Counting and Enumeration in Combinatorial Geometry Günter Rote

Transcription

1 Counting and Enumeration in Combinatorial Geometry Günter Rote Freie Universität Berlin General position: No three points on a line two triangulations

2 Counting and Enumeration in enumeration counting and sampling bounds optimization Combinatorial Geometry Günter Rote Freie Universität Berlin General position: No three points on a line two triangulations

3 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?

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5 Min #Triangulations: Ω(2.43 n ) O(3.455 n )

6 Previous Results on Perfect Matchings P convex position smallest possible number of perfect matchings: Θ (2 n ) Q double-chain previous record: Θ (3 n ) [García, Noy, Tejel 2000] Upper bound: O (10.06 n ) [Sharir, Welzl 2006] = up to a polynomial factor

7 The Double-Zigzag Chain P Q

8 The Proof C = 2(1 + x + x 3 ) 2(1 + x + x 3 ) smallest singularity: 1 9x 3x 2 = 0 ( 1 2x 8x 2 3x 3 + (1 + x) ) (1 x 3x 2 )(1 9x 3x 2 ) 4x(1 + x)(1 + x + x 3 ) 1/ x 0 = 93 x 0 = /( 93 9) #(perfect matchings in P Q) = Θ ( n )

9 Longer Arcs P = nr r = 5 3 n r = 8: Θ ( n ) [ joint work with Andrei Asinowski ]

10 Dynamic Programming Recursion A runners C B runners X n 1 A k runners X n B X n B = # possibilities after n arcs with B crossing runners

11 Example: r = 5 matrix for transforming (X0 n 1, X1 n 1, X2 n 1,...) into (X0 n, X1 n, X2 n,...) row sum = 271 = vectors grow like 271 n /poly(n)

12 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation

13 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

14 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

15 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

16 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

17 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

18 Counting Triangulations Counting, sampling, enumerating [ V. Alvarez, R. Seidel 2013 ] triangulation sequence of x-monotone paths

19 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 )

20 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!

21 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

22 Extension to Perfect Matchings Every point set has at least [ Manuel Wettstein 2014 ] Catalan(n/2) 2 n perfect non-crossing matchings. v 2 v 3 v 4 v 1 v i v n v n 1

23 Extension to Perfect Matchings Every point set has at least [ Manuel Wettstein 2014 ] Catalan(n/2) 2 n perfect non-crossing matchings. v 2 v 3 v 4 i 2 v 1 v i v n v n 1 n i (tight (almost only) for point sets in convex position)

24 Extension to Perfect Matchings Every point set has at least [ Manuel Wettstein 2014 ] Catalan(n/2) 2 n perfect non-crossing matchings. v 1 v 2 v 3 v 4 v i i 2 TRICK to achieve polynomial delay: Output those trivial matchings while preparing the DAG. v n v n 1 n i (tight (almost only) for point sets in convex position)