Deterministic broadcasting time with partial knowledge of the network

We consider the time of deterministic broadcasting in networks whose nodes have limited knowledge of network topology. Each node u knows only the part of the network within knowledge radius r from it, i.e., it knows the graph induced by all nodes at distance at most r from u. Apart from that, each node knows the maximum degree Delta of the network. One node of the network, called the source, has a message which has to reach all other nodes. We adopt the widely studied communication model called the one-way model in which, in every round, each node can communicate with at most one neighbor, and in each pair of nodes communicating in a given round, one can only send a message while the other can only receive it. This is the weakest of all store-and-forward models for point-to-point networks, and hence our algorithms work for other models as well, in at most the same time.We show trade-offs between knowledge radius and time of deterministic broadcasting, when the knowledge radius is small, i.e., when nodes are only aware of their close vicinity. While for knowledge radius 0, minimum broadcasting time is theta(e), where e is the number of edges in the network, broadcasting can be usually completed faster for positive knowledge radius. Our main results concern knowledge radius 1. We develop fast broadcasting algorithms and analyze their execution time. We also prove lower bounds on broadcasting time, showing that our algorithms are close to optimal

Lower bounds on systolic gossip

AbstractGossiping is an extensively investigated information dissemination process in which each processor has a distinct item of information and has to collect all the items possessed by the other processors. In this paper we provide an innovative and general lower bound technique relying on the novel notion of delay digraph of a gossiping protocol and on the use of matrix norm methods. Such a technique is very powerful and allows the determination of new and significantly improved lower bounds in many cases. In fact, we derive the first general lower bound on the gossiping time of systolic protocols, i.e., constituted by a periodic repetition of simple communication steps. In particular, given any network of n processors and any systolic period s, in the directed and the undirected half-duplex cases every s-systolic gossip protocol takes at least log(n)/log(1/λ)−O(loglog(n)) time steps, where λ is the unique solution between 0 and 1 of λ·p⌊s/2⌋(λ)·p⌈s/2⌉(λ)=1, with pi(λ)=1+λ2+⋯+λ2i−2 for any integer i>0. We then provide improved lower bounds in the directed and half-duplex cases for many well-known network topologies, such as Butterfly, de Bruijn, and Kautz graphs. All the results are extended also to the full-duplex case. Our technique is very general, as for s→∞ it allows the determination of improved results even for non-systolic protocols. In fact, for general networks, as a simple corollary it yields a lower bound only an O(loglog(n)) additive factor far from the general one independently proved in [Proc. 1st ACM Symposium on Parallel Algorithms and Architectures (SPAA), 1989, p. 318; Topics in Combinatorics and Graph Theory (1990) 451; SIAM Journal on Computing 21(1) (1992) 111; Discrete Applied Mathematics 42 (1993) 75] for all graphs and any (non-systolic) gossip protocol. Moreover, for specific networks, it significantly improves with respect to the previously known results, even in the full-duplex case. Correspondingly, better lower bounds on the gossiping time of non-systolic protocols are determined in the directed, half-duplex and full-duplex cases for Butterfly, de Bruijn, and Kautz graphs. Even if in this paper we give only a limited number of examples, our technique has wide applicability and gives a general framework that often allows to get improved lower bounds on the gossiping time of systolic and non-systolic protocols in the directed, half-duplex and full-duplex cases

Neighbourhood Broadcasting in Hypercubes

International audienceIn the broadcasting problem, one node needs to broadcast a message to all other nodes in a network. If nodes can only communicate with one neighbor at a time, broadcasting takes at least $\lceil \log_2 N \rceil$ rounds in a network of $N$ nodes. In the neighborhood broadcasting problem, the node that is broadcasting needs to inform only its neighbors. In a binary hypercube with $N$ nodes, each node has $\log_2 N$ neighbors, so neighborhood broadcasting takes at least $\lceil \log_2 \log_2 (N+1) \rceil$ rounds. In this paper, we present asymptotically optimal neighborhood broadcast protocols for binary hypercubes

Multiple message broadcasting and gossiping in the dynamically orientable graphs

This research investigates the problems of gossiping and multiple message broadcasting in dynamically orientable graphs of different network topologies. These are new problems never attempted before. Dynamically orientable graphs and six different network topologies are considered: paths, cycles, stars, binary trees, complete trees and two-dimensional grids. Information dissemination in graphs that are dynamically orientable requires that number of messages sent in each direction along an edge be balanced and therefore necessitates a different approach in gossiping and multiple message broadcasting.;The obvious upper bound for gossiping and multiple message broadcasting in dynamically orientable graphs is twice the best known time for gossiping and multiple message broadcasting in classical graphs. This is obtained by inserting an additional time step t\u27 after each time step t in the classical graph algorithm in which all calls of time step t are repeated with messages moving along the same edges but in the opposite direction to reset the bias of these edges. Finding better bounds for gossiping and multiple message broadcasting in dynamically orientable graphs is the goal of this research.;For each network topology an algorithm is proposed to perform gossiping and multiple message broadcasting. For some network topologies proposed algorithms for dynamically orientable graphs achieved the same upper bound as it is known for classical graphs, for example, gossiping in dynamically orientable grid graphs. In some cases the best time is the twice the best known time for gossiping and multiple message broadcasting in classical graphs, for example, gossiping in dynamically orientable star graphs. In other cases, good time bounds are achieved that are very close to the upper bounds in classical graphs, for example, multiple message broadcasting in dynamically orientable grid graphs. Multiple message broadcasting in dynamically orientable cycle graphs is also a good example of close upper bounds. As number of messages increases bounds become very close to each other

Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1