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Given a set of n colored points in IR2 with a total of m (3 ≤ m ≤ n) colors, the problem of identifying the smallest color-spanning object of some predefined shape is studied in this paper. We shall consider two different shapes: (i) corridor and (ii) rectangle of arbitrary orientation. Our proposed algorithm for identifying the smallest color-spanning corridor is simple and runs in O(n2log n) time using O(n) space. A dynamic version of the problem is also studied, where new points may be added, and the narrowest color-spanning corridor at any instance can be reported in O(mn(α(n))2log m) time. Our algorithm for identifying the smallest color-spanning rectangle of arbitrary orientation runs in O(n3log m) time and O(n) space. [ABSTRACT FROM AUTHOR]

A new type of range searching problem, called the partial enclosure range searching problem, is introduced in this paper. Given a set of geometric objects S and a query region Q , our goal is to identify those objects in S which intersect the query region Q by at least a fixed proportion of their original size. Two variations of this problem are studied. In the first variation, the objects in S are axis-parallel line segments and the goal is to count the total number of members of S so that their intersection with Q is at least a given proportion of their size. Here, Q can be an axis-parallel rectangle or a parallelogram of arbitrary orientation. In the second variation, S is a polygon and Q is an axis-parallel rectangle. The problem is to report the area of the intersection between the polygon S and a query rectangle Q. [ABSTRACT FROM AUTHOR]

In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set of unit disks in . We first present a simple time 5-factor approximation algorithm for this problem, where is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time and respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461-477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to . A minor modification of this algorithm produces a -factor approximation algorithm in time. The same techniques can be applied to have a 3-factor and a -factor approximation algorithms in time and respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present -factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio (1+1/k)2 in nO(k) time. [ABSTRACT FROM AUTHOR]