Polynomial-Time Algorithms for Minimum-Time Broadcast in Trees

Cohen, Johanne; Fraigniaud, Pierre; Riera, Margarida Mitjana

doi:10.1007/s00224-002-1047-5

Cited by 3 publications

(3 citation statements)

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“…Optimal broadcast strategies in trees in the single-port model have been studied in [1,8]. In [2], the problem was enhanced with non uniform edge transmission times and an O(n·log(n)) algorithm was proposed.…”

Section: Related Workmentioning

confidence: 99%

Constrained Content Distribution and Communication Scheduling for Several Restricted Classes of Graphs

Andreica

Țăpuș

2008

2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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Abstract-In this paper we address several problems regarding content distribution (broadcast) and communication optimization and scheduling for some restricted classes of graphs (trees, intersecting cliques). For the broadcast problem in trees we introduce some new extensions and present some new algorithmic results for determining optimal offline broadcast strategies. The communication scheduling problems are also addressed from an offline algorithmic perspective, considering mutual exclusion constraints or geometric aspects.

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Section: Related Workmentioning

confidence: 99%

Constrained Content Distribution and Communication Scheduling for Several Restricted Classes of Graphs

Andreica

Țăpuș

2008

2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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“…Under this latter mode, it is known [6,12] that if the graph is undirected, then an optimal log|D| -round multicast protocol can be computed in polynomial time for any instance (G; s; D), s ∈ D. However, very little is known in the case of directed graphs. It is only known that there exists a polynomial-time 2-approximation algorithm for directed trees [3] (see also [7]) where a directed tree is rooted at the source and has edges directed from the root toward the leaves.…”

Section: Statement Of the Problemmentioning

confidence: 99%

A note on line broadcast in digraphs under the edge-disjoint paths mode

Fraigniaud

2004

Discrete Applied Mathematics

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It is known that unless NP ⊂ DTIME(n log log n ), no polynomial-time approximation algorithm for the multicast problem can have approximation ratio less than (log n) in n-node digraphs under the edge-disjoint paths mode of the line model. In this note, we give a polynomial-time O(( min + log n)=(log( min + log n)))-approximation algorithm, where min is the smallest integer k such that there exists a rooted directed tree of maximum out-degree k, spanning the considered digraph. Statement of the problemGiven a source-node s in a network, and a set D of destination-nodes, multicasting from s to D consists in transmitting a piece of information from s to all nodes in D using the communication facilities of the network. Broadcasting is a particular case of multicasting in which the destination set consists of all nodes of the network. Multicasting and broadcasting are two of the basic operations upon which network applications are frequently based nowadays. They hence gave rise to a vast literature, covering both applied and fundamental aspects of the problem (cf. [8,22] and [19], respectively).As far as graph-theoretic aspects are concerned, the multicast problem can be expressed as follows. We are given a graph G = (V; E), a node s ∈ V , and a set D ⊆ V . We are looking for the most e cient multicast protocol from s to D in G. The nature of the "protocol" and the measure of its "e ciency" depend on the communication model.The local model assumes that transmissions proceed by synchronous calls between the nodes of the graph. It is assumed, moreover, that (1) a call involves exactly two neighboring nodes (locality constraint), (2) a node can participate in at most one call at a time (single-port constraint), and (3) the duration of a call is 1 (atomic constraint). A multicast protocol is then described by the list of calls placed between the nodes of the graphs. The e ciency of the protocol is measured in terms of number of rounds, where round t is deÿned as the set of all calls performed between time t − 1 and time t, t = 1; 2; : : : . This model has been intensively investigated, for both speciÿc and arbitrary topologies (cf. [16,20] and [1,2,[9][10][11]21,23,24], respectively).The line model relaxes the locality constraint, and allows calls to be placed between non-neighboring nodes. A call is then a path in the graph, whose two extremities are the "caller" and the "callee". (Note that, as opposed to the models in [4,5], the intermediate nodes between the caller and the callee do not receive the information which just "cuts through" the routers attached to them.) Several modern technologies (e.g. single-hop WDM for optical networks) support transmissions whose costs are distance-invariant, and are hence good applications for the line model.Two main variants of the line model have been investigated.E-mail address: pierre@lri.fr (P. Fraigniaud).

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“…Cohen et al summarized current results and proposed new schemes for achieving minimum‐time line‐broadcasting in trees and in directed trees. Averbuch et al introduced a line‐broadcasting algorithm in a tree in the k ‐port model, where each vertex can transmit a message to at most k vertices at each time unit.…”

Section: Introductionmentioning

confidence: 99%

Line-broadcasting in complete k -ary trees

Shabtai

Roditty

2015

NETWORKS

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A line‐broadcasting model in a connected graph G = ( V , E ) , | V | = n , is a model in which one vertex, called the originator of the broadcast, holds a message that has to be transmitted to all vertices of the graph through placement of a series of calls over the graph. In this model, an informed vertex can transmit a message through a path of any length in a single time unit, as long as two transmissions do not use the same edge at the same time. Farley [6] has shown that the process can be completed within at most ⌈ log 2 n ⌉ time units from any originator in a tree (and thus in any connected undirected graph) and that the cumulative cost of broadcasting one message from any vertex, that is, the total number of edges used, is at most ( n − 1 ) ⌈ log 2 n ⌉ . In this article, we present lower and upper bounds for the cumulative cost to broadcast one message in a complete k‐ary tree, k ≥ 2 , from any vertex using the line‐broadcasting model. We prove that if B(u) is the minimum cumulative cost of line‐broadcast in a graph G = ( V , E ) from an originator u ∈ V using the line‐broadcasting model, then ( 2 − o ( 1 ) ) n ≤ B ( u ) ≤ ( 2 + o ( 1 ) ) n , where u is any vertex in a complete k‐ary tree. Furthermore, for certain conditions, B(u) is about ( 2 − o ( 1 ) ) n . © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 139–147 2016

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Polynomial-Time Algorithms for Minimum-Time Broadcast in Trees

Cited by 3 publications

References 29 publications

Constrained Content Distribution and Communication Scheduling for Several Restricted Classes of Graphs

Constrained Content Distribution and Communication Scheduling for Several Restricted Classes of Graphs

A note on line broadcast in digraphs under the edge-disjoint paths mode

Line-broadcasting in complete k -ary trees

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