Super-Fast 3-Ruling Sets

A $t$-ruling set of a graph $G = (V, E)$ is a vertex-subset $S \subseteq V$ that is independent and satisfies the property that every vertex $v \in V$ is at a distance of at most $t$ from some vertex in $S$. A \textit{maximal independent set (MIS)} is a 1-ruling set. The problem of computing an MIS on a network is a fundamental problem in distributed algorithms and the fastest algorithm for this problem is the $O(\log n)$-round algorithm due to Luby (SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago. Since then the problem has resisted all efforts to yield to a sub-logarithmic algorithm. There has been recent progress on this problem, most importantly an $O(\log \Delta \cdot \sqrt{\log n})$-round algorithm on graphs with $n$ vertices and maximum degree $\Delta$, due to Barenboim et al. (Barenboim, Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS 2012). We approach the MIS problem from a different angle and ask if O(1)-ruling sets can be computed much more efficiently than an MIS? As an answer to this question, we show how to compute a 2-ruling set of an $n$-vertex graph in $O((\log n)^{3/4})$ rounds. We also show that the above result can be improved for special classes of graphs such as graphs with high girth, trees, and graphs of bounded arboricity. Our main technique involves randomized sparsification that rapidly reduces the graph degree while ensuring that every deleted vertex is close to some vertex that remains. This technique may have further applications in other contexts, e.g., in designing sub-logarithmic distributed approximation algorithms. Our results raise intriguing questions about how quickly an MIS (or 1-ruling sets) can be computed, given that 2-ruling sets can be computed in sub-logarithmic rounds

Automatic analysis of distance bounding protocols

Distance bounding protocols are used by nodes in wireless networks to calculate upper bounds on their distances to other nodes. However, dishonest nodes in the network can turn the calculations both illegitimate and inaccurate when they participate in protocol executions. It is important to analyze protocols for the possibility of such violations. Past efforts to analyze distance bounding protocols have only been manual. However, automated approaches are important since they are quite likely to find flaws that manual approaches cannot, as witnessed in literature for analysis pertaining to key establishment protocols. In this paper, we use the constraint solver tool to automatically analyze distance bounding protocols. We first formulate a new trace property called Secure Distance Bounding (SDB) that protocol executions must satisfy. We then classify the scenarios in which these protocols can operate considering the (dis)honesty of nodes and location of the attacker in the network. Finally, we extend the constraint solver so that it can be used to test protocols for violations of SDB in these scenarios and illustrate our technique on some published protocols.Comment: 22 pages, Appeared in Foundations of Computer Security, (Affiliated workshop of LICS 2009, Los Angeles, CA)

On the Analysis of a Label Propagation Algorithm for Community Detection

This paper initiates formal analysis of a simple, distributed algorithm for community detection on networks. We analyze an algorithm that we call \textsc{Max-LPA}, both in terms of its convergence time and in terms of the "quality" of the communities detected. \textsc{Max-LPA} is an instance of a class of community detection algorithms called \textit{label propagation} algorithms. As far as we know, most analysis of label propagation algorithms thus far has been empirical in nature and in this paper we seek a theoretical understanding of label propagation algorithms. In our main result, we define a clustered version of \er random graphs with clusters $V_1, V_2,..., V_k$ where the probability $p$, of an edge connecting nodes within a cluster $V_i$ is higher than $p'$, the probability of an edge connecting nodes in distinct clusters. We show that even with fairly general restrictions on $p$ and $p'$ ($p = \Omega(\frac{1}{n^{1/4-\epsilon}})$ for any $\epsilon > 0$, $p' = O(p^2)$, where $n$ is the number of nodes), \textsc{Max-LPA} detects the clusters $V_1, V_2,..., V_n$ in just two rounds. Based on this and on empirical results, we conjecture that \textsc{Max-LPA} can correctly and quickly identify communities on clustered \er graphs even when the clusters are much sparser, i.e., with $p = \frac{c\log n}{n}$ for some $c > 1$.Comment: 17 pages. Submitted to ICDCN 201

Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model

Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al. PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming ??(n) memory-per-machine, where n is the number of nodes in the graph (e.g., the O(log log n) MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., O(n^?) for constant 0 < ? < 1. In this paper, we present an algorithm for the 2-ruling set problem, running in O?(log^{1/6} ?) rounds whp, in the low-memory MPC model. Here ? is the maximum degree of the graph. We then extend this result to ?-ruling sets for any integer ? > 1. Specifically, we show that a ?-ruling set can be computed in the low-memory MPC model with O(n^?) memory-per-machine in O?(? ? log^{1/(2^{?+1}-2)} ?) rounds, whp. From this it immediately follows that a ?-ruling set for ? = ?(log log log ?)-ruling set can be computed in in just O(? log log n) rounds whp. The above results assume a total memory of O?(m + n^{1+?}). We also present algorithms for ?-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to O?(m). For ? > 1, these algorithms are all substantially faster than the Ghaffari-Uitto O?(?{log ?})-round MIS algorithm in the low-memory MPC model. All our results follow from a Sample-and-Gather Simulation Theorem that shows how random-sampling-based Congest algorithms can be efficiently simulated in the low-memory MPC model. We expect this simulation theorem to be of independent interest beyond the ruling set algorithms derived here

How Far Must You See To Hear Reliably

We consider the problem of probabilistic reliable communication (PRC) over synchronous networks modeled as directed graphs in the presence of a Byzantine adversary when players\u27 knowledge of the network topology is not complete. We show that possibility of PRC is extremely sensitive to the changes in players\u27 knowledge of the topology. This is in complete contrast with earlier known results on the possibility of perfectly reliable communication over undirected graphs where the case of each player knowing only its neighbours gives the same result as the case where players have complete knowledge of the network. Specifically, in either case, $(2t+1)$-vertex connectivity is necessary and sufficient, where $t$ is the number of nodes that can be corrupted by the adversary \cite{DDWY93:PSMT,SKR05}. We introduce a novel model for quantifying players\u27 knowledge of network topology, denoted by {$\mathcal TK$}. Given a directed graph $G$, influenced by a Byzantine adversary that can corrupt up to any $t$ players, we give a necessary and sufficient condition for possibility of PRC over $G$ for any arbitrary topology knowledge {$\mathcal TK$}. It follows from our main characterization theorem that knowledge of up to $d = \lfloor \frac{n - 2t}{3} \rfloor + 1$ levels is sufficient for the solvability of honest player to honest player communication over any network over which PRC is possible when each player has complete knowledge of the topology. We also show the existence of networks where PRC is possible when players have complete topology knowledge but it is not possible when the players do not have knowledge of up to $d = \lfloor \frac{n - 2t}{3} \rfloor + 1$ levels