Random triangulations of planar point sets

Micha Sharir, Emo Welzl

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publicationProceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06
PublisherAssociation for Computing Machinery (ACM)
Number of pages9
ISBN (Print)1595933409, 9781595933409
StatePublished - 2006
Event22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States
Duration: 5 Jun 20067 Jun 2006

Publication series

NameProceedings of the Annual Symposium on Computational Geometry


Conference22nd Annual Symposium on Computational Geometry 2006, SCG'06
Country/TerritoryUnited States
CitySedona, AZ


  • Charging
  • Counting
  • Crossing-free geometric graphs
  • Crossing-free spanning trees
  • Degree sequences
  • Number of triangulations
  • Random triangulations


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