A Note on Hardness of Diameter Approximation

We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, $\tilde \Omega (n)$ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12], where $\tilde \Omega (\cdot)$ hides polylogarithmic factors. Abboud et al. [DISC 2016] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer $1 \leq \ell \leq \operatorname{polylog} (n)$, distinguishing between networks of diameter $4 \ell + 2$ and $6 \ell + 1$ requires $\tilde \Omega (n)$ rounds. We slightly tighten this result by showing that even distinguishing between diameter $2 \ell + 1$ and $3 \ell + 1$ requires $\tilde \Omega (n)$ rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition.Comment: Accepted to Information Processing Letter

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Distributed Exact Shortest Paths in Sublinear Time

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in $O(n)$ time, where $n$ is the number of vertices in the input graph $G$. Peleg and Rubinovich (FOCS'99) showed a lower bound of $\tilde{\Omega}(D + \sqrt{n})$ for this problem, where $D$ is the hop-diameter of $G$. Whether or not this problem can be solved in $o(n)$ time when $D$ is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires $O((n \log n)^{5/6})$ time, for $D = O(\sqrt{n \log n})$, and $O(D^{1/3} \cdot (n \log n)^{2/3})$ time, for larger $D$. This running time is sublinear in $n$ in almost the entire range of parameters, specifically, for $D = o(n/\log^2 n)$. For the all-pairs shortest paths problem, our algorithm requires $O(n^{5/3} \log^{2/3} n)$ time, regardless of the value of $D$. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset $G"$ of a skeleton graph $G'$ of $G$ without first computing $G'$ itself. We then conduct a Bellman-Ford exploration in $G' \cup G"$, while computing the required edges of $G'$ on the fly. As a result, our algorithm computes exactly those edges of $G'$ that it really needs, rather than computing approximately the entire $G'$

Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question

Improved Hardness of Approximation of Diameter in the CONGEST Model

We study the problem of approximating the diameter D of an unweighted and undirected n-node graph in the congest model. Through a connection to extremal combinatorics, we show that a (6/11 + ?)-approximation requires ?(n^{1/6}/log n) rounds, a (4/7 + ?)-approximation requires ?(n^{1/4}/log n) rounds, and a (3/5 + ?)-approximation requires ?(n^{1/3}/log n) rounds. These lower bounds are robust in the sense that they hold even against algorithms that are allowed to return an additional small additive error. Prior to our work, only lower bounds for (2/3 + ?)-approximation were known [Frischknecht et al. SODA 2012, Abboud et al. DISC 2016]. Furthermore, we prove that distinguishing graphs of diameter 3 from graphs of diameter 5 requires ?(n/log n) rounds. This stands in sharp contrast to previous work: while there is an algorithm that returns an estimate ? 2/3D ? ? D? ? D in O?(?n+D) rounds [Holzer et al. DISC 2014], our lower bound implies that any algorithm for returning an estimate 2/3D ? D? ? D requires ??(n) rounds

The Sparsest Additive Spanner via Multiple Weighted BFS Trees

Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing (+2) and (+4)-all-pairs spanners, the sparsest case of (+6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a (+6)-spanner with the essentially-optimal sparsity of O~(n^{4/3}) edges. We then show a distributed implementation of our algorithm, answering an open problem in [Keren Censor{-}Hillel et al., 2016]. A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in |S|+D-1 rounds, where S is the set of sources and D is the diameter of the network graph