TY - GEN
T1 - Random triangulations of planar point sets
AU - Sharir, Micha
AU - Welzl, Emo
PY - 2006
Y1 - 2006
N2 - Let S be a finite set of n + 3 points in general position in the plane, with 3 extreme points and n interior points. We consider triangulations drawn uniformly at random from the set of all triangulations of S, and investigate the expected number, v̂i, of interior points of degree i in such a triangulation. We provide bounds that are linear in n on these numbers. In particular, n/43 ≤ v̂3 ≤ (2n + 3)/5. Moreover, we relate these results to the question about the maximum and minimum possible number of triangulations in such a set S, and show that the number of triangulations of any set of n points in the plane is at most 43n, thereby improving on a previous bound by Santos and Seidel.
AB - Let S be a finite set of n + 3 points in general position in the plane, with 3 extreme points and n interior points. We consider triangulations drawn uniformly at random from the set of all triangulations of S, and investigate the expected number, v̂i, of interior points of degree i in such a triangulation. We provide bounds that are linear in n on these numbers. In particular, n/43 ≤ v̂3 ≤ (2n + 3)/5. Moreover, we relate these results to the question about the maximum and minimum possible number of triangulations in such a set S, and show that the number of triangulations of any set of n points in the plane is at most 43n, thereby improving on a previous bound by Santos and Seidel.
KW - Charging
KW - Counting
KW - Crossing-free geometric graphs
KW - Crossing-free spanning trees
KW - Degree sequences
KW - Number of triangulations
KW - Random triangulations
UR - http://www.scopus.com/inward/record.url?scp=33748064661&partnerID=8YFLogxK
U2 - 10.1145/1137856.1137898
DO - 10.1145/1137856.1137898
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AN - SCOPUS:33748064661
SN - 1595933409
SN - 9781595933409
T3 - Proceedings of the Annual Symposium on Computational Geometry
SP - 273
EP - 281
BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06
PB - Association for Computing Machinery (ACM)
T2 - 22nd Annual Symposium on Computational Geometry 2006, SCG'06
Y2 - 5 June 2006 through 7 June 2006
ER -