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キーワード:
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要旨:
We describe a new randomized data structure,
the {\em sparse partition}, for solving the
dynamic closest-pair problem. Using this
data structure the closest pair of a set of $n$ points in
$k$-dimensional space, for any fixed $k$, can be found
in constant time. If the points are chosen from a finite universe,
and if the floor function is available at unit-cost, then the
data structure supports insertions into and deletions from the set
in expected $O(\log n)$ time and requires expected $O(n)$ space.
Here, it is assumed that the updates are chosen by an adversary who
does not know the random choices made by the data structure.
The data structure can be modified to run in $O(\log^2 n)$ expected
time per update in the algebraic decision tree model of
computation. Even this version is more efficient than the currently
best known deterministic algorithms for solving the problem for $k>1$.
