3 research outputs found
Partitioning a Polygon Into Small Pieces
We study the problem of partitioning a given simple polygon into a
minimum number of polygonal pieces, each of which has bounded size. We give
algorithms for seven notions of `bounded size,' namely that each piece has
bounded area, perimeter, straight-line diameter, geodesic diameter, or that
each piece must be contained in a unit disk, an axis-aligned unit square or an
arbitrarily rotated unit square.
A more general version of the area problem has already been studied. Here we
are, in addition to , given positive real values such that
the sum equals the area of . The goal is to partition
into exactly pieces such that the area of is .
Such a partition always exists, and an algorithm with running time has
previously been described, where is the number of corners of . We give
an algorithm with optimal running time . For polygons with holes, we
get running time .
For the other problems, it seems out of reach to compute optimal partitions
for simple polygons; for most of them, even in extremely restricted cases such
as when is a square. We therefore develop -approximation algorithms
for these problems, which means that the number of pieces in the produced
partition is at most a constant factor larger than the cardinality of a minimum
partition. Existing algorithms do not allow Steiner points, which means that
all corners of the produced pieces must also be corners of . This has the
disappointing consequence that a partition does often not exist, whereas our
algorithms always produce useful partitions. Furthermore, an optimal partition
without Steiner points may require pieces for polygons where a
partition consisting of just pieces exists when Steiner points are allowed.Comment: 32 pages, 24 figure