Cocoon'02 - Algorithms for Normal Curves and Surfaces...

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Algorithms for Normal Curves and Surfaces Marcus Schaefer 1 , Eric Sedgwick 2 , and Daniel ˇ Stefankoviˇ c 3 1 DePaul University ( mschaefer@cs.depaul.edu ) 2 DePaul University ( esedgwick@cs.depaul.edu ) 3 University of Chicago ( stefanko@cs.uchicago.edu ) Abstract. We derive several algorithms for curves and surfaces rep- resented using normal coordinates. The normal coordinate representa- tion is a very succinct representation of curves and surfaces. For em- bedded curves, for example, its size is logarithmically smaller than a representation by edge intersections in a triangulation. Consequently, fast algorithms for normal representations can be exponentially faster than algorithms working on the edge intersection representation. Normal representations have been essential in establishing bounds on the com- plexity of recognizing the unknot [Hak61, HLP99, AHT02], and string graphs [SS ˇ S02]. In this paper we present efficient algorithms for count- ing the number of connected components of curves and surfaces, deciding whether two curves are isotopic, and computing the algebraic intersec- tion numbers of two curves. Our main tool are equations over monoids, also known as word equations. 1 Introduction Computational topology is a recent area in computational geometry that inves- tigates the complexity of determining properties of topological objects such as curves and surfaces [BE + 99, DEG99]. For example, it is known that we can de- cide whether two curves on a surface are homotopic in linear time if the surface is represented by a triangulation, and the curves as sequences of intersections with the triangulation [DG99]. In 1930 Kneser [Kne30] introduced a representation for curves and surfaces in which these objects are described by their normal coordinates . This led to the theory of normal surfaces which was used by Haken in 1961 to show that the un- knot could be recognized by an algorithm (which, much later, was shown to run in exponential time). Haken’s approach was pushed further by Hass, Lagarias, and Pippenger who showed that the unknot could be recognized in NP [HLP99]. To this end they had to verify in polynomial time that a special type of normal surface was an essential disk. The result of [HLP99] was recently extended by Agol, Hass, and Thurston [AHT02]. The main contribution of [AHT02] was a polynomial time algorithm for computing the number of connected components of a normal surface. This immediately implies polynomial time algorithms for checking whether a normal surface is connected, and whether it is orientable. The theory of normal curves is much simpler than the theory of normal surfaces. Nevertheless, it was one of three essential ingredients in the proof that
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string graphs can be recognized in NP , a problem that had only recently been shown to be decidable at all [SS ˇ S02].
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Cocoon'02 - Algorithms for Normal Curves and Surfaces...

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