Approximation Algorithms for the Minimum Convex Partition Problem

Knauer, Christian; Spillner, Andreas

doi:10.1007/11785293_23

Cited by 8 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Knauer and Spillner [26] recently showed that any planar n-element point set admits a convex partition with at most 15n−24 11 faces (improving an earlier bound of 10n−18 7 by Neumann-Lara et al [36]); García-López and Nicolás [17] gave a lower bound construction of 12n 11 − 2 for n ≥ 4. Knauer and Spillner [26] also gave a polynomial time 30 11 -approximation for the minimum number convex partition problem. No corresponding results are known for the minimum number convex Steiner partition problem.…”

Section: Our Contribution (I)mentioning

confidence: 86%

Minimum Weight Convex Steiner Partitions

Dumitrescu

Tóth

2009

Algorithmica

View full text Add to dashboard Cite

New tight bounds are presented on the minimum length of planar straight line graphs connecting n given points in the plane and having convex faces. Specifically, we show that the minimum length of a convex Steiner partition for n points in the plane is at most O(log n/ log log n) times longer than a Euclidean minimum spanning tree (EMST), and this bound is the best possible. Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudo-triangle. We also show that the minimum length convex Steiner partition of n points along a pseudo-triangle is at most O(log log n) times longer than an EMST, and this bound is also the best possible. Our methods are constructive and lead to O(n log n) time algorithms for computing convex Steiner partitions having O(n) Steiner points and weight within the above worst-case bounds in both cases.

show abstract

Section: Our Contribution (I)mentioning

confidence: 86%

Minimum Weight Convex Steiner Partitions

Dumitrescu

Tóth

2009

Algorithmica

View full text Add to dashboard Cite

show abstract

“…There has been substantial work on estimating f 2 (n), and computing f 2 (S) for a given set S in the plane. It has been shown successively that f 2 (n) ≤ 10n−18 7 by Neumann-Lara et al [34], f 2 (n) ≤ 15n−24 11 by Knauer and Spillner [29], and f 2 (n) ≤ 4n−6 3 for n ≥ 6 by Sakai and Urrutia [36]. From the other direction, García-López and Nicolás [20] proved that f 2 (n) ≥ 12n− 22 11 , for n ≥ 4, thereby improving an earlier lower bound f 2 (n) ≥ n + 2 by Aichholzer and Krasser [1].…”

Section: Introductionmentioning

confidence: 99%

Minimum Convex Partitions and Maximum Empty Polytopes

Dumitrescu

Har-Peled

Tóth

2012

Algorithm Theory – SWAT 2012

View full text Add to dashboard Cite

Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n). Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1] d .

show abstract

“…Spillner [Spi05] has given a fixed-parameter algorithm for the problem, the number of points in the interior of the convex hull being the parameter. For the case that points are in general position, Knauer and Spillner [KS06] have given a simple 3-approximation that runs in O(n log n) time and a more involved 30/11-approximation that runs in O(n 2 ) time.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, there are point sets where any minimum convex decomposition consists of at least 12n/11 − 2 subpolygons [GN06]. (On the other hand, Knauer and Spillner [KS06] showed that every point set can be decomposed into no more than 15n/11 − 24/11 convex polygons. )…”

Section: Introductionmentioning

confidence: 99%

Decomposing a simple polygon into pseudo-triangles and convex polygons

Wolff

2008

Computational Geometry

View full text Add to dashboard Cite

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudotriangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n 3 ) time and space, where n is the number of the vertices of the polygon.

show abstract

Approximation Algorithms for the Minimum Convex Partition Problem

Cited by 8 publications

References 8 publications

Minimum Weight Convex Steiner Partitions

Minimum Weight Convex Steiner Partitions

Minimum Convex Partitions and Maximum Empty Polytopes

Decomposing a simple polygon into pseudo-triangles and convex polygons

Contact Info

Product

Resources

About