Eastern Great Lakes Theory Workshop Talk

Tracing Compressed Curves in Triangulated Surfaces

Amir Nayyeri, Carnegie Mellon University

Saturday, September 30, 11:30am-12:30pm

ABSTRACT

A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to "trace" a normal curve in O(min{X, n^2 log X}) time, where n is the complexity of the surface triangulation and X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in n. Our tracing algorithm computes a new cellular decomposition of the surface with complexity O(n); the traced curve appears as a simple path or cycle in the 1-skeleton of the new decomposition.

We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer, Sedgwick, and Štefankovic [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005].

Joint work with Jeff Erickson (SoCG'12, full version submitted to the special issue of SoCG'12)

Speaker Bio

Amir Nayyeri is a postdoctoral researcher in Carnegie Mellon University working with Gary Miller. He received his PhD in 2012 from University of Illinois at Urbana Champaign under the supervision of Jeff Erickson. He is generally interested in theoretical computer science and particularly in computational topology and geometry.

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