shortest-noncontractible

# shortest-noncontractible - Computational Topology(Jeff...

This preview shows pages 1–2. Sign up to view the full content.

Computational Topology (Jeff Erickson) Shortest Noncontractible Cycles A tie is a noose, and inverted though it is, it will hang a man nonetheless if he’s not careful. — Yann Martel, Life of Pi (2001) (written:) Yakka foob mog. Grug pubbawup zink wattoom gazork. Chumble ¨spuzz. (spoken:) I love loopholes. — Calvin, explaining Newton’s First Law of Motion ‘in his own words’ Bill Watterson, Calvin and Hobbes (January 9, 1995) 8 Shortest Noncontractible Cycles In this lecture, I’ll describe several algorithm to compute the shortest noncontractible cycle in a combi- natorial 2-manifold. The input is a cellularly embedded graph G on a 2-manifold Σ (or equivalently, a polygonal schema for Σ ) with non-negatively weighted edges, and our goal is to compute a cycle in G of minimum length that is not contractible in Σ . This problem has several natural motivations. the length of the shortest non-contractible cycle. In topological graph theory, the length of the shortest non-contractible cycle in an embedded graph is called the handle girth [ 1 ] or edge-width of the embedding [ 15 , 18 ] ; graphs that have embeddings with large edge-width share many useful combinatorial properties with planar graphs. A closely related concept is the face-width or representativity of a graph embedding, which is the minimum number of faces of G intersecting a noncontractible cycle on the surface, or equivalently, half the edge-width of the radial graph G ± [ 17 ] . The length of the shortest noncontractible cycle in a Riemannian manifold is called the systole (to be more speciﬁc, the homotopy 1-systole ) of the manifold [ 2 ] . Shortest non-contractible cycles are good indicators of topological noise of geometric surface models reconstructed from point clouds or volume data. The algorithms described below all require that edge weights are non-negative. We treat the graph as a continuous metric space, where the edge weights represent distance. I will assume throughout the lecture that the shortest path between any two vertices of the graph is unique; this assumption can be enforced if necessary using standard perturbation schemes. However, we do not assume that the edge weights satisfy the triangle inequality. 1 8.1 Thomassen’s 3-Path Property The ﬁrst efﬁcient algorithm to compute shortest non-contractible cycles is an easy consequence of the following observation of Thomassen [ 18 ] . Lemma 8.1. Let x and y be two points on a surface Σ , and let α , β , γ be paths in Σ from x to y . If the loops α · ¯ β and β · ¯ γ are contractible, then the loop α · ¯ γ is also contractible. Proof: The concatenation of any two contractible loops is contractible. ± Corollary 8.2.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

shortest-noncontractible - Computational Topology(Jeff...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online