6.838, Computational Geometry; Fall 2001

Point Location

Sergi Elizalde & David Pritchard

A Tail Estimate

We have seen that the expected query time is good. However, for some points the query time can be bad.

We will see that the probability that this happens is small.

Lemma.

Let l>0 be a parameter. The probability that the search path for q has more than 3lln(n+1) nodes is at most 1/(n+1)^{l ln1.25-1}

Proof

Consider the following acyclic graph:

	The nodes are all the subsets of S
	Arcs go from every subset of size i to every subset of size i+1 containing it.
	Note that directed paths from the empty set to S correspond to permutations
	Every arc represents the insertion of an element

Mark an arc if the insertion of the corresponding element changes the trapezoid containing q.

Using backwards analysis, every node has at most 4 marked incoming arcs.

Mark arbitrary nodes so that every node has exactly 4 marked incoming arcs (with the exception of those having less than 4 incoming arcs, for which we mark all of them).