On Dynamic Range Reporting in One Dimension

TL;DR

This paper introduces a new dynamic range reporting data structure with faster query times than existing methods, using a novel recursion approach and achieving optimality with sublinear space hashing schemes.

Contribution

It presents a novel recursion technique for dynamic range reporting that significantly improves query times and introduces the first sublinear space dynamic perfect hashing scheme.

Findings

Query time is O(lg lglg w), faster than previous structures.

Update time matches van Emde Boas structure at O(lg w).

Develops the first sublinear space dynamic perfect hashing scheme.

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(lglg 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 (an approximate storage scheme for sparse vectors) which uses low space. We strengthen previous lower bounds to show that these results are optimal.