33 research outputs found
Gossiping in chordal rings under the line model
The line model assumes long distance
calls between non neighboring processors. In this sense, the line
model is strongly related to circuit-switched networks, wormhole
routing, optical networks supporting wavelength division
multiplexing, ATM switching, and networks supporting connected mode
routing protocols.
Since the chordal rings are competitors of networks as meshes or
tori because of theirs short diameter and bounded degree, it is of
interest to ask whether they can support intensive communications
(typically all-to-all) as efficiently as these networks. We
propose polynomial algorithms to derive optimal or near optimal
gossip protocols in the chordal ring
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)âO(loglog(n)) time steps, where λ is the unique solution between 0 and 1 of λ·pâs/2â(λ)·pâs/2â(λ)=1, with pi(λ)=1+λ2+âŻ+λ2iâ2 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 sââ 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
Circuit-Switched Gossiping in the 3-Dimensional Torus Networks
In this paper we describe, in the case of short messages, an efficient gossiping algorithm for 3-dimensional torus networks (wrap-around or toroidal meshes) that uses synchronous circuit-switched routing. The algorithm is based on a recursive decomposition of a torus. The algorithm requires an optimal number of rounds and a quasi-optimal number of intermediate switch settings to gossip in an torus
Minimum-cost line broadcast in paths
AbstractUnder the line communication protocol, calls can be placed between pairs of non-adjacent sites over a path of lines connecting them; only one call can utilize a line at any time. This paper addresses questions regarding the cumulative cost, i.e., sum of lengths of calls, of broadcasting under the line protocol in path networks. Let Pn be the path with n vertices, and Cn be the cost of an optimal, line broadcast scheme from a terminal vertex in path Pn. We show that a minimum-cost line broadcast scheme from any source vertex in Pn has cost no more than Cn and no less than Cn â n + 2 for any n â©Ÿ 2 and any time t > [log2n]. We derive a closed-form expression for the minimum cost of a minimum-time line broadcast from a terminal vertex in certain paths and relate this to costs from nearby sources
Minimal contention-free matrices with application to multicasting
In this paper, we show that the multicast problem in trees can be
expressed in term of arranging rows and columns of boolean matrices.
Given a matrix with 0-1 entries, the {\em shadow}
of is defined as a boolean vector of entries such that
if and only if there is no 1-entry in the th column of
, and otherwise. (The shadow can also be seen as the
binary expression of the integer .
Similarly, every row of can be seen as the binary expression of
an integer.) According to this formalism, the key for solving a
multicast problem in trees is shown to be the following. Given a matrix with 0-1 entries, finding a matrix such
that:
1- has at most one 1-entry per column;
2- every row of (viewed as the binary expression of
an integer) is larger than the corresponding row of , ; and
3- the shadow of (viewed as an integer) is minimum.
We show that there is an algorithm that
returns for any boolean matrix .
The application of this result is the following: Given a {\em directed}
tree whose arcs are oriented from the root toward the leaves,
and a subset of nodes , there exists a polynomial-time algorithm
that computes an optimal multicast protocol from the root to all
nodes of in the all-port line model.Peer Reviewe
Problems related to broadcasting in graphs
The data transmission delays become the bottleneck on modern high speed interconnection networks utilized by high performance computing or enterprise data centers. This motivates the study directed towards finding more efficient interconnection topologies as well as more efficient algorithms for information exchange between the nodes of the given network.
Broadcasting is the process of distributing a message
from a node, called the originator, to all other nodes of a communication network. Broadcasting is used as a basic communication primitive by many higher level network operations, which involve a set of nodes in distributed systems. Therefore, it is one the most important operations, which can determine the total efficiency of a given distributed system.
We study interconnection networks via modeling them as graphs. The results described in this work can be used for efficient message routing algorithms in switch based interconnection networks as well as in the choice of the interconnection topologies of such networks.
This thesis is divided into six chapters. Chapter 1 gives a general introduction to the research area and literature overview. Chapter 2 studies the family of graphs for which the broadcast time is equal to the diameter. Chapter 3 studies the routing and broadcasting problem in the Knodel graph. Chapter 4 studies the possible vertex degrees and the possible connections between vertices of different degrees in a broadcast graph. Using this, a new lower bound is obtained on broadcast function. Chapter 5 presents some miscellaneous results. Chapter 6 summarizes the thesis
Broadcasting in Hyper-cylinder graphs
Broadcasting in computer networking means the dissemination of information, which is known initially only at some nodes, to all network members. The goal is to inform every node in the minimal time possible. There are few models for broadcasting; the simplest and the historical model is called the Classical model. In the Classical model, dissemination happens in synchronous rounds, wherein a node may only inform one of its neighbors. The broadcast question is: What is the minimum number of rounds needed for broadcasting, and what broadcast scheme achieves it?
For general graphs, these questions are NP-hard, and it is known to be at least 3 - Δ inapproximable for any real Δ > 0. Even for some very restricted classes of graphs, the questions remain as an NP-hard problem. Little is known about broadcasting in restricted graphs, and only a few classes have a polynomial solution.
Parallel and distributed computing is one of the important domains which relies on efficient broadcasting. Hypercube and torus are the most used network topology in this domain. The widespread use is not only due to their simplicity but also is for their efficiency and high robustness (e.g., fault tolerance) while having an acceptable number of links. In this thesis, it is observed that the Cartesian product of a number of path and cycle graphs produces a valuable set of topologies, we called hyper-cylinders, which contain hypercube and Torus as well. Any hyper-cylinder shares many of the beneficial features of hypercube and torus and might be a suitable substitution in some cases. Some hyper-cylinders are also similar to other practically used topologies such as cube-connected cycles. In this thesis, the effect of the Cartesian product on broadcasting and broadcasting of hyper-cylinders under the Classical and Messy models is studied. This will add a valuable class of graphs to the limited classes of graphs which have a polynomially computable broadcast time. In the end, the relation between worst-case originators and diameters in trees is studied, which may help in the broadcast study of a larger class of graphs where any tree is allowed instead of a path in the Cartesian product
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-É. 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
Optimal broadcasting in treelike graphs
Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator , disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Once informed, the nodes aid the originator in distributing the message. Finding the broadcast time of a vertex in an arbitrary graph is NP-complete. The problem is solved polynomially only for a few classes of graphs. In this thesis we study the broadcast problem in different classes of graphs which have various similarities to trees. The unicyclic graph is the simplest graph family after trees, it is a connected graph with only one cycle in it. We provide a linear time solution for the broadcast problem in unicyclic graphs. We also studied graphs with increasing number of cycles and complexity and provide again polynomial time solutions. These graph families are: tree of cycles, necklace graphs, and 2-restricted cactus graphs. We also define the fully connected tree graphs and provide a polynomial solution and use these results to obtain polynomial solution for the broadcast problem in tree of cliques and a constant approximation algorithm for the hierarchical tree cluster networks
Finding edge-disjoint paths with artificial ant colonies
One of the basic operations in communication networks consists in establishing routes
for connection requests between physically separated network nodes. In many situations,
either due to technical constraints or to quality-of-service and survivability requirements, it is
required that no two routes interfere with each other. These requirements apply in particular
to routing and admission control in large-scale, high-speed and optical networks. The same
requirements also arise in a multitude of other applications such as real-time communications,
vlsi design, scheduling, bin packing, and load balancing. This problem can be modeled as
a combinatorial optimization problem as follows. Given a graph G representing a network
topology, and a collection T = f(s1; t1) : : : (sk; tk)g of pairs of vertices in G representing
connection request, the maximum edge-disjoint paths problem is an NP-hard problem that
consists in determining the maximum number of pairs in T that can be routed in G by
mutually edge-disjoint si - ti paths.
We propose an ant colony optimization (aco) algorithm to solve this problem. aco algo-
rithms are approximate algorithms that are inspired by the foraging behavior of real ants. The
decentralized nature of these algorithms makes them suitable for the application to problems
arising in large-scale environments. First, we propose a basic version of our algorithm in order
to outline its main features. In a subsequent step we propose several extensions of the basic
algorithm and we conduct an extensive parameter tuning in order to show the usefulness of
those extensions. In comparison to a multi-start greedy approach, our algorithm generates
in general solutions of higher quality in a shorter amount of time. In particular the run-time
behaviour of our algorithm is one of its important advantages.Postprint (published version