Three Classical Computational Geometry Problems Approached
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Three Classical Computational Geometry Problems Approached
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INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION
Volume 8, 2014
Three Classical Computational Geometry Problems Approached Using Range Trees Antonio G. Sturzu, Costin A. Boiangiu
Abstract—This article proposes an alternate way for resolving classical computational geometry problems. The particularity is the integration of Range Tree data structures in solving problems like: segment intersections, orthogonal queries and calculation of rectangular areas. For particular scenarios complexity improvements can be observed. Given that these three algorithms were implemented relying on range trees, the research opens the door for introducing similar computational structures in related geometry problems. Keywords—computational geometry, orthogonal rectangular area problem, segment intersections
queries, Fig. 1 graphical representation of a range tree
An important property of range trees is that they are balanced binary trees (the absolute difference between the height of the left and right sub-trees is no greater than 1). The depth of a range tree that contains N intervals will be
I. INTRODUCTION
W
HEN presenting computational geometry classical problems for pure scientific or educational purposes, a huge number of approaches, data structures, algorithms and languages is employed. From teaching by means of using UML diagrams [1] to complex structures like Regions of Point-free Overlapping Circles [2] there are very few areas of the currently available knowledge that are not used. The purpose of this paper is to fill another possible gap by employing the Range-Trees into solving some well-known computational geometry problems. A Range tree [3] is a balanced (each sub-tree is balanced and the height of the two sub-trees differ by at most one) binary search tree (the largest element from the left sub-tree is smaller than the smallest element from the right sub-tree) where each node can have an auxiliary structure associated to it. An example of a visual representation for a range tree may be seen in Fig. 1. Throughout this paper, the following notations will be used: - [ a, b] , to be read as “the range a to b “, represents the
log 2 (N + 1) . Fig. 1 depicts a range tree for the range 0 to
11. A. Updating an Interval in a Range Tree The most efficient method for storing a range tree is using a vector. The storing method is identical to the one for storing a heap. Below is presented the pseudocode for updating a range a to b in a range tree T: Update (node, l, r, a, b) if (a
