6.838, Computational Geometry; Fall 2001
Point Location
Sergi Elizalde & David Pritchard
Proof: Expected Query Time is O(log n)
Fix a query point q, and consider the path in D traversed by the query.
Define
Si = {s1, s2, ..., si}
Xi = number of nodes added to the search path for q during iteration i
Pi = probability that some node on the search path of q is created in iteration i
Dq(Si) = trapezoid containing q in T(Si)
From our construction, Xi £ 3; thus E[Xi] £ 3Pi
Note
Pi = Pr[Dq(Si) ¹ Dq(Si-1)]
Backwards analysis:
How many segments in Si affect Dq(Si) when they are removed? At most 4. Since they have been chosen in random order, each one has probability 1/i of being si. So, Pi £ 4/i.