|Title:||Maximal Sets of K-increasing Subseouences with Applications To Counting Points in Triangles
|Authors:||R. Bar-Yehuda and S. Fogel
|Abstract:||We consider the problem of finding a maximal set of disjoint increasing subsequences of length k of a given sequence of real numbers. We achieve an O(nlogn + mk^2) algorithm, where m is the number of sequences found. Our algorithm is of a geometric nature. Using this result we show how to partition a sequence of n real numbers into 2*lower_bound(sqrt(n)) monotone subsequences in time O(n^1.5), an improvement over the best previous algorithm. An application to a fundamental problem in computational geometry is shown. Given n points in the plane, pteprocess them into an O(n log n) data structure such that counting points inside a query triangle can be done in time O(sqrt(n)*logn). The best preprocessing algorithm for this problem runs in time O(n^1.5*logn). We improve this time to O(n^1.5). In fact, our algorithm permits a linear tradeoff between preprocessing and query time.|
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