Criteria for Reducibility of Moving Objects Closeness Problem

Consider two streams of moving objects inside the square [0, 1](2). We assume that objects in each of streams move in prescribed manner, but have random coordinate and random time of appearance. One of the streams moves from South to North and we call it a stream of queries, and another stream moves from West to East and we call it a stream of objects. Moving objects closeness problem (MOC problem) consists of enumeration for every new query those and only those objects that will be not far than rho from the query in Manhattan metrics at some moment of time during the query's or the objects' movements inside the square. In general case this problem is very hard to solve because of dynamic situation and two-dimensional movements of objects and queries. But, in some cases the MOC problem is equivalent to one-dimensional range searching problem that can be solved effectively with logarithmic search, insertion and deletion time and a linear memory size as functions of the number of objects inside the square. In this paper we present and prove criteria for reducibility of the MOC problem to one-dimensional range searching problem.