Call q
and q' equivalent if they follow the same path through the search structure
D. 6.838, Computational Geometry; Fall 2001
Point Location
Sergi Elizalde & David Pritchard
Lemma.
The probability that the maximum search path in D has more than 3lln(n+1) nodes is at most 2/(n+1)lln1.25-3.
Proof.
Call q
and q' equivalent if they follow the same path through the search structure
D.
Partition the
plane into vertical slabs by passing a vertical line through every endpoint
of S.
Partition every
slab into trapezoids by intersecting it with all possible segments in S.
This decomposes
the plane into at most 2(n+1)2 trapezoids.
Any two points
in the same trapezoid are equivalent in any search structure for S.
Take a point
in each trapezoid.
For each of the
points we can apply the previous lemma.
So, the probability
that the length of the search path for any of the 2(n+1)2
points exceeds the bound is at most 2/(n+1)lln1.25-3.
