Succinct Geometric Indexes Supporting Point Location Queries

We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence, to support geometric queries in optimal time. 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 $\lg n + 2\sqrt{\lg n} + O(\lg^{1/4} n)$ 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 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 use O(n) words or O(n lg n) bits, while saving drastic amounts of space. We then 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 $O(\lg² n)$ time.