We define the notion of a well-separated pair decomposition of points in d -dimensional space. We then develop efficient sequential and parallel algorithms for computing such a decomposition. We apply the resulting decomposition to the efficient computation of k -nearest neighbors and n -body potential fields.

A convincing proof of the decidability of reachability in vector addition systems is presented.No drastically new ideas beyond those in Sacerdote and Tenney, and Mayr are made use of. The complicated tree constructions in the earlier proofs are completely eliminated.

No abstract

No abstract

Two sets of planar points S1 and S2 are circularly separable i f there is a circle that encloses S1 but excludes S2. We show that deciding whether two sets are circularly separable can be accomplished in O(n) time via Megiddo's linear programming algorithm. We also show that a smallest separating circle can be found in O(n) time, and largest separating circles can be found in O(n1ogn) time. Finally we establish that all these results are optimal.