Persistent Predecessor Search and Orthogonal Point Location on the Word RAM

Abstract

We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1, . . . , U } under an arbitrary sequence of n insertions and deletions, with O(log log U ) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linear-space structures and improves previous near-O((log log U )2 ) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U ) worst-case query time and linear space. This result is again optimal in U for linear-space structures and improves the previous O((log log U )2 ) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM.Our key technique is an interesting new van-EmdeBoas-style recursion that alternates between two strategies, both quite simple.

Linear-Space Data Structures for Range Minority Query in Arrays

“…The overall query time would then be O(log log n + k) if we use the best known linear-space data structure for orthogonal point location of Chan [4]. To reduce the query time to O(k), our key idea is to observe that there are only n distinct vertical rays with which we query the hive graph, and hence only n distinct points with which we do point location.…”

Computing Discriminating and Generic Words

“…Our problem is then to identify all horizontal segments that intersect with the vertical segment (d, [minord(v), maxord(v)]). These horizontal segments can be found in time O(log log n + output) and space O(n) [2]. Therefore, the following theorem holds.…”

Linear-Space Data Structures for Range Minority Query in Arrays

“…The cost of Chazelle's query algorithm is O(t PL (n)+k) time, where t PL (n) denotes the cost of a point location query in an orthogonal subdivision of size O(n). The overall query time would then be O(log log n + k) if we use the best known linear-space data structure for orthogonal point location of Chan [4].…”