Get Algorithms and Computation: 18th International Symposium, PDF

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

ISBN-10: 3540771182

ISBN-13: 9783540771180

This e-book constitutes the refereed lawsuits of the 18th foreign Symposium on Algorithms and Computation, ISAAC 2007, held in Sendai, Japan, in December 2007.

The seventy seven revised complete papers awarded 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, facts constitution, online game concept, database purposes, on-line algorithms, I/O algorithms, networks, geometric purposes, and string.

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Extra resources for Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings

Example text

A nonempty proper subset X of V is called an extreme subset of a graph (G = (V, E), w) if d(G,w) (Y ) > d(G,w) (X) for all nonempty proper subsets Y of X. (4) We denote by X (G, w) the family of all extreme subsets of (G, w). Any singleton {v}, v ∈ V is an extreme subset. Note that at least one of extreme subsets corresponds to a minimum cut, and no two extreme subsets cross each other. Figure 2(a) shows the extreme subsets of the graph (G, w) in Fig. 1(a), and Figure 2(b) shows its tree representation that indicates the inclusionwise relation among the extreme subsets in such a way that X ∈ X (G, w) is a child of Y ∈ X (G, w) if X is the largest set properly contained in Y .

2 u4 3 u4 u5 1 (3) u5 1 5 u1 7 3 u2 4 u6 u3 9 7 u1 4 u2 5 u6 u3 1 (a) (G,w) (b) (T,w’) Fig. 1. (a) An edge-weighted graph (G = (V, E), w) and (b) A Gomory-Hu tree (T = (V, F ), w ) of the graph (G, w), where the numbers beside edges indicate their weights Minimum Degree Orderings 19 For example, σ = (u1 , u3 , u4 , u5 , u6 , u2 ) is an MA ordering of the graph (G, w) in Figure 1(a), indicating that {u6 , u2 } is a pendent pair. Based on property (3), one can design an algorithm to compute a minimum cut in a graph (G, w) by repeatedly identifying a pendent pair and contract the pair into a single vertex (see [14,21]).

X∈X (15) Since f0 (X) = 2f (X), it suffices to show that (15) holds for i = 0. We easily see that (15) holds for i = n − 2. Now we assume that (15) holds for i = j. We prove that (15) holds for i = j − 1. Let X be an arbitrary subset of V − Vj−1 that separates vn−1 and vn . Case-1. vj ∈ X: Let x∗ = argmin{fj−1 (x) | x ∈ X}. For two crossing sets Vj−1 ∪ X and Vj + x∗ , we have by submodularity of f f (Vj−1 ∪ X) + f (Vj + x∗ ) ≥ f (Vj ∪ X) + f (Vj−1 + x∗ ). From this and induction hypothesis fj (X) ≥ fj (x∗ ), we have fj−1 (X) − fj−1 (x∗ ) = f (X) + f (Vj−1 ∪ X) − f (Vj−1 + x∗ ) − f (x∗ ) ≥ f (X) + f (Vj ∪ X) − f (Vj + x∗ ) − f (x∗ ) = fj (X) − fj (x∗ ) ≥ 0.

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Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings by Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)


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