Instead of maintaining such a decomposition explicitly, it is convenient to construct a compressed quadtree T of the points of P , and every pair, (Ai, Bi) is just a pair of nodes (vi, ui) of T , such that Ai = Pvi and Bi = Pui , where Pv denote the points of P stored in the subtree of v, where v is a node of T . This gives us a compact representation of the distances. We slightly modify the co...