How to search in history

Abstract

This paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times. We present a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history. This scheme requires O(n + h) space and O(log n log h) query response-time, which saves a factor of log n space over previous structures. Aside from its improved performance, the novelty of our method is to allow the set S to be only partially ordered with respect to queries and the time-measure to be multi-dimensional. The generality of our method makes it useful to a number of problems in three-dimensional geometry. For example, we are able to give fast algorithms for locating a point in a 3d-complex, using linear space, or for finding which of n given points is closest to a query plane. Using a simpler, yet conceptually similar technique, we show that with only O(n²) preprocessing, we can determine in O(log ² n) time which of n given points in E³ is closest to an arbitrary query point.