A Doubly Exponentially Crumbled Cake

Christ, Tobias; Francke, Andrea; Gebauer, Heidi; Matoušek, Jiřı́; Uno, Takeaki

doi:10.48550/arxiv.1104.0122

Cited by 2 publications

(1 citation statement)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Very little is known about anchored rectangle packing. Recently, Christ et al [9] proved that if Alice can force Bob's share to be less than 1 r , then n ≥ 2 2 Ω(r) . Our result indicates that this condition does not materialize for large r, since Bob can always cover at least a constant fraction of the area for any n ∈ N.…”

Section: Introductionmentioning

confidence: 99%

Packing anchored rectangles

Dumitrescu

Tóth

2015

Combinatorica

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Packing anchored rectangles

Dumitrescu

Tóth

2015

Combinatorica

View full text Add to dashboard Cite

show abstract

Packing anchored rectangles

Dumitrescu

Tóth

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

Let S be a set of n points in the unit square [0, 1] 2 , one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2. IntroductionWe consider a rectangle packing problem popularized by Peter Winkler [18,19,20], which has been open for decades. It is a one-round game between Alice and Bob. First, Alice chooses a finite point set S in the unit square U = [0, 1] 2 in the plane, including the origin, that is, (0, 0) ∈ S ( Fig. 1(a)). Then Bob chooses an axisparallel rectangle r(s) ⊆ U for each point s ∈ S such that s is the lower left corner of r(s), and the interior of r(s) is disjoint from all other rectangles ( Fig. 1(b)). The rectangle r(s) is said to be anchored at s, but r(s) contains no point from S in its interior. It is conjectured that for any finite set S ⊂ U , (0, 0) ∈ S, Bob can choose such rectangles that jointly cover at least half of U [1,18,19,20]. However, it has not even been known whether Bob can always cover at least a positive constant area. It is clear that Bob cannot always cover

show abstract

A Doubly Exponentially Crumbled Cake

Cited by 2 publications

References 4 publications

Packing anchored rectangles

Packing anchored rectangles

Packing anchored rectangles

Contact Info

Product

Resources

About