testing-contractibility - Computational Topology(Jeff...

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Computational Topology (Jeff Erickson) Testing Homotopic Paths in the Plane When you turn the corner And you run into yourself Then you know that you have turned All the corners that are left. — Langton Hughes, “Final Curve” (1951) 3 Testing Homotopic Paths in the Plane In the previous lecture, we saw an algorithm to compute the shortest path in a polygon with holes that is homotopic to a given path. Here we consider an apparently easier problem: Given two paths in P , are they homotopic? Or equivalently, given a loop in P , is it contractible. We can solve both problems by computing the reduced crossing sequence(s) of the input path(s). Two paths are homotopic if and only if they have the same reduced crossing sequence, and a loop is contractible if and only if its reduced crossing sequence is the empty string. We can test both of these conditions in time O ( nk ) using the first two phases of our previous algorithm. In this lecture, I’ll describe a faster algorithm for both of these problems for simple paths and loops, originally due to Cabello et al. [ 4 ] , with some simplifications by Efrat et al. [ 6 ] . Most of the lecture will concentrate on testing whether a simple loop in a polygon with holes is contractible. We will briefly return to testing homotopy of simple paths at the end. 3.1 Sentinel Points If the number of holes in P is small, we can test contractibility more quickly by applying a little more topology. Let S = { s 1 , s 2 ,..., s h } denote a set of h sentinel points, one chosen arbitrarily inside each hole of P . Because P is a proper subset of the space R 2 \ S , any function from [ 0 , 1 ] 2 to P is also a function from [ 0 , 1 ] 2 to R 2 \ S . It follows that any contractible loop in P is also contractible in R 2 \ S . In fact, the converse is true as well. Lemma 3.1. Let be a loop in P . If is contractible in R 2 \ S , then is contractible in P . Proof: Let P 0 denote the outer boundary of P , and let P 1 ,..., P h denote the boundaries of its holes, indexes so that each sentinel point s i lies in the interior of the corresponding polygon P i . The Jordan- Schönflies theorem implies that for each i , there is a homeomorphism φ i : R 2 R 2 such that the restriction of φ i to S 1 is the cycle P i . Without loss of generality, we can assume that φ i ( 0 ) = s i , where 0 denotes the origin. Let u : R 2 \ 0 S 1 be the function u ( x ) = x / k x k ; this function is clearly continuous. Then the function Φ i = φ i u φ - 1 i maps R 2 \ s i continuously onto P i . Finally, let Φ : R 2 \ S P denote the function Φ( x ) = x if x P , Φ 0 ( x ) if x is outside P 0 , Φ i ( x ) if x is inside P i . This function is continuous and obviously fixes P . Thus, for any homotopy h : [ 0 , 1 ] 2 R 2 \ S between two paths in P , the function Φ h : [ 0,1 ] 2 P is a homotopy in P between the same two paths.
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