By Sue Whitesides (auth.), Peter Eades, Tadao Takaoka (eds.)
This publication constitutes the refereed court cases of the twelfth foreign convention on Algorithms and Computation, ISAAC 2001, held in Christchurch, New Zealand in December 2001.
The sixty two revised complete papers awarded including 3 invited papers have been conscientiously reviewed and chosen from a complete of 124 submissions. The papers are prepared in topical sections on combinatorial new release and optimization, parallel and disbursed algorithms, graph drawing and algorithms, computational geometry, computational complexity and cryptology, automata and formal languages, computational biology and string matching, and algorithms and information constructions.
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Additional info for Algorithms and Computation: 12th International Symposium, ISAAC 2001 Christchurch, New Zealand, December 19–21, 2001 Proceedings
Proof. For every range query of the form [xl , xh ] × [yl , yh ], our query algorithm performs a one dimensional generalized range reporting query on the instance of the data structure D stored in each of the O(log n) candidate canonical nodes. D has a query time of O(log n + c), where c is the number of colors reported. ≈ ⊗ Hence our method requires O(log2 n + c log n) time per query. Thus we have the following result: 42 N. Moidu et al. Theorem 4. Let P be a set of colored points in two dimensions.
Proof. Consider any line σ. Clearly the distance function from a query point q is convex along σ. Since SDIST is the sum of these convex distance functions, SDIST is also convex . Consider the set of points in Pt that lie on a line segment of C. These points are already sorted by their x-coordinates in the preprocessing phase. Because of convexity of SDIST in Lemma 5, we sort them in the ascending order of SDIST in time linear to the number of points. We do this for the remaining points of Pt in a similar way, and get the final list of points in Pt sorted in the ascending order of SDIST in time O(|Pt |).
2 that the path Pmc will consist of O(log n) light edges and O(log n) heavy paths. 1. This takes O(log n + c) time per heavy path. e. those that are not part of a heavy path, we simply report the colors when such a point is encountered. This takes a total of O(log n) time since there are O(log n) light edges. Thus reporting the colors on the maximal chain for a three sided query can be done in O(log2 n + c log n). However, there still remains the issue of duplicates. To ensure that each color in the maxima is reported only once, we leverage the fact that colors are encoded as integers from 1 to C.