6.838, Computational Geometry; Fall 2001

Point Location

Sergi Elizalde & David Pritchard

Definite Linear Storage, Logarithmic Queries with n log n Construction Time

The probability that D has maximum query time O(log n) is high.

Similarly one can show that the probability that the size of D is O(n) is high.

So, the probability that both things happen simultaneously (and hence the running time of the algorithm is O(n log n)) is high as well.

Theorem.
There exists a D that uses O(n) storage and has O(log n) query time in the worst case.

To obtain such a D:

- Run the algorithm on the set S, keeping track of the size and maximum length of a query path on the search structure being created.

- As soon as the size or depth get too big, stop the algorithm and start again from the beginning with a fresh random permutation.

- The probability that we get the desired size and depth is high (bounded from below by a constant).

- So, the expected number of trials before we get a good D is constant.