By Kazuo Iwama (auth.), Tetsuo Asano (eds.)

This e-book constitutes the refereed court cases of the seventeenth overseas Symposium on Algorithms and Computation, ISAAC 2006, held in Kolkata, India in December 2006.

The seventy three revised complete papers provided have been rigorously reviewed and chosen from 255 submissions. The papers are equipped in topical sections on algorithms and knowledge buildings, on-line algorithms, approximation set of rules, graphs, computational geometry, computational complexity, community, optimization and biology, combinatorial optimization and quantum computing, in addition to allotted computing and cryptography.

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Extra resources for Algorithms and Computation: 17th International Symposium, ISAAC 2006, Kolkata, India, December 18-20, 2006. Proceedings

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In the rest of the proof we upper bound the running time of this algorithm. It is essential to provide a good bound on the width of the produced path decomposition of G. The following lemma gives us the desired bounds on the pathwidth. Its proof is easy and is based on the bound on the pathwidth given in Lemma 1. Lemma 3. Let G (V E) be the input graph and (G H C) be a branch node of the search tree of our algorithm then the pathwidth of the graph is bounded by pw(H) · C . In particular, 0. (a) If ¡(H) 3, then pw(G) ( 16 · ) V(H) · C for any (b) If ¡(H) 2, then pw(G) C · 1.

1 ε Proof: Instead of using algorithm 1 as a subroutine, one can use algorithm 2 and then continue recursively needing two additional markers for the low and high values in each recursive step. Computing the optimal value for t by the formula above resolves to 1−x n n x =t ⇒ n1−x = t2−x ⇒ t = n 2−x . t t Applying the proof of lemma 3 with this t yields an algorithm with distance 1−x 1 n = Ω n1− 2−x = Ω n 2−x Ω t for a subroutine with a guarantee of Ω(nx ). In particular for a subroutine with a 1 a guarantee Ω n a+1 the guarantee is raised to Ω n 2− a+1 ing with 12 , a a+1 a+1 = Ω n a+2 .

A single marker will always end at the boundary in the worst case. In this setting two markers are already sufficient to achieve a distance of n3 by always keeping the median between the two markers and their distance as small as possible. This strength is due to the unrealistic assumption that the 30 T. Lenz add here add here add here Fig. 1. The horizontal line represents the sorted order of the stream elements seen so far, the vertical line is the marker algorithms always know the exact position of a new element in the sorted data.

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