Consider a clique of n nodes, where in each synchronous round each pair of nodes can exchange O(log n) bits. We provide deterministic constant-time solutions for two problems in this model. The first is a routing problem where each node is source and destination of n messages of size O(log n). The second is a sorting problem where each node i is given n keys of size O(log n) and needs to receive the i th batch of n keys according to the global order of the keys. The latter result also implies deterministic constant-round solutions for related problems such as selection or determining modes.

No abstract

Given a simple graph G = (V, E) and a set of sources S ⊆ V , denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection,Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k = ∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d = ∞ and S = V requires constructing a partial BFS tree comprising at least k nodes from every node in V .In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d + k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions.

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/ω ) round matrix multiplication algorithm, where ω < 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 ) 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 theÕ(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.

Tis a lesson you should heed: Try, try, try again. If at first you don't succeed, Try, try, try again. (William Edward Hickson, 19th century educational writer) AbstractLet G = (V, E) be an n-vertex graph and M d a d-vertex graph, for some constant d. Is M d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n (d−2)/d / log n) communication rounds is presented. For the special case that M d is a triangle, we present a probabilistic algorithm that requires an expected O( n 1/3 /(t 2/3 + 1) ) rounds of communication, where t is the number of triangles in the graph, and O(min{n 1/3 log 2/3 n/(t 2/3 + 1), n 1/3 }) with high probability.We also present deterministic algorithms specially suited for sparse graphs. In any graph of maximum degree ∆, we can test for arbitrary subgraphs of diameter D in O( ∆ D+1 /n ) rounds. For triangles, we devise an algorithm featuring a round complexity of O(A 2 /n + log 2+n/A 2 n), where A denotes the arboricity of G.

No abstract

Today's hardware technology presents a new challenge in designing robust systems. Deep submicron VLSI technology introduced transient and permanent faults that were never considered in low-level system designs in the past. Still, robustness of that part of the system is crucial and needs to be guaranteed for any successful product. Distributed systems, on the other hand, have been dealing with similar issues for decades. However, neither the basic abstractions nor the complexity of contemporary fault-tolerant distributed algorithms match the peculiarities of hardware implementations.This paper is intended to be part of an attempt striving to overcome this gap between theory and practice for the clock synchronization problem. Solving this task sufficiently well will allow to build a very robust high-precision clocking system for hardware designs like systems-on-chips in critical applications. As our first building block, we describe and prove correct a novel Byzantine fault-tolerant self-stabilizing pulse synchronization protocol, which can be implemented using standard asynchronous digital logic. Despite the strict limitations introduced by hardware designs, it offers optimal resilience and smaller complexity than all existing protocols.

Given a distributed system of n balls and n bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires Θ(log log n) rounds, and any such algorithm running for r ∈ O(1) rounds incurs a bin load of Ω((log n/ log log n) 1/r ). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in log * n + O(1) communication rounds using O(n) messages in total. Our main result, however, is a matching lower bound of (1 − o(1)) log * n on the time complexity of symmetric algorithms that guarantee small bin loads. The essential preconditions of the proof are (i) a limit of O(n) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls need not be globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time.An extended abstract of preliminary work appeared at STOC 2011 [25] and the corresponding article has been published on arxiv [24].