A simple entropy-based algorithm for planar point location

TitleA simple entropy-based algorithm for planar point location
Publication TypeJournal Articles
Year of Publication2007
AuthorsArya S, Malamatos T, Mount D
JournalACM Transactions on Algorithms (TALG)
Date Published2007/05//
ISBN Number1549-6325
KeywordsEntropy, expected-case complexity, Point location, polygonal subdivision, randomized algorithms, trapezoidal maps

Given a planar polygonal subdivision S, point location involves preprocessing this subdivision into a data structure so that given any query point q, the cell of the subdivision containing q can be determined efficiently. Suppose that for each cell z in the subdivision, the probability pz that a query point lies within this cell is also given. The goal is to design the data structure to minimize the average search time. This problem has been considered before, but existing data structures are all quite complicated. It has long been known that the entropy H of the probability distribution is the dominant term in the lower bound on the average-case search time. In this article, we show that a very simple modification of a well-known randomized incremental algorithm can be applied to produce a data structure of expected linear size that can answer point-location queries in O(H) average time. We also present empirical evidence for the practical efficiency of this approach.