1. 2015/9/81 Advisor : Chih-Hung Yen Student : Chia-Ying Wu 2015/9/811

2. 2 1. Fundamental Concepts 1.1 Polyominoes, Enumerations, and Tessellations 1.2 Skewing Schemes and Data Templates 1.3 Surroundings 1.4 Labelings 2. Main Results 2.1 Some Results 2.2 N 2 -Skewing Schemes 3. Conclusion 2015/9/82 Outline

3. 3 333

4. 4 444

5. 5 Let the Euclidean plane (or the plane) be divided into unit squares, that is, the four corners of a square have coordinates (x,y), (x+1,y), (x,y+1), (x+1,y+1) for some integers x and y. And, for each unit square, we use the coordinate of the lower left corner to name itself. 2015/9/855

6. 6 For example, we use unit square (1,1) to represent the unit square whose four corners have coordinates (1,1), (2,1), (1,2) and (2,2). 2015/9/866 (1,1)(2,1) (1, 2) (2, 2) y x

7. 2015/9/87 A polyomino is defined as a finite, nonempty, and connected set of unit squares. A configuration generalizes the notation of polyomino by dropping the requirement “being connected”. 2015/9/8777 y x

8. 8 Consider a polyomino P (or a configuration C) of size N, namely, P (or C) consists of N unit squares. A polyomino of size N is also called an N-omino. 2015/9/88

9. 9 In 1907, the notion of “polyomino” appeared in a puzzle involving 5 unit squares was posed in the book 「 Canterbury Puzzles 」. Between the years 1937 to 1957, many results with the polyominoes of size 1 to 6 were first published in 「 Fairy Chess Review 」, a puzzle journal issued in British, under the name “dissection problems”. In 1953, the name polyomino was invented by Golomb and popularized by Gardner. 2015/9/899

10. 10 Polyominoes are sources of combinatorial problems. The most basic one is to enumerate distinct polyominoes of size N for a given positive integer N. 2015/9/810

11. 2015/9/811 2015/9/811 Another important problem on the subject of polyomino is the following: 『 Given a polyomino P, can P tessellate the plane? 』 11 2015/9/8

12. 122015/9/812 x (1,1)(2,1) (1, 2) (3,1) (6,2)(7,2) (6, 3) (8,2) v =(v x,v y ) = (5,1) Q P y

13. 2015/9/813 y x P Q

14. 2015/9/814 y x Q P

15. 2015/9/815 Consider two polyominoes P and Q in the plane. If Q satisfies one of the following conditions, then Q is said to be equivalent to P. 1. Q is a translation, a rotation, or a reflection of P. 2. Q is a rotation or a reflection of some translation of P. 3. Q is a reflection of some rotation of P. 4. A translation of Q is a reflection of some rotation of P. 2015/9/815

16. 2015/9/8162015/9/816 We say that a polyomino P tiles the plane, or there exists a tiling of the plane using a polyomino P, if the plane can be composed of polyominoes that are equivalent to P and do not overlap except along their sides. 16

17. 2015/9/8172015/9/817

18. 2015/9/8182015/9/818 We say that a polyomino P tessellates the plane, or there exists a tessellation of the plane using a polyomino P, if the plane can be composed of polyominoes that are translations of P and do not overlap except along their sides. 18

19. 2015/9/8192015/9/8192015/9/819 x y O 2015/9/819

20. 2015/9/8202015/9/820 If a polyomino P tessellates the plane, then P also tiles the plane. Conversely, if P tiles the plane, then we cannot guarantee that P tessellates the plane. 20

21. 2015/9/8212015/9/821 It is known that every N-omino for 1  N  6 tiles the plane and only four of all 7-ominoes cannot tile the plane. 21

22. 2015/9/8222015/9/822 Golomb proved that the problem of determining whether an arbitrary finite set of polyomino tiles the plane is undecidable; that is, there is no NP-algorithm for this problem. On the other hand, there does exist several methods for determining whether a (single) polyomino tessellates the plane. 22

23. 2015/9/8232015/9/823

24. 2015/9/824 A single instruction-stream, multiple data-stream (SIMD) computer contains one control unit, t arithmetic processors, and one memory unit of N (independent) memory modules. All arithmetic processor receive the same instruction from the control unit, but operate on different items of data stored in different memory modules. 2015/9/824

25. 2015/9/825 control unit A1A1 A1A1 A2A2 A2A2 AtAt AtAt memory-processor connection network M1M1 M1M1 M2M2 M2M2 MNMN MNMN arithmetic processors memory modules 252015/9/8

26. 262015/9/826 We consider to store an N  N matrix A into the N memory modules of an SIMD computer, where A = [A i, j ] for i = 0,1,…,N － 1 and j = 0,1,…,N － 1.

27. 2015/9/827 If the element A i, j is stored in memory module j for all i and j, then it will be possible to simultaneously fetch all the elements of any row of A, since distinct elements of a row of A lie in distinct memory modules. 272015/9/8 A 0,0 A 1,0 A 2,0. A N  1,0 A 0,0 A 1,0 A 2,0. A N  1,0 A 0,1 A 1,1 A 2,1. A N  1,1 A 0,1 A 1,1 A 2,1. A N  1,1 A 0,2 A 1,2 A 2,2. A N  1,2 A 0,2 A 1,2 A 2,2. A N  1,2 A 0, N  1 A 1, N  1 A 2, N  1. A N  1, N  1 A 0, N  1 A 1, N  1 A 2, N  1. A N  1, N  1 M0M0 M1M1 M2M2 MN1MN1

28. 2015/9/828 Fetching all the elements of any column of A, however, will result in delays, since more than one element of the column of A (in fact all) will lie in the same memory module. 282015/9/8 A 0,0 A 1,0 A 2,0. A N  1,0 A 0,0 A 1,0 A 2,0. A N  1,0 A 0,1 A 1,1 A 2,1. A N  1,1 A 0,1 A 1,1 A 2,1. A N  1,1 A 0,2 A 1,2 A 2,2. A N  1,2 A 0,2 A 1,2 A 2,2. A N  1,2 A 0, N  1 A 1, N  1 A 2, N  1. A N  1, N  1 A 0, N  1 A 1, N  1 A 2, N  1. A N  1, N  1 M0M0 M1M1 M2M2 MN1MN1

29. 2015/9/829 A 0,0 A 1, N  1 A 2, N  2. A N  1,1 A 0,0 A 1, N  1 A 2, N  2. A N  1,1 A 0,1 A 1,0 A 2, N  1. A N  1,2 A 0,1 A 1,0 A 2, N  1. A N  1,2 A 0,2 A 1,1 A 2,0. A N  1,3 A 0,2 A 1,1 A 2,0. A N  1,3 A 0, N  1 A 1, N  2 A 2, N  3. A N  1, 0 A 0, N  1 A 1, N  2 A 2, N  3. A N  1, 0 M0M0 M1M1 M2M2 MN1MN1 2015/9/8 If we adopt a different storage strategy instead, where the element A i, j is stored in memory module i + j (mod N), then all the elements of any row or any column of A will lie in distinct memory modules and can be fetched simultaneously.

30. 2015/9/830 Problem 1.2.1 (Shapiro; 1978) Given an M  M matrix, a collection of desirable matrix subparts such as rows, columns, or square blocks, and an SIMD computer with N memory modules, how do we store the matrix so that all the elements comprising any desirable matrix subpart are stored in different memory modules? 302015/9/8

31. 31 2015/9/8 A 0,0 A 1, N  1 A 2, N  2. A N  1,1 A 0,0 A 1, N  1 A 2, N  2. A N  1,1 A 0,1 A 1,0 A 2, N  1. A N  1,2 A 0,1 A 1,0 A 2, N  1. A N  1,2 A 0,2 A 1,1 A 2,0. A N  1,3 A 0,2 A 1,1 A 2,0. A N  1,3 A 0, N  1 A 1, N  2 A 2, N  3. A N  1, 0 A 0, N  1 A 1, N  2 A 2, N  3. A N  1, 0 M0M0 M1M1 M2M2 MN1MN1

32. 2015/9/832 COLUMN0COLUMN0 COLUMN1COLUMN1 ROW 0 ROW 1 32 ROW 2 012 2015/9/8 COLUMN2COLUMN2 COLUMN3COLUMN3 COLUMNN1COLUMNN1 3  N1N1 1 2 ROW 3 ROW N  1 3 N1N1 2 3 3 …

33. 2015/9/833 For any two positive integers M and N, an (M, N)-skewing scheme is defined as a 2-dimentional funtion S : ℤ M × ℤ M  ℤ N, namely, for each (i, j )  ℤ M × ℤ M, there exists a k  ℤ N such that S(i, j ) = k. An N-skewing scheme is a 2-dimentional funtion S : ℤ × ℤ  ℤ N, namely, for each k  ℤ N such that S(i, j ) = k. 332015/9/8

34. 34 A data template T is a set of ordered pairs of nonnegative integers in which no two components identical, namely, T = { (x 1, y 1 ), (x 2, y 2 ),..., (x t, y t ) }, where x i  0, y i  0, and (x i, y i )  (x j, y j ) for any i  j. An instance of a data template T through a vector v  ℤ × ℤ, denoted by T+v, is a set of ordered pairs of integers which formed by componentwise addition of v to T. For example, if T = {(x 1, y 1 ), (x 2, y 2 ),..., (x t, y t ) } and v = (v x,v y ), then T+v = { (x 1 +v x, y 1 +v y ), (x 2 +v x, y 2 +v y ),..., (x t +v x, y t +v y ) }. 342015/9/8

35. 35 In fact, for any data template T or any instance T+v of T, there exists uniquely a polyomino or a configuration which is corresponding to T or T+v, and vice versa. 352015/9/8

36. 362015/9/836 y x (0,0)(1,0) (0,1) (2,0) For example, a data template T = { (0,0), (1,0), (2,0), (0,1) }. P

37. 2015/9/837 We say that an (M, N)-skewing scheme or an N-skewing scheme, denoted by S, is valid for a collection of data templates, denoted by T 1, T 2,..., T r, if and only if, for any two ordered pairs (i 1, j 1 ) and (i 2, j 2 ) satisfying S(i 1, j 1 ) = S(i 2, j 2 ), there exists no l  {1, 2,..., r } such that an instance of T l contains both (i 1, j 1 ) and (i 2, j 2 ) as components. 372015/9/8

38. 382015/9/838 Problem 1.2.2 Consider a collection of data templates, how do we determine if there is a valid (M, N)-skewing scheme for this collection of data templates, and if a valid (M, N)-skewing scheme exists, how do we determine what it is? 38

39. 2015/9/8392015/9/839 Problem 1.2.3 Consider a data template T of size N satisfying that a polyomino corresponds to T, how do we determine whether there is a valid N-skewing scheme for T, if a valid N-skewing scheme for T exists, how do we determine what it is? Problem 1.2.4 Consider a data template T of size N satisfying that a configuration corresponds to T, how do we determine whether there is a valid N-skewing scheme for T, if a valid N-skewing scheme for T exists, how do we determine what it is? 39

40. 2015/9/8402015/9/840 Theroem 1.2.5 (Shapiro; 1978) (Wijshoff and van Leeuwen; 1984) Consider a data template T of size N. Then there exists a valid N- skewing scheme for T if and only if the polyomino or the configuration corresponding to T tessellates the plane. 40

41. 2015/9/8412015/9/841

42. 42 2015/9/8 P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P

43. 432015/9/843 Theorem 1.3.1 (Beauquier and Nivat; 1991) Consider a polyomino P. Then the plane can be tessellated by P if and only if there exists a surrounding of P. 43

44. 2015/9/8442015/9/844

45. 2015/9/845 Consider a polyomino P (or a configuration C) of size N in the plane. Then P (or C) has a 1-linear labeling if we can label the unit squares of P (or C) by using the elements of ℤ N exactly once in such a way that the labels of unit squares in each row is an arithmetic sequence with skip parameter a and the labels of unit squares in each column is an arithmetic sequence with skip parameter b, where a, b  ℤ N and a = b is allowed. 2015/9/845 0 0 1 1 2 2 4 4 3 3 5 5 6 6 7 7

46. 2015/9/846 Consider a polyomino P (or a configuration C) of size N in the plane. Then P (or C) has a 2-linear labeling if we can label the unit squares of P (or C) by using the elements of ℤ f × ℤ N / f exactly once for some f  1 and f | N (i.e., f divides N ) in such a way that the labels of unit squares in each row is an arithmetic sequence with skip parameter A=(a 1,a 2 ) and the labels of unit squares in each column is an arithmetic sequence with skip parameter B=(b 1,b 2 ), where A, B  ℤ f × ℤ N / f and A = B is allowed. 2015/9/846

47. 2015/9/847 (0,2) (1,1) (0,0) (1,2) (1,3) (1,0) (0,1) (0,3)

48. 2015/9/848 If a polyomino P (or a configuration C) of size N has a 1-linear labeling with skip parameter a and b in ℤ N, then P (or C) also has a 2-linear labeling with skip parameters A = (0,a) and B = (b, 0)  ℤ 1 × ℤ N. But the converse is not. 2015/9/848 (0,2) (1,1) (0,0) (1,2) (1,3) (1,0) (0,1) (0,3)

49. Theorem 1.4.1 (Chen, Hwang and Yen; 2006) A polyomino P of size N tessellates the plane if and only if P has a 2-linear labeling. 49 2015/9/8 0 0 1 1 2 2 4 4 3 3 5 5 6 6 7 7 (0,2) (1,1) (0,0) (1,2) (1,3) (1,0) (0,1) (0,3)

50. 2015/9/8502015/9/850

51. 2015/9/8512015/9/851

52. 2015/9/852 Lemma 2.1.1 An N-skewing scheme S is N-periodic if and only if S(i, j) = S(i + N, j) = S(i, j + N) for any (i, j )  ℤ × ℤ. Proof. (  ) It is trivial. (  ) We want to show that S(i, j) = S(i +  N, j +  N) for any (i, j )  ℤ × ℤ and any two integers  and . Without loss of generality, we suppose that   0 and   0. For any (i, j )  ℤ × ℤ, since S(i, j) = S(i + N, j ) = S(i, j + N), we have S(i, j) = S(i + N, j ) = S(i + 2N, j ) = … = S(i + (  1)N, j ) = S(i +  N, j ) = S(i +  N, j + N) = S(i +  N, j + 2N) = … = S(i +  N, j + (   1)N) = S(i +  N, j +  N). Thus S is N-periodic. 2015/9/852

53. 2015/9/853 Lemma 2.1.2 A linear N-skewing scheme is also N-periodic. Proof. Let S be a linear N-skewing scheme. By Lemma 2.1.1, it suffices to show that S(i, j) =S(i + N, j) = S(i, j + N) for any (i, j )  ℤ × ℤ. Since S is linear, there exist a and b in ℤ N such that S(i, j) = ia + jb (mod N) for any (i, j )  ℤ × ℤ, where a = b is allowed. Hence, for any (i, j )  ℤ × ℤ, we have S(i, j) = ia + jb (mod N) = ia + Na + jb (mod N) = (i + N)a + jb (mod N) =S(i + N, j) and S(i, j) = ia + jb (mod N) = ia + jb + Nb (mod N) =ia + (j + N)b (mod N) = S(i, j + N). Therefore, S is an N-periodic N-skewing scheme. 2015/9/853

54. 2015/9/854 Theorem 2.1.3 Problem 1.1.1, Problem 1.1.2, Problem 1.2.3, Problem 1.3.1, and Problem 1.4.2 are pairwise equivalent. Problem 1.1.1. For what classes of polyominoes do the existence of tessellations of the plane? Problem 1.1.2. For what classes of polyominoes do the existence of periodic tessellations of the plane? 2015/9/854

55. 2015/9/855 Problem 1.2.3. Consider a data template T of size N satisfying that a polyomino corresponds to T, how do we determine whether there is a valid N-skewing scheme for T, if a valid N-skewing scheme for T exists, how do we determine what it is? Problem 1.3.1. For what classes of polyominoes do the existence of surroundings? Problem 1.4.2. For what classes of polyominoes do the existence of 2–linear labelings? 2015/9/855

56. 2015/9/856 Theorem 2.1.4 Problem 1.1.3, and Problem 1.2.4 are equivalent. Problem 1.1.3. Does the existence of tessellations of the plane using a polyomino P imply the extence of periodic tessllations of the plane using P? Problem 1.2.4. Consider a data template T of size N satisfying that a configuration corresponds to T, how do we determine whether there is a valid N-skewing scheme for T, if a valid N-skewing scheme for T exists, how do we determine what it is? 2015/9/856

57. 2015/9/857 One feasible method for determine the existence of such a (A, B) is to test each ordered pairs of elements in ℤ f × ℤ N/f for any f  1 and f | N. However, it is inefficient. Let T N denote the total number of tests performed and f N denote the number of factors of N. Then 2015/9/857 N234510100 fNfN 223249 TNTN 14181624378408

58. 2015/9/858 On the other hand, if N is a prime number, then f N = 2 and T N  (N  1) 2 ; otherwise, f N  N and T N  (N  1) 3. Therefore, a method to find (A, B) quickly is required. 2015/9/858

59. 2015/9/8592015/9/859

60. 2015/9/8602015/9/860

61. 2015/9/8612015/9/861

62. 2015/9/862 An N 2 -skewing scheme is a mapping S: ℤ  ℤ  ℤ f × ℤ N/f for some f  1 and f | N, namely, for each (i, j)  ℤ  ℤ, there exists a (k 1, k 2 )  ℤ f × ℤ N/f such that S(i, j) = (k 1, k 2 ). An N 2 -skewing scheme S is valid for a collection of data templates, denoted by T 1, T 2,..., T r, if and only if, for any two ordered pairs (i 1, j 1 ) and (i 2, j 2 ) satisfying S(i 1, j 1 ) = S(i 2, j 2 ), there exists no l  {1, 2,..., r} such that an instance of T l contains both (i 1, j 1 ) and (i 2, j 2 ) as components. 622015/9/8

63. 63 An N 2 -skewing scheme S is N-periodic if S(i, j) = S( i +  N, j +  N ) for any (i, j)  ℤ  ℤ and any two integers  and . An N 2 -skewing scheme S is periodic if there exist u = (u 1, u 2 ) and v = (v 1, v 2 ) in ℤ  ℤ such that S(i, j) = S( i +  u 1 +  v 1, j +  u 2 +  v 2 ) for any (i, j)  ℤ  ℤ and any two integers  and . 632015/9/8

64. 64 An N 2 -skewing scheme S is linear if there exist A = (a 1, a 2 ) and B = (b 1, b 2 ) in ℤ f × ℤ N/f such that S(i, j) = iA + jB ( mod ( f, N / f ) ) = (ia 1 + jb 1 (mod f ), ia 2 + jb 2 (mod N / f )) for any (i, j)  ℤ  ℤ, where A = B is allowed. If an N-skewing scheme S is linear, then it is easy to obtain a linear N 2 -skewing scheme S : ℤ  ℤ  ℤ 1 × ℤ N by letting S(i, j)=(0, S(i, j)) for any (i, j)  ℤ  ℤ. 642015/9/8

65. 65 Lemma 2.2.1 An N 2 -skewing scheme S is N-periodic if and only if S(i, j) = S(i + N, j) = S(i, j + N) for any (i, j )  ℤ × ℤ. Proof. (  ) It is trivial. (  ) We want to show that S(i, j) = S(i +  N, j +  N) for any (i, j )  ℤ × ℤ and any two integers  and . Without loss of generality, we suppose that   0 and   0. For any (i, j )  ℤ × ℤ, since S(i, j) = S(i + N, j ) = S(i, j + N), we have S(i, j) = S(i + N, j ) = S(i + 2N, j ) = … = S(i + (  1)N, j ) = S(i +  N, j ) = S(i +  N, j + N) = S(i +  N, j + 2N) = … = S(i +  N, j + (   1)N) = S(i +  N, j +  N). Thus S is N-periodic. 2015/9/865

66. 2015/9/866 Lemma 2.2.2 A linear N 2 -skewing scheme is also N-periodic. Proof. Let S be a linear N 2 -skewing scheme. By Lemma 2.2.1, it suffices to show that S(i, j) =S(i + N, j) = S(i, j + N) for any (i, j )  ℤ × ℤ. Since S is linear, there exist A = (a 1, a 2 ) and B = (b 1, b 2 ) in ℤ f × ℤ N/f such that S(i, j) = iA + jB (mod ( f, N / f )) = (ia 1 + jb 1 (mod f ), ia 2 + jb 2 (mod N / f )) for any (i, j)  ℤ  ℤ, where A = B is allowed. 2015/9/866

67. 2015/9/867 Hence, for any (i, j )  ℤ × ℤ, we have S(i, j) = iA + jB (mod ( f, N / f )) = (ia 1 + jb 1 (mod f ), ia 2 + jb 2 (mod N / f )) = (ia 1 + f  N/f  a 1 + jb 1 (mod f ), ia 2 + f  N/f  a 2 + jb 2 (mod N / f )) = (ia 1 + Na 1 + jb 1 (mod f ), ia 2 + Na 2 + jb 2 (mod N / f )) = ((i + N)a 1 + jb 1 (mod f ), (i + N)a 2 + jb 2 (mod N / f )) = (i + N)A+ jB (mod ( f, N / f )) = S(i + N, j) and 2015/9/867

68. 2015/9/868 S(i, j) = iA + jB (mod ( f, N / f )) = (ia 1 + jb 1 (mod f ), ia 2 + jb 2 (mod N / f )) = (ia 1 + jb 1 + f  N/f  b 1 (mod f ), ia 2 + jb 2 + f  N/f  b 2 (mod N / f )) = (ia 1 + jb 1 + Nb 1 (mod f ), ia 2 + jb 2 + Nb 2 (mod N / f )) = (ia 1 + (j + N)b 1 (mod f ), ia 2 + (j + N)b 2 (mod N / f )) = iA+ (j + N)B (mod ( f, N / f )) = S(i, j+ N) Therefore, S is an N-periodic N 2 -skewing scheme. 2015/9/868

69. 2015/9/8692015/9/869

70. 2015/9/870 The study of tessellations of the plane using polyominoes has a long history in mathematics. And the motivation of our study comes from the labelings of polyominoes. We revise some definitions and terminologies, and give some results on the correlations between different methods of determining whether a polyomino tessellates the plane. However, there are still some unsolved problems. 2015/9/870

71. 2015/9/8712015/9/871