On Generalized Planar Skyline and Convex Hull Range Queries Nadeem Moidu, Jatin Agarwal, Sankalp Khare, Kishore Kothapalli, and Kannan Srinathan Center for Security, Theory and Algorithmic Research (CSTAR), IIIT Hyderabad (India) Abstract. We present output sensitive techniques for the generalized reporting versions of the planar range maxima problem and the planar range convex hull problem. Our solutions are in the pointer machine model, for orthogonal range queries on a static point set. We solve the planar range maxima problem for two-sided, three-sided and four-sided queries. We achieve a query time of O(log n + c) using O(n) space for the two-sided case, where n denotes the number of stored points and c the number of colors reported. For the three-sided case, we achieve query time O(log 2 n + c log n) using O(n) space while for four-sided queries we answer queries in O(log 3 n + c log 2 n) using O(n log n) space. For the planar range convex hull problem, we provide a solution that answers queries in O(log 2 n + c log n) time, using O(n log 2 n) space. 1 Introduction Generalized intersection searching was introduced by Janardan and Lopez [14]. Since then there has been a considerable amount of work on generalized search- ing and reporting problems [9][10][11][3][12][13][1][7][22][20]. A comprehensive survey of developments in the area can be found in [8]. In the generalized ver- sion of a problem, points are associated with colors. Colors capture the idea of membership, dividing objects into groups based on some common property. Such categorization has practical applications in databases, spatial information systems and other areas where objects are separable into classes and queries involve membership checking in these classes. We present here solutions for the generalized (colored) range-query versions of two classic problems in computational geometry – Convex Hull and Skyline 1 – in the two dimensional setting, for a static set of points. Both these problems, in our knowledge, have not been tackled before in literature. Generalized intersection searching problems are broadly divided into two kinds, reporting and counting. In the former, the goal is to report the distinct colors whose points fall in the query range, while in the latter the goal is to count the number of such colors. Our solutions are for the reporting versions of both problems. 1 We will use the terms maxima and skyline interchangeably in this paper. S.P. Pal and K. Sadakane (Eds.): WALCOM 2014, LNCS 8344, pp. 34–43, 2014. c  Springer International Publishing Switzerland 2014