Quasi-Valid range querying and its implications for nearest neighbor problems

We define a new formalism called the quasi-valid range aggregation. This formalism leads to a new and quite simple method for reducing non-range query-like problems to range queries and often to orthogonal range queries, with immediate applications to the ATTRACTED NEIGHBOR and the planar ALL-PAIRS NEAREST NEIGHBORS problems (the latter being solved optimally on both parallel and sequential machines). A point of special interest is that our new formalism permits operators “+” that are neither associative nor abelian (unlike traditional range query theory). The technique described in this paper should therefore have significant other applications besides those we consider, especially in the context of parallel computational geometry.