TY - GEN

T1 - The visible perimeter of an arrangement of disks

AU - Nivasch, Gabriel

AU - Pach, János

AU - Tardos, Gábor

PY - 2013

Y1 - 2013

N2 - Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n1/2), then there is a stacking order for which the visible perimeter is Ω(n2/3). We also show that this bound cannot be improved in the case of the n1/2 × n 1/2 piece of a sufficiently small square grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n3/4) with respect to any stacking order. This latter bound cannot be improved either. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.

AB - Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n1/2), then there is a stacking order for which the visible perimeter is Ω(n2/3). We also show that this bound cannot be improved in the case of the n1/2 × n 1/2 piece of a sufficiently small square grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n3/4) with respect to any stacking order. This latter bound cannot be improved either. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.

KW - Visible perimeter

KW - dense set

KW - disk

KW - unit disk

UR - http://www.scopus.com/inward/record.url?scp=84874119705&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-36763-2_33

DO - 10.1007/978-3-642-36763-2_33

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AN - SCOPUS:84874119705

SN - 9783642367625

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 364

EP - 375

BT - Graph Drawing - 20th International Symposium, GD 2012, Revised Selected Papers

T2 - 20th International Symposium on Graph Drawing, GD 2012

Y2 - 19 September 2012 through 21 September 2012

ER -