Well Separated Pairs Decomposition
نویسنده
چکیده
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 construction of the compressed quadtree, such that for every nodes v ∈ T , it also stores a representative point repv, which is a point in Pv. Furthermore, repv is one of the representative points of one of the children of v. This can be easily computed in linear time, once the compressed quadtree is computed. Before presenting the algorithm for computing WSPD, we remind the reader that `(v) is lgκ( v), where v is the cell (i.e., square, cube or hypercube depending on the dimension) which is the region that the node v corresponds to, and κ( v) is the sidelength of v. Since, the root node corresponds to the unit-square/hypercube, `(v) is always a non-positive integer number. For techincal reasons, we induce an aribtrary ordering of the nodes of the given quadtree. Let denote this ordering.
منابع مشابه
INDEX TERMS: none.
Well-separated pair decomposition, introduced by Callahan and Kosaraju [3], has found numerous applications in solving proximity problems for points in the Euclidean space. A pair of point sets (A, B) is c-well-separated if the distance between A,B is at least c times the diameters of both A and B. A well-separated pair decomposition of a point set consists of a set of well-separated pairs that...
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