Message Reduction in the LOCAL Model Is a Free Lunch

A new spanner construction algorithm is presented, working under the LOCAL model with unique edge IDs. Given an n-node communication graph, a spanner with a constant stretch and O(n^{1 + epsilon}) edges (for an arbitrarily small constant epsilon > 0) is constructed in a constant number of rounds sending O(n^{1 + epsilon}) messages whp. Consequently, we conclude that every t-round LOCAL algorithm can be transformed into an O(t)-round LOCAL algorithm that sends O(t * n^{1 + epsilon}) messages whp. This improves upon all previous message-reduction schemes for LOCAL algorithms that incur a log^{Omega (1)} n blow-up of the round complexity

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

Probably Approximately Knowing

Whereas deterministic protocols are typically guaranteed to obtain particular goals of interest, probabilistic protocols typically provide only probabilistic guarantees. This paper initiates an investigation of the interdependence between actions and subjective beliefs of agents in a probabilistic setting. In particular, we study what probabilistic beliefs an agent should have when performing actions, in a protocol that satisfies a probabilistic constraint of the form: 'Condition C should hold with probability at least p when action a is performed'. Our main result is that the expected degree of an agent's belief in C when it performs a equals the probability that C holds when a is performed. Indeed, if the threshold of the probabilistic constraint should hold with probaility p=1-x^2 for some small value of x then, with probability 1-x, when the agent acts it will assign a probabilistic belief no smaller than 1-x to the possibility that C holds. In other words, viewing strong belief as, intuitively, approximate knowledge, the agent must probably approximately know (PAK-know) that C is true when it acts.Comment: 23 pages, 2 figures, a full version of a paper whose extended abstract appears in the proceeding of PODC 202

Distributed Approximation on Power Graphs

We investigate graph problems in the following setting: we are given a graph $G$ and we are required to solve a problem on $G^2$. While we focus mostly on exploring this theme in the distributed CONGEST model, we show new results and surprising connections to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on $G^2$ would be quite difficult to solve efficiently on $G$, due to congestion. However, we show that the picture is both more complicated and more interesting. Specifically, we encounter two phenomena acting in opposing directions: (i) slowdown due to congestion and (ii) speedup due to structural properties of $G^2$. We demonstrate these two phenomena via two fundamental graph problems, namely, Minimum Vertex Cover (MVC) and Minimum Dominating Set (MDS). Among our many contributions, the highlights are the following. - In the CONGEST model, we show an $O(n/\epsilon)$-round $(1+\epsilon)$-approximation algorithm for MVC on $G^2$, while no $o(n^2)$-round algorithm is known for any better-than-2 approximation for MVC on $G$. - We show a centralized polynomial time $5/3$-approximation algorithm for MVC on $G^2$, whereas a better-than-2 approximation is UGC-hard for $G$. - In contrast, for MDS, in the CONGEST model, we show an $\tilde{\Omega}(n^2)$ lower bound for a constant approximation factor for MDS on $G^2$, whereas an $\Omega(n^2)$ lower bound for MDS on $G$ is known only for exact computation. In addition to these highlighted results, we prove a number of other results in the distributed CONGEST model including an $\tilde{\Omega}(n^2)$ lower bound for computing an exact solution to MVC on $G^2$, a conditional hardness result for obtaining a $(1+\epsilon)$-approximation to MVC on $G^2$, and an $O(\log \Delta)$-approximation to the MDS problem on $G^2$ in \mbox{poly}\log n rounds.Comment: Appears in PODC 2020. 40 pages, 7 figure