Low‐cost minimum‐time line‐broadcasting schemes in complete binary trees

Averbuch, Amir; Gaber, Iris; Roditty, Yehuda

doi:10.1002/net.1036

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…In this article, (for

k \geq 3

) and in (for k = 2) it was shown that the cumulative cost of line‐broadcast, from any originator in a k ‐ary tree, is O ( n ). Thus, in addition to the problem we posed at the beginning of our article we pose the following conjecture.…”

Section: Discussionmentioning

confidence: 95%

“…Averbuch et al studied line‐broadcasting in complete binary trees. They provided a minimum‐time line‐broadcasting scheme originating from any vertex of the tree.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, keeping in mind the result of Fujita and Farley , concerning paths, on one hand, and that of Averbuch et al and Perez Alon on the other hand, we pose the following problem: Problem Characterize the trees

T = (V, E), | V | = n

, such that the cumulative cost using the line‐broadcasting model from any vertex is linear in terms of n .…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we provide some insight into the solution to this problem by extending the work of Averbuch et al . Specifically, we introduce lower and upper bounds for the cumulative cost of a line‐broadcasting process in a complete k ‐ary tree,

k \geq 2

.…”

Section: Introductionmentioning

confidence: 99%

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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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“…In this article, (for

k \geq 3

Section: Discussionmentioning

confidence: 95%

“…Averbuch et al studied line‐broadcasting in complete binary trees. They provided a minimum‐time line‐broadcasting scheme originating from any vertex of the tree.…”

Section: Introductionmentioning

confidence: 99%

T = (V, E), | V | = n

, such that the cumulative cost using the line‐broadcasting model from any vertex is linear in terms of n .…”

Section: Introductionmentioning

confidence: 99%

k \geq 2

.…”

Section: Introductionmentioning

confidence: 99%

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Line-broadcasting in complete k -ary trees

Shabtai

Roditty

2015

NETWORKS

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“…An available path is defined as any simple path in which none of the edges are involved in another call. Some examples in which line broadcasting schemes have been investigated include cycles [KP98], grids [W99] and complete binary trees [AGR01].…”

Section: Related Workmentioning

confidence: 99%

Broadcasting in cycles with chords

Scoy

Kovalchick

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Broadcasting is the process of information dissemination in which one node, the originator, knows a single piece of information and using a series of calls must inform every other node in the network of this information. We assume that at any given time, a node can communicate the message to another node, with which it shares an edge, by acting as either a sender or receiver, but not both. Multiple message broadcasting considers the case when the originator has m messages, where m > 1, to disseminate. Whereas broadcasting limits the communication of a message from one node to another node via a single edge, line broadcasting allows one node to send a message to any other node in the network as long as a simple path exists between the sending node and the receiving node and every edge along the path is not in use. In this dissertation, we consider the problem of broadcasting in a cycle with chords and we develop broadcast schemes for this type of network. We begin by investigating the problem of broadcasting in a cycle with one and two chords, respectively. Then, we consider the problem of multiple message broadcasting in cycles with one and two chords. Finally, we consider the problem of line broadcasting in cycles with chords. Through our investigations, we develop two algorithms for the problem of broadcasting in a cycle with one and two chords, respectively and we analyze the correctness and complexity of these algorithms. Then, we discuss problems associated with multiple message broadcasting in cycles with one and two chords. Finally, we use techniques developed for line broadcasting in cycles to create minimum time broadcast schemes for cycles through the addition of chords. Using techniques developed in this dissertation, we are able to broadcast in minimum time in cycles with chords. In cycles whose size is a power of 2, we have proved that the number of chords that we add to the cycle is the minimum number of chords required to broadcast in minimum time in such a cycle. Committee. Finally, I wish to thank my parents, James and Dorothy Kovalchick and my sister, Amanda Kovalchick for their support and encouragement.

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