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Erschienen in:
2010 | OriginalPaper | Buchkapitel
A Randomized Algorithm for Weighted Approximation of Points by a Step Function
verfasst von : Jin-Yi Liu
Erschienen in: Combinatorial Optimization and Applications
Verlag: Springer Berlin Heidelberg
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The problem considered in this paper is: given an integer
k
> 0 and a set
P
of
n
points in the plane each with a corresponding non-negative weight, find a step function
f
with
k
steps that minimize the maximum weighted vertical distance between
f
and all the points in
P
. We present a randomized algorithm to solve the problem in
O
(
n
log
n
) expected running time. The bound is obviously optimal for the unsorted input. The previously best known algorithm runs in
O
(
n
log
2
n
) worst-case time. Another merit of the algorithm is its simplicity. The algorithm is just a randomized implementation of Frederickson and Johnson’s matrix searching technique, and it only exploits a simple data structure.