12 research outputs found
Covering many points with a small-area box
Let be a set of points in the plane. We show how to find, for a given
integer , the smallest-area axis-parallel rectangle that covers points
of in time. We also consider the problem of,
given a value , covering as many points of as possible with an
axis-parallel rectangle of area at most . For this problem we give a
probabilistic -approximation that works in near-linear time:
In time we find an
axis-parallel rectangle of area at most that, with high probability,
covers at least points, where
is the maximum possible number of points that could be
covered
Computing the smallest k-enclosing circle and related problems
AbstractWe present an efficient algorithm for solving the âsmallest k-enclosing circleâ (kSC) problem: Given a set of n points in the plane and an integer k â©œ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments
Randomized Data Structures for the Dynamic Closest-Pair Problem
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 points in -dimensional space, for any fixed , 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 time and requires expected 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 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
Static and dynamic algorithms for k-point clustering problems
Let be a set of points in -space and let be an integer. A unified approach is given for solving the problem of finding a subset of of size that minimizes some closeness measure, such as the diameter, perimeter or the circumradius. Moreover, data structures are given that maintain such a subset under insertions and deletions of points