An optimal construction method for generalized convex layers

Abstract

Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O(log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O((log n)2) time. In this dynamic data structure, a query with output size k can be solved in O(log n+k(log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O((log n)2) amortized update time and O(log n+k) query time.