Formulae for Polyominoes on Twisted Cylinders Gadi Aleksandrowicz - PowerPoint PPT Presentation

Formulae for Polyominoes on Twisted Cylinders Gadi Aleksandrowicz (CS, Technion) Andrei Asinowski (Math, Technion, now in CS, Free Univ. Berlin) Gill Barequet (CS, Technion) Ronnie Barequet (CS, Tel Aviv Univ.) LATA’ 14, Madrid, March 2014 1 Center for Graphics and Geometric Computing, Technion

Polyominoes  A polyomino of size n is an edge-connected set of n squares in Z 2  We refer only to “fixed” polyominoes: distinct if they differ in shape or orientation ≠  Notation: A( n ) = # of polyominoes of size (# of squares) n 2 Center for Graphics and Geometric Computing, Technion

 : Growth Rate  Widely believed: ,  =-1    n A ( n ) Cn  Klarner ’ 67: exists   lim n A ( n )   n  A ( n 1 )  Madras ’ 99: also exists, and so is equal to  lim   A ( n ) n  Bounds: [B, Moffie, Ribo , & Rote ’ 06]   3 . 9801 ... (new lower bound: 4.00253, B, Rote, & Shalah, unpublished) [Klarner & Rivest ’ 73]   4 . 6496 ... [Gaunt, Sykes, & Ruskin ’ 76 , …]    4 . 06 0 . 02  A( n ) is currently known till n =56 [Jensen ’ 03] 3 Center for Graphics and Geometric Computing, Technion

Motivations  Statistical physics: Percolation processes: Fluid flow in random media Collapse of branched polymer molecules in dilute solutions  Mathematics and Computer Science: Hard enumeration problem Implementation challenge  Fun: Puzzles in 2D and 3D 4 Center for Graphics and Geometric Computing, Technion

Twisted Cylinders  New Idea: Consider polyominoes on a twisted cylinder (a spiral square lattice)  Here we can also count fixed polyominoes and estimate their growth rates. Notation: = # of polyominoes of size n on a twisted cylinder A w ( n ) of “width” (perimeter) w  A ( n 1 )   Respective growth rate w lim : w   A ( n ) n  Advantage: more structure than in the plane! w  Originally used for proving   3.9801 … (now 4.00253 …) 5

Main Results  When w tends to infinity, approaches   w (Previously it was only known that (𝜇 𝑥 ) is monotone increasing and that 𝜇 𝑥  𝜇 for all w )  Formulae enumerating polyominoes on twisted cylinders satisfy linear recurrences  Formulae computed up to w =10 6 Center for Graphics and Geometric Computing, Technion

Large Cylinders Thm:    lim . w   w Proof:      By Madras (1999), lim A ( n 1 ) / A ( n ) lim A ( n ) . n     n n  Similarly (BMRR’ 06),     lim A ( n 1 ) / A ( n ) lim A ( n ) n w w w w     and  1 ≤  2 ≤ ⋯ ≤  . n n  Thus, exists and Goal: Prove   lim        * * * . . w   w  Idea: For every find s.t.   (       ) 0 , w .  w ( )  So fix   0 . 7 Center for Graphics and Geometric Computing, Technion

   lim w   w Large Cylinders (cont.) Proof (cont.):              lim n A ( n ) n n ( ) s.t. n n , n A ( n ) . 0 0 0   n      In particular, n A ( n ) . 0 0  The desired cylinder is of width w  n : 0 For obviously  k  A n ( k ) A ( k ). n , 0 0 In addition, there exists a monotone increasing subsequence of By concatenation A n k ( ) : k 0    i.e., that is, 2 A ( n ) A ( 2 n ) , A ( m ) A ( n ) A ( m n ), n n w w w 0 0 hence, is monotone  i i A ( n ) A ( 2 n ) , 2 n A ( 2 n 0 ) n 2 n 0 n n n 0 0 0 and found below its limit. Hence,         n lim A ( n ) n A ( n ) A ( n ) . n 0 0 0 0 n n n   0 0 0 n Q.E.D. 8 Center for Graphics and Geometric Computing, Technion

Large Cylinders (take 2) Thm:    lim . w   w Ideas behind the proof:  Main difficulty: Order of limits ( 𝑜 and then 𝑋 ).  If limit on 𝑋 is taken first, proof becomes easy.  So calculus machinery is used to show rigorously that exchanging the order of limits is permitted. 9 Center for Graphics and Geometric Computing, Technion

Plot of  w  27  4 . 0025 ... [B, Rote, Shalah, unpublishd ] 10 Center for Graphics and Geometric Computing, Technion

Recall:  =4.06 ± 0.02 widely-believed Estimating  , I  Assuming that has a 1/w-expansion, approximate  w   it by first 20 terms:    19 i f ( w ) c / w w i i 0  Consider only (for error is large)   w  w 1 , 2 , 3 4 22   Solve LP problem :     f ( w ) , . w   Minimize  , subject to , 4 w 22     f ( w ) w  Solution: 𝑑 0 ≈ 4.06766 (𝑑 1 ≈ −0.878, 𝑑 2 ≈ −27.3, 𝑑 3 ≈ 108, 𝑑 4 ≈ −361, 𝑑 5 ≈ 891, 𝑑 6 ≈ −978, 𝑑 7 = ⋯ = 𝑑 19 = 0 → 𝜁 ≈ 4.29 ⋅ 10 −6 ) 11 Center for Graphics and Geometric Computing, Technion

Recall:  =4.06 ± 0.02 widely-believed Estimating  , II  Similarly, solve a linear least-square problem: where and  x  ,    j A b , A 1 / i b . i j i i  Again, consider only and, this time, only  w  4 22 ,  j  0 6 .  Solution: LP solution:   4 . 06789 4 . 06766 x c 0 0     x 0 . 892 c 0 . 878 1 1     x 27 . 0 c 27 . 3 2 2   x 105 c 108 3 3     x 339 c 361 4 4 12 Center for Graphics and Geometric Computing, Technion

Recall:  =4.06 ± 0.02 widely-believed Estimating  , III  Wynn’s epsilon algorithm  Result: 4.04161 (All three estimations were performed by Mathematica.) 13 Center for Graphics and Geometric Computing, Technion

0 Signature System 4 3 0  Invented in [Jensen ‘ 01 + ’ 03] 0 3 for non -twisted cylinders 3 0  Signatures of polyominoes describe 4 connectivity of boundary cells 2 0  5-letter alphabet: 1  0: empty  2: lowest in component  0 3  1: singleton  3: middle in component 3 2  4: highest in component  M w+1 -1 valid signatures [B, Moffie, Ribó , & Rote, ’ 06], where M k is the k th Motzkin number (~3 k /k 3/2 )  Adding cells (either occupied or empty) ad infinitum model the growth of polyominoes 14 Center for Graphics and Geometric Computing, Technion

Finite Automaton Example ( w =3):  Each state is a signature (initial: 000)  An edge labeled i means concatenating an  i  ( 0 w ) occupied cell followed by i empty cells  Dead-end state (and edges leading to it) omitted  Initial state 000 and accepting state 100 united  Goal: Count accepted words of size n . 15 Center for Graphics and Geometric Computing, Technion

Transfer Matrix  k x k matrix, k = number of signatures  Recall: k = ( w +1)st Motzkin number (~3 w+1 )  A ij : # of edges leading from state i to state j 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 16 Center for Graphics and Geometric Computing, Technion

Transfer Matrix to Recurrence  Method described by Stanley [Enumerative Combinatorics, vol. I, 1997, sec. 4.7]  Route 1 (direct): Compute minimal polynomial of transfer matrix  linear recurrence  Route 2 (a bit more tedious): Compute generating function for accepting state  linear recurrence 17 Center for Graphics and Geometric Computing, Technion

Twisted Cylinder of Width 3    3 2 x x x 1  Generating function:    3 2 2 x x 2 x 1  Recurrence:    a 2 a a 2 a    n n 1 n 2 n 3  Sequence: (1,2,6,16,42,112,298,792,2106 ,…) cylinder polyomino subgraph of host graph 18 Center for Graphics and Geometric Computing, Technion

Twisted Cylinder of Any Width Thm : For any satifies a linear recurrence.  w 2 , A ( n ) w Proof : As demonstrated above. Polyominoes are in bijection with words accepted by a finite automaton that models growth of polyominoes on a twisted cylinder. Recurrence is obtained from the transfer matrix. 19 Center for Graphics and Geometric Computing, Technion