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

Center for Graphics and Geometric Computing, Technion

1

Gadi Aleksandrowicz (CS, Technion) Andrei Asinowski (Math, Technion, now in CS, Free Univ. Berlin) Gill Barequet (CS, Technion) Ronnie Barequet (CS, Tel Aviv Univ.)

Formulae for Polyominoes

• n Twisted Cylinders

LATA’14, Madrid, March 2014

Center for Graphics and Geometric Computing, Technion

2

Polyominoes

 A polyomino of size n is an edge-connected set of n squares in Z2  We refer only to “fixed” polyominoes: distinct if they differ in shape or orientation ≠  Notation: A(n) = # of polyominoes of size (# of squares) n

Center for Graphics and Geometric Computing, Technion

3

: Growth Rate

 Widely believed: ,  =-1  Klarner ’67: exists  Madras ’99: also exists, and so is equal to  Bounds: [B, Moffie, Ribo, & Rote ’06] (new lower bound: 4.00253, B, Rote, & Shalah, unpublished) [Klarner & Rivest ’73] [Gaunt, Sykes, & Ruskin ’76, …]  A(n) is currently known till n =56 [Jensen ’03]  ) ( A ) 1 ( A lim n n

n



  n n

n) ( A lim

 

 

... 9801 . 3   ... 6496 . 4   02 . 06 . 4   

n

Cn n 



 ) ( A

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

Center for Graphics and Geometric Computing, Technion

4

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

• f “width” (perimeter) w

Respective growth rate

 Advantage: more structure than in the plane!  Originally used for proving   3.9801…

(now 4.00253…)

5

) (n Aw

: ) ( A ) 1 ( A lim n n

w w n w

 

 



Center for Graphics and Geometric Computing, Technion

6

Main Results

 When w tends to infinity, approaches (Previously it was only known that (𝜇𝑥) is monotone increasing and that 𝜇𝑥𝜇 for all w)  Formulae enumerating polyominoes on twisted cylinders satisfy linear recurrences  Formulae computed up to w =10

w

 

Large Cylinders

Thm: Proof:  By Madras (1999),  Similarly (BMRR’06), and 1 ≤ 2 ≤ ⋯ ≤ .  Thus, exists and Goal: Prove  Idea: For every find s.t.  So fix

Center for Graphics and Geometric Computing, Technion

7

. lim   

  w w

. ) ( lim ) ( / ) 1 ( lim

n n n

n A n A n A

   

   

n w n w w n w

n A n A n A ) ( lim ) ( / ) 1 ( lim

   

   

w w

 

 

 lim

*

.

*

   .

*

   ,   ) ( w .

) (

  



 

w

.  

Large Cylinders (cont.)

Proof (cont.):   In particular,  The desired cylinder is of width For obviously In addition, there exists a monotone increasing subsequence of By concatenation i.e., that is, hence, is monotone and found below its limit. Hence,

Q.E.D.

Center for Graphics and Geometric Computing, Technion

8

  

  w w

lim . ) ( , s.t. ) ( ) ( lim            

  n n n

n A n n n n n A ). ( ) ( k A k An  . ) ( ) ( ) ( lim        

  n n n n n n n

n A n A n A . ) (    

n

n A : n w  , n k  : ) (

k n k

A ), ( ) ( ) ( n m A n A m A

w w w

  , ) 2 ( ) (

2

n A n A

n n



2 0)

2 (

n i n

i

n A , ) 2 ( ) (

2 n n n n

n A n A 

Large Cylinders (take 2)

Thm: 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.

Center for Graphics and Geometric Computing, Technion

9

. lim   

  w w

Plot of w

Center for Graphics and Geometric Computing, Technion

10

] unpublishd Shalah, Rote, [B, ... 0025 . 4

27 



Estimating , I

 Assuming that has a 1/w-expansion, approximate it by first 20 terms:  Consider only (for error is large)  Solve LP problem: Minimize , subject to  Solution: 𝑑0 ≈ 4.06766 (𝑑1 ≈ −0.878, 𝑑2 ≈ −27.3, 𝑑3 ≈ 108, 𝑑4 ≈ −361,

𝑑5 ≈ 891, 𝑑6 ≈ −978, 𝑑7 = ⋯ = 𝑑19 = 0 → 𝜁 ≈ 4.29 ⋅ 10−6)

Center for Graphics and Geometric Computing, Technion

11

w



 

 

19

/ ) (

i i i w

w c w f 

22 4   w 3 , 2 , 1  w



.

22 4 , , ) ( ) (       w w f w f

w w

   

Recall: =4.06±0.02 widely-believed

Estimating , II

 Similarly, solve a linear least-square problem: where and  Again, consider only and, this time, only  Solution: LP solution:

Center for Graphics and Geometric Computing, Technion

12

j j i

i A / 1

, 

.

i i

b   , b A  x , 22 4   w . 6   j

361 339 108 105 3 . 27 . 27 878 . 892 .

4 4 3 3 2 2 1 1

06766 . 4 06789 . 4

                c x c x c x c x c x

Recall: =4.06±0.02 widely-believed

Estimating , III

 Wynn’s epsilon algorithm  Result: 4.04161 (All three estimations were performed by Mathematica.)

Center for Graphics and Geometric Computing, Technion

13

Recall: =4.06±0.02 widely-believed

Signature System

 Invented in [Jensen ‘01 + ’03] for non-twisted cylinders  Signatures of polyominoes describe connectivity of boundary cells  5-letter alphabet:  0: empty  2: lowest in component   1: singleton  3: middle in component

 4: highest in component

 Mw+1-1 valid signatures [B, Moffie, Ribó, & Rote, ’06], where Mk is the kth Motzkin number (~3k/k3/2)  Adding cells (either occupied or empty) ad infinitum model the growth of polyominoes

Center for Graphics and Geometric Computing, Technion

14

4 3 3 3 4 2 1 3 3 2

Finite Automaton

Example (w =3):  Each state is a signature (initial: 000)  An edge labeled i means concatenating an

• ccupied 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.

Center for Graphics and Geometric Computing, Technion

15

) ( w i  

Transfer Matrix

 kxk matrix, k = number of signatures  Recall: k = (w +1)st Motzkin number (~3w+1)  Aij: # of edges leading from state i to state j

Center for Graphics and Geometric Computing, Technion

16

0 1 2 3 4 5 6 7 1 2 3 4 5 6 7

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

Center for Graphics and Geometric Computing, Technion

17

Twisted Cylinder of Width 3

 Generating function:  Recurrence:  Sequence: (1,2,6,16,42,112,298,792,2106,…) cylinder polyomino subgraph of host graph

Center for Graphics and Geometric Computing, Technion

18

1 2 2 1

2 3 2 3

      x x x x x x

3 2 1

2 2

  

  

n n n n

a a a a

Twisted Cylinder of Any Width

Thm: For any satifies a linear recurrence. 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.

Center for Graphics and Geometric Computing, Technion

19

) ( , 2 n A w

w



Recurrences for 2w6

Above method allowed computation for

Center for Graphics and Geometric Computing, Technion

20

,9,-2) 03,34,-1,7 161,-201,1 45,147, 299,731,-2 1,330,120, ,-183,-110 1200,-1622

• ,1016,

1924,-2893 472,-2952, 93,-2537,2 1109,367,7

• ,1941,

2815,-2102 181,-2416, 87,-1411,2 435,-550,7 46,-208, ,204,-107, 0,217,-260

• 21,32,-10

(8,-23,30, : 6

• 14,0,-2)

6,22,23,1, ,10,-14,-1 ,0,25,9,18 (4,-1,-6,5 : 5 ) 555 , 181 , 59 , 19 , 6 , 2 , 1 ( : init ), 2 , 6 , 2 , 2 , 7 , 8 , 5 ( 2 6 2 2 7 8 5 : 4 ) 6 , 2 , 1 ( : init , ) 2 , 1 , 2 ( 2 2 : 3 ) 1 ( : init , ) 2 ( 2 : 2

7 6 5 4 3 2 1 3 2 1 1

                   

          

w w a a a a a a a a w a a a a w a a w

n n n n n n n n n n n n n n

Recurrences for 7w10

 Computation of minimal polynomial or generating function from transfer matrix is not feasible (takes too much time, requires too much memory)  BUT: We have enough memory for building the automata (actually, for w much greater than 10)  Thus, we can generate values from the beginning

• f the sequences as much as we want!

Center for Graphics and Geometric Computing, Technion

21

Recurrences for 7w10 (cont.)

Variant of Berlekamp-Massey alg (originally for LSR):  Choose big enough where L is the largest coefficient of recurrence in absolute value  Compute “modulo sequences” si = s mod pi  Recover recurrences of all si; since original non-negative coefficients are recovered in the range [0,N/2] and original negative coefficients in (N/2,N]  Use CRT to recover each original coefficient independently; if >N/2, subtract N  Caveat: L unknown… Remedy: Start from nominal L; test if procedure succeeded by checking enough values; if not, double L and repeat.

Center for Graphics and Geometric Computing, Technion

22

 

 

k i i

L p N

1

, 2 , 2L N 

Recurrence Data

w Order Mw+1 | Largest Coefficient | Value Bits Time (seconds, Home Laptop) 1 1    2 1 2 2 1  3 3 4 2 1  4 7 9 8 3  5 18 21 25 5  6 48 51 2.95  103 12  7 121 127 8.74  107 27  8 315 323 5.48  1019 66 < 1 9 826 835 5.18  1051 172 20 10 2,168 2,188 6.39  10129 432 300

Center for Graphics and Geometric Computing, Technion

23

B.-M. + CRT

Exponent blow-up by a factor of ~2.5?

Doubly-Exponential Growth

• f Largest Coefficient

Center for Graphics and Geometric Computing, Technion

24

Conclusion

 We used automata in order to find recurrence formulae enumerating polyominoes on twisted cylinders

(The purpose of this was to find the growth rates of polyomiones on twisted cylinders -> lower bounds on 𝜇.)

 CRT was used in order to recover formulae for 𝑥 ≤ 10

Center for Graphics and Geometric Computing, Technion

25

Center for Graphics and Geometric Computing, Technion

26

Thank You! ¡Gracias!