From Tetris to polyominoes generation. June 3rd, 2016 GASCOM 2016 La Marana, Bastia, France.
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1 From Tetris to polyominoes generation June 3rd, 2016 GASCOM 2016 La Marana, Bastia, France.
2 Authors Enrico Formenti Laboratoire I3S Université Nice Sophia Antipolis Paolo Massazza Università dell Insubria Varese, Italy. 2
3 Tetris All variants of tetrominos fall one-by-one from top to bottom. 3
4 Classic Tetris 4
5 Classic Tetris 4
6 Classic Tetris 4
7 Classic Tetris 4
8 Classic Tetris 4
9 Classic Tetris 4
10 Classic Tetris 4
11 Classic Tetris 4
12 Classic Tetris 4
13 Classic Tetris 4
14 Classic Tetris 4
15 Classic Tetris 4
16 Classic Tetris 4
17 Classic Tetris 4
18 Classic Tetris 4
19 Classic Tetris 4
20 Classic Tetris 4
21 Classic Tetris 4
22 Classic Tetris 4
23 Classic Tetris 4
24 Classic Tetris Cancelling rule 4
25 Classic Tetris Gravity rule 4
26 Classic Tetris 4
27 Bad Tetris and bad players Pieces: any bar of length k (bad Tetris) 5
28 Bad Tetris and bad players Pieces: any bar of length k (bad Tetris) Area: fix s N (bad player) 5
29 Bad Tetris and bad players Pieces: any bar of length k (bad Tetris) Area: fix s N (bad player) -Rules: No cancelling (bad Tetris) 5
30 Bad Tetris and bad players Pieces: any bar of length k (bad Tetris) Area: fix s N (bad player) -Rules: No cancelling (bad Tetris) +Rules: New pieces and old ones have to be 4-connected (bad player) 5
31 Bad Tetris and bad players Pieces: any bar of length k (bad Tetris) Area: fix s N (bad player) -Rules: No cancelling (bad Tetris) +Rules: New pieces and old ones have to be 4-connected (bad player) -Rules: No more left and right walls (bad Tetris) 5
32 Bad Tetris (and bad player) 6
33 Bad Tetris (and bad player) 6
34 Bad Tetris (and bad player) 6
35 Bad Tetris (and bad player) 6
36 Bad Tetris (and bad player) 6
37 Bad Tetris (and bad player) 6
38 Bad Tetris (and bad player) 6
39 Bad Tetris (and bad player) 6
40 Bad Tetris (and bad player) 6
41 Bad Tetris (and bad player) 6
42 Polyominoes Polyomino Any finite 4-connected subset of Z 2 7
43 Polyominoes Polyomino Any finite 4-connected subset of Z 2
44 Polyominoes Polyomino Any finite 4-connected subset of Z 2 8
45 Polyominoes: history and motivations Introduction: Golomb (1954) Polyominoes are widely studied in many fields: Enumerative combinatorics Bijective combinatorics Two dimensional language theory Tilings Discrete tomography 9
46 Polyominoes: the open questions Exhaustive generation Can polyominoes of a given area n be efficiently generated? Enumeration How many polyominoes of a given area n are there? Closed formula Is there a closed formula for the number of polyominoes of a given area n? 10
47 Exhaustive generation: known results & CATs 2011 Parallel polyominoes [Mantaci, Massazza] 2012 L-convex polyominoes [Massazza] 2014 Z-convex polyominoes [Castiglione, Massazza] 2015 Convex-polyominoes [Massazza] 2015 k-convex polyominoes [Brocchi, Castiglione, Massazza] 2016 Prefix-closed polyominoes [this talk] 11
48 Polyomino representation Polyomino Ordered sequence of columns. Column Sequence of integers {α 1, x 1, α 2, x 2,..., α k, x k } st. α 1 = vertical displacement wrt previous column x i = segment length α i = hole length (i > 1) 12
49 Some notation Given P = {C 1, C 2,..., C i,... C k } with C 1 = {α 1, x 1, α 2, x 2,..., α k, x k }, C 2 = { α 1, x 1, α 2, x 2,..., α l, x l},... P i = C i (i-th column of P) P i = {C 1,..., C i } C i C i 1 iff any segment of C i is edge adjacent to some segment of C i 1 13
50 Comparing columns C = {α 1, x 1, α 2, x 2,..., α k, x k } C = {α 1, x 1, α 2, x 2,..., α l, x l } C C if one of the following holds k i=1 x i < l j=1 x i k i=1 x i = l j=1 x i but α 1 > α 1 k i=1 x i = l j=1 x i and α 1 = α 1 but e s.t. 3.1 x i = x i and α i = α i for 1 < i < e 3.2 x e > x e if e is even 3.3 α e < α e if e is odd 14
51 Comparing polyominoes P = {c 1, c 2,..., c i,... c k } P = { c 1, c 2,..., c i,... } c l P P iff i s.t. c i < c i and c h = c h for h < i. 15
52 Prefix-closed polyominoes Prefix-closed polyomino P = {c 1, c 2,..., c i,... c k } is prefix-closed iff for all i {1,..., k}, P i is a polyomino.
53 Prefix-closed polyominoes Prefix-closed polyomino P = {c 1, c 2,..., c i,... c k } is prefix-closed iff for all i {1,..., k}, P i is a polyomino. Prefix-closed Non Prefix-closed 16
54 Bad Tetris + bad player = Prefix-closed polyominoes 17
55 Bad Tetris + bad player = Prefix-closed polyominoes 17
56 Bad Tetris + bad player = Prefix-closed polyominoes 17
57 The result Theorem There exists a CAT algorithm for the exhaustive generation of PCPol(n) which uses O(n) space. 18
58 Generating columns: split move j Split(a, b, j) a b a b 19
59 Generating columns: shift move j Shift(a, b, j) a b a b 20
60 Generating columns: the grand ancestor Grand ancestor C = {α 1, x 1, α 2, x 2,..., α k, x k } C = {α 1, x 1, α 2, x 2,..., α l, x l } G(C, C) = column which is 1. the smallest -compat. with C 2. identical to C up to j = max {M(C )} 3. admits a move at j 4. has same area as C 21
61 Generating columns: the grand ancestor Shift Shift Split Split a b a G M(b) = {2, 6, 8, 9} 22
62 Generating columns: the dynamical system C(a, r) = set of all columns of area r and -compatible with a b j b if b = Split(a, b, j) or b = Shift(a, b, j) f a,r (b) = { b (α 1 h(a) + 1, r) otherwise b, GA(b, a) j b, j = max M(b) 23
63 Generating columns: the dynamical system (cont.) Lemma Fix an integer r and a column a = (α 1, x 1, α 2, x 2,..., α k, x k ) then for all b, d C(a, r) it holds: 1. f a,r (b) a 2. b < d implies f a,r (b) < f a,r (d); 3. f n a,r(b) < f n+1 a,r (b) for b (α 1 h(a) + 1, r); 4. n N fn a,r((r 1, r)) = C(a, r). 24
64 Generating PCPol(n): the algorithm 1: Procedure PCPolGen(n) 2: for r := 1 to n 1 do 3: P 1 := (0, r); ColGen(2, n r); 4: end for 5: P := (0, n); Output(P); 25
65 Generating PCPol(n): the algorithm (columns) 1: Procedure ColGen(i, r) 2: for d := 1 to r do 3: InitColumn(i); {i.e. P i := ((d), d 1)} 4: Init(S, i); {init stack} 5: if d < r then ColGen(i + 1, r d); else Output(P); endif 6: while not IsEmpty(S) do 7: j:=gran(i, S); {restore the grand ancestor} 8: Move(j, S); {execute a move at j and update S} 9: if d < r then ColGen(i + 1, r d); else Output(P); endif 10: end while 11: end for 26
66 Generating PCPol(n): the algorithm (columns) 27
67 Generating PCPol(n): the algorithm (columns) w x, y, z 27
68 Generating PCPol(n): the algorithm (columns) w x, y, z length 27
69 Generating PCPol(n): the algorithm (columns) w x, y, z length position of lowest cell # of cells below length of segment 27
70 Generating PCPol(n): the algorithm (columns) w x, y, z ptr to smallest compat. segment of prev. column ptr to largest compat. segment of prev. column 27
71 Generating PCPol(n): the algorithm (columns) w x, y, z ptr to smallest compat. segment of prev. column ptr to largest compat. segment of prev. column 27
72 Generating PCPol(n): an example b a 1 1, 5, 2 3 3, 2, 8 3 2, 0, , 6, 3 1 2, 4, 6 1 1, 3, 8 1 3, 0, 12 28
73 Conclusions and perspectives CAT algorithm for PCPol(n) 29
74 Conclusions and perspectives CAT algorithm for PCPol(n) Done 29
75 Conclusions and perspectives CAT algorithm for PCPol(n) Closed formula for PCPol(n) Done 29
76 Conclusions and perspectives CAT algorithm for PCPol(n) Closed formula for PCPol(n) Done To do! 29
77 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) 29
78 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) Done (?) 29
79 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) Done (?) Closed formula for k-pcpol(n) 29
80 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) Done (?) Closed formula for k-pcpol(n) To do!! 29
81 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) Done (?) Closed formula for k-pcpol(n) To do!! Extension to Pol(n) 29
82 Conclusions and perspectives CAT algorithm for PCPol(n) Done Closed formula for PCPol(n) To do! Extension to k-pcpol(n) (k N fixed) Done (?) Closed formula for k-pcpol(n) To do!! Extension to Pol(n) To do!!! 29
83 Thank you.
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