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 -