By Sumit Ganguly, Ramesh Krishnamurti
This ebook collects the refereed lawsuits of the 1st overseas convention onon Algorithms and Discrete utilized arithmetic, CALDAM 2015, held in Kanpur, India, in February 2015. the amount includes 26 complete revised papers from fifty eight submissions besides 2 invited talks offered on the convention. The workshop lined a various variety of subject matters on algorithms and discrete arithmetic, together with computational geometry, algorithms together with approximation algorithms, graph thought and computational complexity.
Read or Download Algorithms and Discrete Applied Mathematics: First International Conference, CALDAM 2015, Kanpur, India, February 8-10, 2015. Proceedings PDF
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Additional info for Algorithms and Discrete Applied Mathematics: First International Conference, CALDAM 2015, Kanpur, India, February 8-10, 2015. Proceedings
We will show that bS2 (w) ≤ bS1 (w) = b(w, Gk ). According to scheme S1 , w informs its adjacent vertex along the shorter path at time two. Now we construct a new broadcast scheme S2 where w informs its adjacent vertex along the shorter path at time one. The order in which u broadcasts along the remaining k−1 cycles is the same in both schemes. However, under S2 , every vertex along the longer path towards vertex u from w will receive the message exactly one time unit later compared to S1 . To prove that bS2 (w) = b(w, Gk ) we consider two cases: Case 1: under S1 , u is informed along the shorter path at time b1 ≤ b(w, Gk ): Under S2 all the vertices along the shorter path will be informed exactly one time unit earlier.
The broadcasting problem becomes very diﬃcult when two cycles intersect.  presents a (4 − )-approximation algorithm to ﬁnd the broadcast time in simple graphs where intersection of two cycles is a path, called a k-path graph. In this paper we consider broadcasting in simple graphs where cycles intersect at single vertex. The simplest such graph where several cycles have only one intersecting vertex is called a k-cycle graph. A k-cycle graph is a collection of k cycles of arbitrary lengths connected by a central vertex on one end.
All the vertices in the remaining p − 1 cycles must have been informed within 2k + 1 − p time units. From the proof in part a) it is clear that the time taken to inform any of the k − p + 1 cycles will be lk +2k−1 2 = 2k − p + lk −1−2k+2p 2 . Now, lk − 1 − 2k + 2p is the number of uninformed vertices in Cp before time Constant Approximation for Broadcasting in k-cycle Graph 31 unit 2k + 1 − p. Since u informs Cp at time 2k + 1 − p, then lk − 1 − 2k + 2p ≥ 1. Thus, 2k − p + lk −1−2k+2p 2 ≥ 2k + 1 − p.