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Pants Decomposition of the Punctured Plane

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Talk given at the MSRI Workshop on Topological Methods, 2006

Talk given at the MSRI Workshop on Topological Methods, 2006

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Pants Decomposition of the Punctured Plane Pants Decomposition of the Punctured Plane Presentation Transcript

  • MSRI Workshop on Topological Methods October 2–6, 2006, Berkeley, CA Pants Decomposition of the Punctured Plane Shripad Thite sthite@win.tue.nl Department of Mathematics and Computer Science Technische Universiteit Eindhoven The Netherlands Joint work with Sheung-Hung Poon 1-1
  • Surface ≡ 2-manifold Σ boundary of Σ Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 2-1
  • Pant ≡ a sphere with 3 holes waist W leg R leg L Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 3-1
  • Pant ≡ a sphere with 3 holes Every simple cycle on a pant is contractible to a point or to a boundary Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 3-2
  • Pants decomposition Definition: A set of simple cycles that decompose a surface into disjoint pants To understand the topology of the surface and to compute its various properties Every compact orientable surface has at least one pants decomposition* * except sphere, cylinder, disk, or torus A surface can have many pants decompositions Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 4-1
  • Essential cycle Simple cycle not contractible to a point or to a boundary Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 5-1
  • Example: decomposing into pants Σ Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 6-1
  • Example: decomposing into pants Σ β α essential cycles on Σ Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 6-2
  • Example: decomposing into pants Σ cut! Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 6-3
  • Example: decomposing into pants Σ γ essential cycle on Σ Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 6-4
  • Example: decomposing into pants π1 π2 cut! Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 6-5
  • The big open problem Computing an exact or approximate shortest pants decomposition of a general combinatorial surface Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 7-1
  • The big open problem Computing an exact or approximate shortest pants decomposition of a general combinatorial surface We consider a variant in the Euclidean plane . . . Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 8-1
  • Punctured plane the plane a c e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 9-1
  • Punctured plane the plane a c e d b f point punctures (zero-length cycles) Surface Σ is the plane minus n points Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 9-2
  • Decomposing the punctured plane Σ c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-1
  • Decomposing the punctured plane essential cycle on Σ c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-2
  • Decomposing the punctured plane c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-3
  • Decomposing the punctured plane c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-4
  • Decomposing the punctured plane Σ essential cycle on Σ c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-5
  • Decomposing the punctured plane c a e d b f Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 10-6
  • Properties c a e d b c a f c c a a e d b e e d d b b f f c c a a f e e d d b b n − 1 disjoint simple cycles f f Nested in a binary tree Decompose the plane into a set of pants and an un- bounded component Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 11-1
  • Shortest pants decomposition Input: A set P of n points in the plane E2 Remove an ε-disk Di centered at each point pi ∈ P n 2 Let Π be a pants decomposition of E Di i=1 Imagine tightening the cycles of Π as ε → 0 The limit is a non-crossing pants decomposition Π Problem: Compute a non-crossing pants decompo- sition Π∗ of E2 P of minimum total length Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 12-1
  • Points on a line not optimal shorter Lemma: Every cycle in a shortest pants decomposi- tion of collinear points encloses an interval of points Compute shortest pants decomposition in O(n2 ) time using dynamic programming with Yao’s speedup Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 13-1
  • A lower bound Every cycle in a shortest pants decomposition is a simple polygon with vertices in P (no Steiner points) A shortest pants decomposition Π∗ of E2 P contains a TSP tour of P So, |Π∗ | ≥ |T SP (P )| Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 14-1
  • O(log n) approximation Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 15-1
  • O(log n) approximation Construct an O(1)- 2 approximate TSP 1 3 tour T 8 4 7 Start with the n T points in order along the tour T 6 5 Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 15-2
  • O(log n) approximation Repeatedly enclose 2 pairs of smaller cy- 1 3 cles by a bigger cy- cle until we have 8 a pants decomposi- tion Π 4 7 T 6 5 Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 15-3
  • O(log n) approximation Each cycle of Π 2 is obtained by dou- 1 3 bling the edges of a sub-tour of T 8 4 7 T 6 5 Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 15-4
  • O(log n) approximation Each edge of T be- 2 longs to O(log n) 1 3 cycles of Π 8 4 7 So, T |Π| ≤ O(log n) |T | ≤ O(log n) |Π∗ | 6 5 Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 15-5
  • PTAS For every ε > 0, compute a (1+ε)-approximate short- est pants decomposition in polynomial time Extension of PTAS for Euclidean TSP Uses Mitchell’s guillotine rectangular subdivisions Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 16-1
  • Previous work Allen Hatcher. Pants Decompositions of Surfaces. arxiv.org/abs/math.GT/9906084 The pants decomposition complex of a given surface is simply connected—vertices are isotopy classes of pants decompositions, edges correspond to elementary moves S-move A-move Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 17-1
  • Previous work ´ Eric Colin de Verdi`re and Francis Lazarus. e Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface. Graph Drawing, pp. 478–490, 2003 (+EuroCG’03) Show how to shorten a given pants decomposition Given a pants decomposition of a general combinatorial surface, they compute a homotopic pants decomposi- tion in which each cycle is shortest in its homotopy class Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 18-1
  • Work in progress NP-complete for general surfaces? . . . I believe so! NP-complete for the punctured plane? . . . I don’t know David Eppstein recently obtained an O(n log n)-time algorithm to compute an O(1)-approximation for the punctured plane using quadtrees [to appear in SODA 2007] Pants Decomposition of the Punctured Plane /// Shripad Thite, TU-Eindhoven 19-1
  • Thank you! 20-1