Sparse networks supporting efficient reliable broadcasting

Parallel Computing, 1996

We consider broadcasting from a fault-free source to all nodes of a completely connected n-node network in the presence of k faulty nodes. Every node can communicate with at most one other node in a unit of time and during this period every pair of communicating nodes can exchange information packets. Faulty nodes can receive information but cannot send it. Broadcasting is adaptive, i.e. a node schedules its next communication on the basis of information currently available to it. We study worst case running time of fault-tolerant broadcasting algorithms in two models: in the wake-up model only nodes already having the source message can call other nodes and in the unrestricted model all fault-free nodes can call. In the rst model we give an optimal algorithm working in time k +dlog(n?k)e. In the second model we give an algorithm working in time O(log 2 n), whenever the fraction of faulty nodes is bounded by a constant smaller than 1.

Automata, Languages and Programming, 2010

In the problem of almost-everywhere agreement (denoted a.e. agreement), introduced by Dwork, Peleg, Pippenger, and Upfal [STOC '86], n parties want to reach agreement on an initially held value, despite the possible disruptive and malicious behavior of up to t of them. So far this is reminiscent of the classical Byzantine agreement problem, except that in the alternative formulation the underlying connectivity graph is sparse-i.e., not all parties share point-to-point reliable channels-thus modeling the actual connectivity of real communication networks. In this setting, one may not be able to guarantee agreement amongst all honest parties, and some of them, say x, must be given up. Thus, in this line of work the goal is to be able to tolerate a high value for t (a constant fraction of n is the best possible) while minimizing x. As shown in the original paper, the dependency on d, the degree of the network, to achieve this goal is paramount. Indeed, the best polynomial-time a.e. agreement protocol tolerating a constant fraction of corruptions t = αn, for some constant α > 0 (presented in the original paper, over two decades ago) has parameters, d = n for constant > 0 and x = µt for some constant µ > 1. In this work, we significantly improve on the above parameters obtaining a protocol with t = αn, d = O(log q n), for constant q > 0 and x = O(t log n). Our approach follows that of Dwork et al. of reducing the a.e. agreement problem to constructing a reliable transmission scheme between pairs of nodes for a large fraction of them; however, given our setting's more stringent conditions-poly-logarithmic degree and a linear number of corruptions, such a task is significantly harder. We also consider the problem of secure computation on partially connected networks, as formulated by Garay and Ostrovsky [Eurocrypt '08], and as a corollary to our main result obtain an almost-everywhere secure multi-party computation protocol with the same parameters as above, again significantly improving on the bound on the number of left-out parties-x = O(t log n) for t ≤ αn corruptions, as opposed to x = O(t) in the original work.

Discrete Applied Mathematics, 1992

,srmond, J.-C., P . Hell, A.L. Liestman plied Mathematics 36 (1992) 97-130. and J.G. Peters, Sparse broadcast graphs, Discrete AP-Broadcasting is an information dissemination process in which a message is to be sent from a single originator to ail members of a network by placing calls over the communication lines of the network. Several previous papers have investigated ways to construct sparse graphs (networks) on n vertices in which this process can be completed in minimum time from any originator. In this paper, we describe four techniques to construct graphs of this type and show that they produce the sparsest known graphs for several values of n. For r= 18. n = 19, n = 30 and n = 31 we also show that our new graphs are rninirmtr~l broudcusr graphs (i.e., !hat no graph with fewer edges is possible). These new graphs can be used with other techniques to improve the best known results for many larger values of n.

Networks, 1996

The problem of broadcasting in a network is to disseminate information from one node to all other nodes by transmitting it over communication links that connect nodes.

Discrete Applied Mathematics, 1998

We consider broadcasting with a linearly bounded number of transmission failures. For a constant parameter 0 < CI < 1 we assume that at most ai faulty transmissions can occur during the first i time units of the communication process, for every natural number i. Every informed node can transmit information to at most one neighbor in a unit of time. Faulty transmissions have no effect. We investigate worst-case optimal non-adaptive broadcasting time under this fault model. for several communication networks. We show, e.g., that for the n-node line network this time is linear in n, if x < l/2, and exponential otherwise. For the hypercube and the complete graph, broadcasting in the linearly bounded fault model can be performed in time logarithmic in the number of nodes.

Networks, 1995

Broadcasting refers to the process of dissemination of a set of messages originating from one node to all other nodes in a communication network. We assume that, at any given time, a node can transmit a message along at most one incident link and simultaneously receive a message along at most one incident link. We first present an algorithm for determining the amount of time needed to broadcast k messages in an arbitrary tree. Second, we show that, for every n, There exists a graph with n nodes whose k-message broadcast time matches the trivial lower bound ⌈ log n⌉ + k − 1 by designing a broadcast scheme for complete graphs. We call those graphs minimal broadcast graphs. Finally, we construct an n node minimal broadcast graph with fewer than (⌈log n⌉ + 1)2⌈ log n⌉ −1 edges.

Parallel Processing Letters, 2009

In a I-fault tolerant minimal broadcast network, a node of a network, called thc originator, has a message which is to be transmitted to all othcr nodes of the network in niininium time rcgar~llcss ol' the fhilure of a single communication line. In some instanccs, it is advantagous to use time-rclaxcd broadcast networks that rcquirc slightly more than thc minimum transmission time, but havc sparscr ctlgc scts. This paper presents a general compounding algorithm to constr~~ct spanc, time-rclaxctl. I-hull tolerant broadcast networks. In the algorithm, copies of a broadcast network without Ihults are interconnected with additional edges according to the structure of a I-fault tolerant broaclcast nctwork with two special properties. Both the I-fault tolerant broadcast nctwork and the broadcast nctwork without faults may bc time-rclaxcd. Clomputational results show that tlic algorithm yicl~ls sparser networks by allowing additional time units.

Discrete Applied Mathematics, 1998

In this paper we give new results on the fault-tolerance capabilities of the star graph. We first consider the problem of determining the maximum number v(n) of vertices in a n! vertices star graph S,, such that by removing any set of vertices and/or edges from S, of cardinality at most r-(n) the diameter of S,, does not increase. Subsequently, we give an algorithm for broadcasting a message in S,, in optimal time (the diameter of S,,) and using the minimum possible number of message transmissions, i.e., it! -1, in presence of up to r(n) vertex or edge faults, assuming the set of faults is known to all vertices of the network. We also extend this result to the case in which there is no global knowledge on the faulty elements. 0 199X Elsevier Science B.V. All rights reserved.

SIAM Journal on Discrete Mathematics, 1992

Broadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network.

Theoretical Computer Science, 1998

This paper considers problems of fault{tolerant information di usion in a network with cost function. We show that the problem of determining the minimumcost necessary to perform fault{ tolerant gossiping among a given set of participants is NP-hard and give approximate (with respect to the cost) fault-tolerant gossiping algorithms. We also analyze the communication time and communication complexity of fault-tolerant gossiping algorithms. Finally, we give an optimal cost fault tolerant broadcasting algorithm and apply our results to the atomic commitment problem.