Succinct geometric indexes supporting point location queries

Abstract

We propose designing data structures called succinct geometric indexes of negligible space (more precisely, o ( n ) bits) that support geometric queries in optimal time, by taking advantage of the n points in the dataset permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O (lg n ) time. We also design three variants of this index. The first supports point location using lg n + 2√lg n + O (lg 1/4 n ) point-line comparisons. The second supports point location in o (lg n ) time when the coordinates are integers bounded by U . The last variant can answer point location queries in O ( H + 1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O ( n ) words or O(n lg n ) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O (lg 2 n ) time.