Distributed (Δ+1)(\Delta+1)-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for $(\Delta+1)$-coloring in the LOCAL model, running in $O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})}$ rounds in a graph of maximum degree~$\Delta$. This implies that the $(\Delta+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of $\Omega \left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta} \right) \right)$ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains $\Delta+1$ colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts

An Improved Distributed Algorithm for Maximal Independent Set

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some recent techniques, also leads to a near-optimal global complexity. Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after $O(\log n)$ rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node $v$ terminates after $O(\log \mathsf{deg}(v)+\log 1/\epsilon)$ rounds, with probability at least $1-\epsilon$. The guarantee holds even if the randomness outside $2$-hops neighborhood of $v$ is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of $O(\log \Delta) + 2^{O(\sqrt{\log \log n})}$, where $\Delta$ denotes the maximum degree. This improves over the $O(\log^2 \Delta) + 2^{O(\sqrt{\log \log n})}$ result of Barenboim et al., and gets close to the $\Omega(\min\{\log \Delta, \sqrt{\log n}\})$ lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lov\'asz Local Lemma

Congested Clique Algorithms for Graph Spanners

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling schemes and routing to solving linear systems and spectral sparsification. A k-spanner maintains pairwise distances up to multiplicative factor of k. It is a folklore that for every n-vertex graph G, one can construct a (2k-1) spanner with O(n^{1+1/k}) edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model using O(k^2) rounds, when randomization is allowed. In this work, we consider spanner constructions in the congested clique model, and show: - a randomized construction of a (2k-1)-spanner with O~(n^{1+1/k}) edges in O(log k) rounds. The previous best algorithm runs in O(k) rounds; - a deterministic construction of a (2k-1)-spanner with O~(n^{1+1/k}) edges in O(log k +(log log n)^3) rounds. The previous best algorithm runs in O(k log n) rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest; - a deterministic construction of a O(k)-spanner with O(k * n^{1+1/k}) edges in O(log k) rounds

Log Diameter Rounds Algorithms for 2-Vertex and 2-Edge Connectivity

Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data - data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM model. In this paper, we study two fundamental problems, 2-edge connectivity and 2-vertex connectivity (biconnectivity). PRAM algorithms which run in O(log n) time have been known for many years. We give algorithms using roughly log diameter rounds in the MPC model. Our main results are, for an n-vertex, m-edge graph of diameter D and bi-diameter D\u27, 1) a O(log D log log_{m/n} n) parallel time 2-edge connectivity algorithm, 2) a O(log D log^2 log_{m/n}n+log D\u27log log_{m/n}n) parallel time biconnectivity algorithm, where the bi-diameter D\u27 is the largest cycle length over all the vertex pairs in the same biconnected component. Our results are fully scalable, meaning that the memory per processor can be O(n^{delta}) for arbitrary constant delta>0, and the total memory used is linear in the problem size. Our 2-edge connectivity algorithm achieves the same parallel time as the connectivity algorithm of [Andoni et al., 2018]. We also show an Omega(log D\u27) conditional lower bound for the biconnectivity problem

GYM: A Multiround Distributed Join Algorithm

Multiround algorithms are now commonly used in distributed data processing systems, yet the extent to which algorithms can benefit from running more rounds is not well understood. This paper answers this question for several rounds for the problem of computing the equijoin of n relations. Given any query Q with width w, intersection width iw, input size IN, output size OUT, and a cluster of machines with M=Omega(IN frac{1}{epsilon}) memory available per machine, where epsilon > 1 and w ge 1 are constants, we show that: 1. Q can be computed in O(n) rounds with O(n(INw + OUT)2/M) communication cost with high probability. Q can be computed in O(log(n)) rounds with O(n(INmax(w, 3iw) + OUT)2/M) communication cost with high probability. Intersection width is a new notion we introduce for queries and generalized hypertree decompositions (GHDs) of queries that captures how connected the adjacent components of the GHDs are. We achieve our first result by introducing a distributed and generalized version of Yannakakis\u27s algorithm, called GYM. GYM takes as input any GHD of Q with width w and depth d, and computes Q in O(d + log(n)) rounds and O(n (INw + OUT)2/M) communication cost. We achieve our second result by showing how to construct GHDs of Q with width max(w, 3iw) and depth O(log(n)). We describe another technique to construct GHDs with longer widths and lower depths, demonstrating other tradeoffs one can make between communication and the number of rounds

Distributed Lower Bounds for Ruling Sets

Given a graph $G = (V,E)$, an $(\alpha, \beta)$-ruling set is a subset $S \subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $\beta$. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a $(2, \beta)$-ruling set in the LOCAL model, we show the following, where $n$ denotes the number of vertices, $\Delta$ the maximum degree, and $c$ is some universal constant independent of $n$ and $\Delta$. $\bullet$ Any deterministic algorithm requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta n \right\}$. By optimizing $\Delta$, this implies a deterministic lower bound of $\Omega\left(\sqrt{\frac{\log n}{\beta \log \log n}}\right)$ for all $\beta \le c \sqrt[3]{\frac{\log n}{\log \log n}}$. $\bullet$ Any randomized algorithm requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta \log n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta \log n \right\}$. By optimizing $\Delta$, this implies a randomized lower bound of $\Omega\left(\sqrt{\frac{\log \log n}{\beta \log \log \log n}}\right)$ for all $\beta \le c \sqrt[3]{\frac{\log \log n}{\log \log \log n}}$. For $\beta > 1$, this improves on the previously best lower bound of $\Omega(\log^* n)$ rounds that follows from the 30-year-old bounds of Linial [FOCS'87] and Naor [J.Disc.Math.'91]. For $\beta = 1$, i.e., for the problem of computing a maximal independent set, our results improve on the previously best lower bound of $\Omega(\log^* n)$ on trees, as our bounds already hold on trees

Using Read-k Inequalities to Analyze a Distributed MIS Algorithm

Until recently, the fastest distributed MIS algorithm, even for simple graphs, e.g., unoriented trees has been the simple randomized algorithm discovered the 80s. This algorithm (commonly called Luby's algorithm) computes an MIS in $O(\log n)$ rounds (with high probability). This situation changed when Lenzen and Wattenhofer (PODC 2011) presented a randomized $O(\sqrt{\log n}\cdot \log\log n)$-round MIS algorithm for unoriented trees. This algorithm was improved by Barenboim et al. (FOCS 2012), resulting in an $O(\sqrt{\log n \cdot \log\log n})$-round MIS algorithm. The analyses of these tree MIS algorithms depends on "near independence" of probabilistic events, a feature of the tree structure of the network. In their paper, Lenzen and Wattenhofer hope that their algorithm and analysis could be extended to graphs with bounded arboricity. We show how to do this. By using a new tail inequality for read-k families of random variables due to Gavinsky et al. (Random Struct Algorithms, 2015), we show how to deal with dependencies induced by the recent tree MIS algorithms when they are executed on bounded arboricity graphs. Specifically, we analyze a version of the tree MIS algorithm of Barenboim et al. and show that it runs in O(\mbox{poly}(\alpha) \cdot \sqrt{\log n \cdot \log\log n}) rounds in the $\mathcal{CONGEST}$ model for graphs with arboricity $\alpha$. While the main thrust of this paper is the new probabilistic analysis via read-$k$ inequalities, for small values of $\alpha$, this algorithm is faster than the bounded arboricity MIS algorithm of Barenboim et al. We also note that recently (SODA 2016), Gaffari presented a novel MIS algorithm for general graphs that runs in $O(\log \Delta) + 2^{O(\sqrt{\log\log n})}$ rounds; a corollary of this algorithm is an $O(\log \alpha + \sqrt{\log n})$-round MIS algorithm on arboricity-$\alpha$ graphs.Comment: To appear in PODC 2016 as a brief announcemen

Optimal Gossip with Direct Addressing

Gossip algorithms spread information by having nodes repeatedly forward information to a few random contacts. By their very nature, gossip algorithms tend to be distributed and fault tolerant. If done right, they can also be fast and message-efficient. A common model for gossip communication is the random phone call model, in which in each synchronous round each node can PUSH or PULL information to or from a random other node. For example, Karp et al. [FOCS 2000] gave algorithms in this model that spread a message to all nodes in $\Theta(\log n)$ rounds while sending only $O(\log \log n)$ messages per node on average. Recently, Avin and Els\"asser [DISC 2013], studied the random phone call model with the natural and commonly used assumption of direct addressing. Direct addressing allows nodes to directly contact nodes whose ID (e.g., IP address) was learned before. They show that in this setting, one can "break the $\log n$ barrier" and achieve a gossip algorithm running in $O(\sqrt{\log n})$ rounds, albeit while using $O(\sqrt{\log n})$ messages per node. We study the same model and give a simple gossip algorithm which spreads a message in only $O(\log \log n)$ rounds. We also prove a matching $\Omega(\log \log n)$ lower bound which shows that this running time is best possible. In particular we show that any gossip algorithm takes with high probability at least $0.99 \log \log n$ rounds to terminate. Lastly, our algorithm can be tweaked to send only $O(1)$ messages per node on average with only $O(\log n)$ bits per message. Our algorithm therefore simultaneously achieves the optimal round-, message-, and bit-complexity for this setting. As all prior gossip algorithms, our algorithm is also robust against failures. In particular, if in the beginning an oblivious adversary fails any $F$ nodes our algorithm still, with high probability, informs all but $o(F)$ surviving nodes