Counting and Enumeration in Combinatorial Geometry G unter Rote - PowerPoint PPT Presentation

Counting and Enumeration in Combinatorial Geometry G¨ unter Rote Freie Universit¨ at Berlin two triangulations General position: No three points on a line G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting and Enumeration in Combinatorial Geometry G¨ unter Rote Freie Universit¨ at Berlin • enumeration • counting and sampling • bounds • optimization • · · · two triangulations General position: No three points on a line G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Background Given a set of n points in the plane in general position, how many • triangulations • non-crossing spanning trees • non-crossing Hamiltonian cycles • non-crossing matchings • non-crossing perfect matchings • . . . • [your favorite straight-line geometric graph structure] can it have? G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Min #Triangulations: Ω(2 . 43 n ) O (3 . 455 n ) G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Previous Results on Perfect Matchings P Q convex position double-chain smallest possible number of previous record: Θ ∗ (3 n ) perfect matchings: Θ ∗ (2 n ) [Garc´ ıa, Noy, Tejel 2000] Upper bound: O ∗ (10 . 06 n ) [Sharir, Welzl 2006] ∗ = up to a polynomial factor G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

The Double-Zigzag Chain P Q G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

The Proof � � 1 − 2 x − 8 x 2 − 3 x 3 + (1 2(1 + x + x 3 ) − 2(1 + x + x 3 ) C = 4 x (1 + x )(1 + x + x smallest singularity: 1 − 9 x − 3 x 2 = 0 √ 93 − 3 x 0 = 6 2 √ 1 / √ x 0 = � 6 / ( 93 − 9) ≈ 3 . 0532 #( perfect matchings in P ∪ Q ) = Θ ∗ (3 . 0532 n ) G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Longer Arcs | P | = nr + 1 r = 5 n 1 2 3 r = 8 : Θ ∗ (3 . 0930 n ) [ joint work with Andrei Asinowski ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Dynamic Programming Recursion A runners B runners C k runners X n − 1 X n B A X n B = # possibilities after n arcs with B crossing runners G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Example: r = 5 matrix for transforming ( X n − 1 , X n − 1 , X n − 1 , . . . ) into 0 1 2 ( X n 0 , X n 1 , X n 2 , . . . )   10 30 30 20 5 1 0 0 0 0 0 · · · 30 40 50 35 21 5 1 0 0 0 0 · · ·     30 50 45 51 35 21 5 1 0 0 0 · · ·     20 35 51 45 51 35 21 5 1 0 0 · · ·     5 21 35 51 45 51 35 21 5 1 0 · · ·     1 5 21 35 51 45 51 35 21 5 1 · · ·     0 1 5 21 35 51 45 51 35 21 5 · · ·     0 0 1 5 21 35 51 45 51 35 21 · · ·     0 0 0 1 5 21 35 51 45 51 35 · · ·     0 0 0 0 1 5 21 35 51 45 51 · · ·     0 0 0 0 0 1 5 21 35 51 45 · · ·   . . . . . . . . . . .  ...  . . . . . . . . . . . . . . . . . . . . . . ⇒ vectors grow like 271 n / poly( n ) row sum = 271 = G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation → sequence of x -monotone paths → path in a DAG of size O ∗ (2 n ) G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation ↔ sequence of x -monotone paths MARKED paths → path in a DAG of size O ∗ (2 n ) always choose the LEFTmost triangle! G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation ↔ sequence of x -monotone paths MARKED paths → path in a DAG of size O ∗ (2 n ) always choose the LEFTmost triangle! O (1) -delay enumeration, with O ∗ (2 n ) preprocessing G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Extension to Perfect Matchings [ Manuel Wettstein 2014 ] Every point set has at least Catalan( n/ 2) ∼ 2 n perfect non-crossing matchings. v 3 v 2 v 4 v 1 v i v n v n − 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Extension to Perfect Matchings [ Manuel Wettstein 2014 ] Every point set has at least Catalan( n/ 2) ∼ 2 n perfect non-crossing matchings. v 3 v 2 i − 2 v 4 v 1 v i v n v n − 1 n − i (tight (almost only) for point sets in convex position) G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden

Extension to Perfect Matchings [ Manuel Wettstein 2014 ] Every point set has at least Catalan( n/ 2) ∼ 2 n perfect non-crossing matchings. v 3 v 2 i − 2 v 4 TRICK to achieve polynomial v 1 delay: v i Output those “trivial” matchings while preparing the DAG. v n v n − 1 n − i (tight (almost only) for point sets in convex position) G¨ unter Rote, Freie Universit¨ at Berlin Counting and Enumeration in Combinatorial Geometry Enumeration Algorithms Using Structure, August 24–28, 2015, Leiden