sublinear space: we present the first deterministic $(Δ+1)$-list coloring algorithm designed for sublinear-space MPC, which runs in $O(\log Δ+ \log\log n)$ rounds.

cs.DS↗

arXiv2020

Small Cuts and Connectivity Certificates: A Fault Tolerant Approach

We revisit classical connectivity problems in the CONGEST model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with $\widetilde{O}(1)$ rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of $Ω(Diam+\sqrt{n})$ rounds). Our main results are: (1) For $D$-diameter unweighted graphs with constant edge connectivity, we show an exact distributed deterministic computation of the minimum cut in $poly(D)$ rounds. This resolves one the open problems recently raised in Daga, Henzinger, Nanongkai and Saranurak, STOC'19. (2) For $D$-diameter unweighted graphs, we present a deterministic algorithm that computes of all edge connectivities up to constant in $poly(D)\cdot 2^{O(\sqrt{\log n\log\log n})}$ rounds. (3) Computation of sparse $λ$ connectivity certificates in $\widetilde{O}(λ)$ rounds. Previous constructions where known only for $λ\leq 3$ and required $O(D)$ rounds. This resolves the problem raised by Dori PODC'18.

cs.DS↗

arXiv2020

On Packing Low-Diameter Spanning Trees

Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing $\cal{T}$ is the largest diameter of any tree in $\cal{T}$. A desirable property of a tree packing, that is both sufficient and necessary for leveraging the high connectivity of a graph in distributed communication, is that its diameter is low. Yet, despite extensive research in this area, it is still unclear how to compute a tree packing, whose diameter is sublinear in $|V(G)|$, in a low-diameter graph $G$, or alternatively how to show that such a packing does not exist. In this paper we provide first non-trivial upper and lower bounds on the diameter of tree packing. First, we show that, for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, there is a tree packing $\cal{T}$ of size $Ω(k)$, diameter $O((101k\log n)^D)$, that causes edge-congestion at most $2$. Second, we show that for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, the diameter of $G[p]$ is $O(k^{D(D+1)/2})$ with high probability, where $G[p]$ is obtained by sampling each edge of $G$ independently with probability $p=Θ(\log n/k)$. This provides a packing of $Ω(k/\log n)$ edge-disjoint trees of diameter at most $O(k^{(D(D+1)/2)})$ each. We then prove that these two results are nearly tight. Lastly, we show that if every pair of vertices in a graph has $k$ edge-disjoint paths of length at most $D$ connecting them, then there is a tree packing of size $k$, diameter $O(D\log n)$, causing edge-congestion $O(\log n)$. We also provide several applications of low-diameter tree packing in distributed computation.

cs.DS↗

arXiv2020

Exponentially Faster Shortest Paths in the Congested Clique

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain $poly(\log\log n)$-round algorithms for the following problems in unweighted undirected $n$-vertex graphs: -- $(1+ε)$-approximation of multi-source shortest paths (MSSP) from $O(\sqrt{n})$ sources. -- $(2+ε)$-approximation of all pairs shortest paths (APSP). -- $(1+ε,β)$-approximation of APSP where $β=O(\frac{\log\log n}ε)^{\log\log n}$. These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold $t = O(\fracβε)$ where $β=O(\frac{\log\log n}ε)^{\log\log n}$. Handling the long distances is done by devising a new algorithm for computing sparse $(1+ε,β)$ emulator with $O(n\log\log n)$ edges. For the short distances, we provide distance-sensitive variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius $t$, their round complexities get improved from $poly(\log n)$ to $poly(\log t)$. Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.

cs.DS↗

arXiv2020

Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

The Massively Parallel Computation (MPC) model is an emerging model which distills core aspects of distributed and parallel computation. It has been developed as a tool to solve (typically graph) problems in systems where the input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in $n$, the number of nodes in the input graph) memory, with randomized algorithms presented for fundamental graph problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding \emph{deterministic} algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all the edges incident to a single node. This poses a considerable obstacle compared to the classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier we introduce a new \emph{graph sparsification technique} that \emph{deterministically} computes a low-degree subgraph with additional desired properties. Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP'86], we obtain $O(\log Δ+\log\log n)$-round \emph{deterministic} MPC algorithms for solving the fundamental problems of \emph{Maximal Matching} and \emph{Maximal Independent Set} with $O(n^ε)$ space on each machine for any constant $ε> 0$. Based on the recent work of Ghaffari et al. [FOCS'18], this additive $O(\log\log n)$ factor is \emph{conditionally} essential. These algorithms can also be shown to run in $O(\log Δ)$ rounds in the closely related model of \congc, improving upon the state-of-the-art bound of $O(\log^2 Δ)$ rounds by Censor-Hillel et al. [DISC'17].

cs.DS↗

arXiv2020

New $(α,β)$ Spanners and Hopsets

An $f(d)$-spanner of an unweighted $n$-vertex graph $G=(V,E)$ is a subgraph $H$ satisfying that $dist_H(u, v)$ is at most $f(dist_G(u, v))$ for every $u,v \in V$. We present new spanner constructions that achieve a nearly optimal stretch of $O(\lceil k /d \rceil)$ for any distance value $d \in [1,k^{1-o(1)}]$, and $d \geq k^{1+o(1)}$. We show the following: 1. There exists an $f(d)$-spanner $H \subseteq G$ with $f(d)\leq 7k$ for any $d \in [1,\sqrt{k}/2]$ with expected size $O_{k}(n^{1+1/k})$. This in particular gives $(α,β)$ spanners with $α=O(\sqrt{k})$ and $β=O(k)$. 2. For any $ε\in (0,1/2]$, there exists an $(α,β)$-spanner with $α=O(k^ε)$, $β=O_ε(k)$ and of expected size $O_{k}(n^{1+1/k})$. This implies a stretch of $O(\lceil k/d \rceil)$ for any $d \in [\sqrt{k}/2, k^{1-ε}]$, and for every $d\geq k^{1+ε}$. In particular, it provides a constant stretch already for vertex pairs at distance $k^{1+o(1)}$ (improving upon $d=(\log k)^{\log k}$ that was known before). Up to the $o(1)$ factor in the exponent, and the constant factor in the stretch, this is the best possible by the girth argument. 3. For any $ε\in (0,1)$ and integer $k\geq 1$, there is a $(3+ε, β)$-spanner with $β=O_ε(k^{\log(3+8/ε)})$ and $O_{k,ε}(n^{1+1/k})$ edges. We also consider the related graph concept of hopsets introduced by [Cohen, J. ACM '00]. We present a new family of $(α,β)$ hopsets with $\widetilde{O}(k \cdot n^{1+1/k})$ edges and $α\cdot β=O(k)$. Most notably, we show a construction of $(3+ε,β)$ hopset with $\widetilde{O}_{k,ε}(n^{1+1/k})$ edges and hop-bound of $β=O_ε(k^{\log(3+9/ε)})$, improving upon the state-of-the-art hop-bound of $β=O(\log k /ε)^{\log k}$ by [Elkin-Neiman, '17] and [Huang-Pettie, '17].

cs.DS↗

arXiv2020

$(Δ+1)$ Coloring in the Congested Clique Model

In this paper, we present improved algorithms for the $(Δ+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its incident edges, and per round each two nodes can exchange $O(\log n)$ bits of information. Our key result is a randomized $(Δ+1)$ vertex coloring algorithm that works in $O(\log\log Δ\cdot \log^* Δ)$-rounds. This is achieved by combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the \local\ model and a degree reduction technique. We also get the following results with high probability: (1) $(Δ+1)$-coloring for $Δ=O((n/\log n)^{1-ε})$ for any $ε\in (0,1)$, within $O(\log(1/ε)\log^* Δ)$ rounds, and (2) $(Δ+Δ^{1/2+o(1)})$-coloring within $O(\log^* Δ)$ rounds. Turning to deterministic algorithms, we show a $(Δ+1)$-coloring algorithm that works in $O(\log Δ)$ rounds.

cs.DS↗