Algorithms and Computation: 18th International Symposium, by Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

By Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

This publication constitutes the refereed complaints of the 18th foreign Symposium on Algorithms and Computation, ISAAC 2007, held in Sendai, Japan, in December 2007.

The seventy seven revised complete papers offered including 2 invited talks have been rigorously reviewed and chosen from 220 submissions. The papers are prepared in topical sections on graph algorithms, computational geometry, complexity, graph drawing, allotted algorithms, optimization, information constitution, online game thought, database purposes, on-line algorithms, I/O algorithms, networks, geometric purposes, and string.

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Example text

We first start with characterizing such cases through the following preparatory lemmas. Lemma 8. Let Wi and Wj denote deficient sets in W0 with Wi ∩ Wj = ∅. (i) |NG (Wi ∪ Wj )| ≥ 1. (ii) Wi ∩ NG (Wj ) = ∅ = Wj ∩ NG (Wi ) holds; |NG (Wi ∩ Wj )| ≥ 2. (iii) If |NG (Wi ∩ Wj )| = 2, then no set W ∈ W0 − {Wi , Wj } satisfies W ∩ Wi ∩ Wj = ∅. (iv) If |NG (Wi ∪ Wj )| = 1, then at most one set W ∈ W0 − {Wi , Wj } satisfies W ∩ Wi ∩ Wj = ∅. (v) If |NG (Wi ∪Wj )| = 2, then for every W ∈ W0 −{Wi , Wj } with Wi ∩Wj ∩W = ∅, we have NG (Wi ∪ Wj ) ∩ W ∩ S0 = ∅.

Wj ∩ W are disconnected, which contradicts Lemma 3, and |Wj ∩ NG (W )| = 1 would indicate that the vertex v with Wj ∩ NG (W ) = {v} satisfies v = sj by Lemmas 4 and 6(ii) (note that W ∩ NG (Wj ) = ∅ holds by Lemma 3). Consider the directed graph H = (V1 , E1 ) such that each vertex vi ∈ V1 corresponds to a set in Wi ∈ W , and that a directed edge (vi , vj ) belongs to E1 if and only if sj ∈ NG (Wi ) (see Fig. 2). From the above claim, the outdegree of each vertex in V1 is at most d∗ − 3. On the other hand, |E1 | ≥ ( − 1)/2 holds, since Lemmas 5 and 6 imply that for every two sets Wi , Wj ∈ W , we have si ∈ NG (Wj ) 36 T.

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