Peyman Afshani

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We improve the previous results by Aronov and Har-Peled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected sizewhich can answer 3D approximate halfspace range counting queries in O(log n/k) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard(More)
We prove the existence of an algorithm <i>A</i> for computing 2D or 3D convex hulls that is optimal for <i>every point set</i> in the following sense: for every sequence &sigma; of <i>n</i> points and for every algorithm <i>A</i>&#8242; in a certain class <i>A</i>, the running time of <i>A</i> on input &sigma; is at most a constant factor times the running(More)
We study the following problem: Given an array <i>A</i> storing <i>N</i> real numbers, preprocess it to allow fast reporting of the <i>K</i> smallest elements in the subarray <i>A</i>[<i>i, j</i>] in sorted order, for any triple (<i>i, j, K</i>) with 1 &#8804; <i>i</i> &#8804; <i>j</i> &#8804; <i>N</i> and 1 &#8804; <i>K</i> &#8804; <i>j &minus; i</i> + 1.(More)
Orthogonal range reporting is the problem of storing a set of <i>n</i> points in <i>d</i>-dimensional space, such that the <i>k</i> points in an axis-orthogonal query box can be reported efficiently. While the 2-d version of the problem was completely characterized in the pointer machine model more than two decades ago, this is not the case in higher(More)
We investigate the problem of finding an unknown cut through querying vertices of a graph G. Our complexity measure is the number of submitted queries. To avoid some worst cases, we make a few assumptions which allow us to obtain an algorithm with the worst case query complexity of O(k) + 2k log n k in which k is the number of vertices adjacent to(More)
In orthogonal range reporting we are to preprocess N points in d-dimensional space so that the points inside a d-dimensional axis-aligned query box can be reported efficiently. This is a fundamental problem in various fields, including spatial databases and computational geometry. In this paper we provide a number of improvements for three and higher(More)
Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point's uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution.(More)
We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex range reporting as well as some other geometric problems. Simplex range reporting is the problem of storing(More)