Title: Computing Optimum Homotopy on Surfaces

                 	Shripad Thite
   Center for the Mathematics of Information
       California Institute of Technology

ABSTRACT:  Last week, I defined the Homotopic Fréchet Distance between two
curves A and B embedded in a metric space as the minimum cost of a homotopy
(`leash map') between some re-parameterization of A and some
re-parameterization of B. In the case that the metric space is the Euclidean
plane with polygonal obstacles, I hinted at a polynomial-time algorithm to
compute an optimum leash map.  This algorithm enumerated a polynomial number
of candidate homotopy classes. I will finish describing the algorithm to
comput an optimum leash map in a given homotopy class h by sketching its two
main
ingredients:

1) Given a real number d, decide whether there exists a leash map of length
at most d.  The key ingredient for solving this decision problem is a data
structure which stores the funnel of geodesics (shortest paths) between
relevant pairs of vertices and edges.

2) Given an algorithm for the decision problem above, we obtain the minimum
value of d for which there exists a leash map of length at most d (and thus
we also obtain the Homotopic Fréchet Distance) by using a very general and
powerful algorithmic technique called Parametric Search. I will describe
parametric search in as much generality as time permits as well as how it is
applied to the current problem.

I will also try to motivate the general problem of computing an optimum
homotopy between curves and cycles (closed loops) on general combinatorial
surfaces. The following problem is especially
interesting:

"Given two cycles A and B on a triangulated surface embedded in E^3, find a
homotopy between them that minimizes the maximum length of every
intermediate cycle."

I will sketch some algorithms that have been used to decide efficiently
whether two cycles on a surface are homotopic, and sketch how to extend them
to solve the above optimization problem.