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 quantity comprises 26 complete revised papers from fifty eight submissions besides 2 invited talks provided on the convention. The workshop coated a various diversity of themes on algorithms and discrete arithmetic, together with computational geometry, algorithms together with approximation algorithms, graph conception and computational complexity.
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Additional info for Algorithms and Discrete Applied Mathematics: First International Conference, CALDAM 2015, Kanpur, India, February 8-10, 2015. Proceedings
5 and 5 will take another O(|V |) time to ﬁnish. Thus, the complexity of the algorithm is O(|V | + k log k). Theorem 1. Algorithm S is a (2 − )-approximation for any originator in the k-cycle graph Gk . Proof. , Ck having maximum number of uninformed vertices in X0 and X1 respectively. Assume that under scheme S, Cj is one of the cycles where broadcasting ﬁnishes at time unit bS (u). In scheme S, Cj has been informed from u at time 2j −1 or sooner along its ﬁrst branch. Let u informs its second adjacent vertex in Cj at time tj , where 2j − 1 < tj ≤ 2k.
Let there are r cycles such that l10 ≥ l20 ≥ ... ≥ lr0 , where lj0 is the length of the cycle Cj0 in X0 and 1 ≤ j ≤ r. , Ck . 2 X1 : It consists of the cycles where at least one vertex has been informed along one branch from u. Let there are m cycles such that l11 ≥ l21 ≥ ... ≥ lm1 , where lj1 is the number of uninformed vertices in the cycle Cj1 in X1 at time i and 1 ≤ j ≤ m. , Ck but not in set X0 . 3 X2 : It consists of the cycles which has been informed from u along both directions. Let there are p such cycles and m + r + p = k.
Alam and A. Mukhopadhyay s Z Y r2 q2 r1 Q B A q1 P r3 q3 p2 p1 X p3 Fig. 5. The 3-path graph Fig. 4. The jewel graph By Theorem 2, a 7-cycle has 42 diﬀerent layer graph representations. We ﬁnd conditions that make these impossible. By Theorem 1, these constitute the set of conditions for the line rigidity of the 7-cycle. The 42 layer graphs for a 7cycle form 6 groups, based on the number of edges on each side. For the 7-cycle (p1 , q1 , r1 , s, r2 , q2 , p2 ), Fig. 6 shows a representative layer graph for each group.