On dynamic range reporting in one dimension

Abstract

We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM with w-bit words, we show how to perform updates in O(lg w) time and answer queries in O(lg lg w) time. The update time is identical to the van Emde Boas structure, but the query time is exponentially faster. Existing lower bounds show that achieving our query time for predecessor search requires doubly-exponentially slower updates. We present some arguments supporting the conjecture that our solution is optimal.Our solution is based on a new and interesting recursion idea which is "more extreme" that the van Emde Boas recursion. Whereas van Emde Boas uses a simple recursion (repeated halving) on each path in a trie, we use a nontrivial, van Emde Boas-like recursion on every such path. Despite this, our algorithm is quite clean when seen from the right angle. To achieve linear space for our data structure, we solve a problem which is of independent interest. We develop the first scheme for dynamic perfect hashing requiring sublinear space. This gives a dynamic Bloomier filter (a storage scheme for sparse vectors) which uses low space. We strengthen previous lower bounds to show that these results are optimal.

Persistent Predecessor Search and Orthogonal Point Location on the Word RAM

“…Note that in the on-line setting, 1-d range emptiness queries are known to have lower complexity than predecessor search (O(1) query time in the static case [6], and O(log log log U ) in the dynamic case with polylogarithmic update time [50]). The offline problem is equivalent to another basic geometric problem, orthogonal 2-d segment intersection: report the k intersections among n horizontal and vertical line segments.…”

“…Or, if the coordinates have been pre-sorted, can the problem be solved in O(n) time? (Known dynamic 1-d range searching results [50] only imply a lineartime solution for an asymmetric special case with n horizontal segments and at most O(n/ log ε n) vertical segments, or vice versa. )…”

Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time

“…Range queries in 1-d (reporting any element inside a query interval) can be solved with O(1) query time by a linear-space data structure [2]. Even for the dynamic problem, exponential improvements over successor search are known [50].…”

A dynamic data structure for flexible molecular maintenance and informatics

“…(proved in [37]) On a RAM with w-bit words the fully dynamic one dimensional integer range reporting problem can be solved in linear space, and with high probability bounds of O (tu) and O (tq + k) on update time and query time, respectively, where k is the number of items reported, and (i) tu = O (log w) and tq = O (log log w) using the data structure in [37]; and (ii) tu = O (log n/log log n) and tq = O (log log n) using the data structure in [37] for small w and a fusion tree [21] for large w.…”

“…At the heart of our data structure is a fully dynamic one dimensional integer range reporting data structure for word RAM described in [37]. The data structure in [37] maintains a set S of integers under updates (i.e., insertions and deletions), and answers queries of the form: report any or all points in S in a given interval. The following theorem summarizes the performance bounds of the data structure which are of interest to us.…”