Invariants for efficiently computing the autotopism group of a partial Latin rectangle

Transcription

1 Invariants for efficiently computing the autotopism group of a partial Latin rectangle Daniel Kotlar Tel-Hai College, Israel joint work with Eiran Danan, Raúl M. Falcón, Trent G. Marbach, Rebecca J. Stones Symmetry vs Regularity, Pilsen,

2 Partial Latin Rectangle 2

3 Partial Latin Rectangle L PLR( r, s, n) r= number of rows s = number of columns n= number of symbols 2

4 Partial Latin Rectangle PLR(,, ) L r s n r= number of rows s = number of columns n= number of symbols PLR(6,9, 7) 2

5 Partial Latin Rectangle PLR(,, ) L r s n r= number of rows s = number of columns n= number of symbols PLR(6,9, 7) 2 Ent( ) : {(,, [, ]) : [ ], [ ], [, ] [ ]} L i j L i j i r j s L i j n = The entry set of L

6 Isotopisms and autotopisms Θ= ( α, β, γ ) S S S r s n Θ : PLR( r, s, n) PLR( r, s, n) α permutes the rows β permutes the columns γ permutes the symbols an isotopism If Θ ( L) = L then Θ is an autotopismof L Atop( L ) = The autotopygroup of L 3

7 Computing Atop( L) The partial Latin rectangle graph (Falcon and Stones, Disc. Math. 2017) Example: Compute Atop( L) by computing graph automorphisms. Use nauty (McKay, Meynert, Myrvold, JCD 2007) and its variants 4

8 Two stages in Computing Atop( L) 1. Use invariants to partition rows, columns, symbols and entries in order to narrow down the search - polynomial time complexity. 5

9 Two stages in Computing Atop( L) 1. Use invariants to partition rows, columns, symbols and entries in order to narrow down the search - polynomial time complexity. 2. Compute Atop( L) using backtracking methods - mostly non-polynomial time complexity. 5

10 Two stages in Computing Atop( L) 1. Use invariants to partition rows, columns, symbols and entries in order to narrow down the search - polynomial time complexity. 2. Compute Atop( L) using backtracking methods - mostly non-polynomial time complexity. 6

11 System of partitions ( ) row, col, sym P= P P P 7

12 System of partitions ( ) row, col, sym P= P P P P is invariantif and only if each of its components is invariant under Atop( L) 7

13 System of partitions ( ) row, col, sym P= P P P P is invariantif and only if each of its components is invariant under Atop( L) Example: the types system P ( ) T = P1, P2, P3 : P1 is defined by the number of entries in the rows P2 is defined by the number of entries in the columns P3 is defined by the number of appearances of each symbol 7

14 How to obtain invariant systems? 8

15 How to obtain invariant systems? Aim: find the finest possible invariant system of partitions 8

16 How to obtain invariant systems? Aim: find the finest possible invariant system of partitions How? 1) Start with P= P ( L) S the trivial system of partitions 2) Apply refinement methods 8

17 The natural refinement ( ) row, col, sym ( ) P= P P P induces a partition E P on Ent( L) : 9

18 The natural refinement ( ) row, col, sym ( ) P= P P P induces a partition E P on Ent( L) : i ~ i P row ( i, j, L[ i, j])) ~ E( ) ( i, j, L[ i, j ]) j ~ j P Pcol L[ i, j] ~ L[ i, j ] Psym 9

19 The natural refinement ( ) row, col, sym ( ) P= P P P induces a partition E P on Ent( L) : i ~ i P row ( i, j, L[ i, j])) ~ E( ) ( i, j, L[ i, j ]) j ~ j P Pcol L[ i, j] ~ L[ i, j ] Psym 1) Label the elements of Ent( L) by their part in E( P) 9

20 The natural refinement ( ) row, col, sym ( ) P= P P P induces a partition E P on Ent( L) : i ~ i P row ( i, j, L[ i, j])) ~ E( ) ( i, j, L[ i, j ]) j ~ j P Pcol L[ i, j] ~ L[ i, j ] Psym 1) Label the elements of Ent( L) by their part in E( P) ( row col sym ) 2) Define the natural refinement where NP ( row ) is define by the multisets of labels in the rows NP ( ) is define by the multisets of labels in the columns col N( P) = N( P ), N( P ), N( P ) col ( ) NP is define by the multisets of labels corresponding to the symbols 9

21 The natural refinement 10

22 The natural refinement N( ) P P 10

23 The natural refinement N( ) P P If is invariant so is P NP ( ) 10

24 The natural refinement N( ) P P If is invariant so is P NP ( ) If ' then P P N( P' ) N( P) 10

25 The natural refinement N( ) P P If is invariant so is P NP ( ) If ' then P P N( P' ) N( P) Can continue the process until it stops: [ ] ( ) ( ) ( ), N P N N P N P P 10

26 The natural refinement N( ) P P If is invariant so is P NP ( ) If ' then P P N( P' ) N( P) Can continue the process until it stops: [ ] ( ) ( ) ( ), N P N N P N P P Examples: N ( S) = T P P N( N ( PS )) = PSEI the types partition the Strong Entry Invariants partition (Falcon and Stones, 2017) 10

27 The natural refinement N( ) P P If is invariant so is P NP ( ) If ' then P P N( P' ) N( P) Can continue the process until it stops: [ ] ( ) ( ) ( ), N P N N P N P P Examples: N ( S) = T P P N( N ( PS )) = PSEI the types partition the Strong Entry Invariants partition (Falcon and Stones, 2017) Disadvantage: when starting from P S it is useless for very dense PLRs 10

28 Two-line graphs for two rows r1, r2 in a PLR, construct a vertex-and-edgecolored bipartite graph G as illustrated here: r, r ( L)

29 Two-line graphs for two rows r1, r2 in a PLR, construct a vertex-and-edgecolored bipartite graph G as illustrated here: r, r ( L) 1 2 Similar constructions for two columns and two symbols. (1,3) (for symbols consider L, the PLR obtained by swapping the 1 st and 3 rd coordinates in Ent( L) ) 11

30 Two-line graphs for two rows r1, r2 in a PLR, construct a vertex-and-edgecolored bipartite graph G as illustrated here: r, r ( L) 1 2 Similar constructions for two columns and two symbols. (1,3) (for symbols consider L, the PLR obtained by swapping the 1 st and 3 rd coordinates in Ent( L) ) Can try it here: 11

31 Two-line graphs for two rows r1, r2 in a PLR, construct a vertex-and-edgecolored bipartite graph G as illustrated here: r, r ( L) 1 2 Similar constructions for two columns and two symbols. (1,3) (for symbols consider L, the PLR obtained by swapping the 1 st and 3 rd coordinates in Ent( L) ) Can try it here: Remark: can be viewed as a generalization of cycles in partial permutations 11

32 The two-line representations R row ( L) 0 0 rkl = rij 0 12

33 The two-line representations r = r G G ij kl i, j k, l R row ( L) 0 0 rkl = rij 0 12

34 The two-line representations r = r G G ij kl i, j k, l R row ( L) 0 0 rkl = rij 0 (Define R L and R ( L) sym analogously) col ( ) 12

35 The two-line graph (TLG) refinement 13

36 The two-line graph (TLG) refinement P= P P P an adequate system r, c, s ( ) 13

37 The two-line graph (TLG) refinement P= P P P an adequate system r, c, s ( ) suppose P = { P, P,, P k } r

38 The two-line graph (TLG) refinement P= P P P an adequate system r, c, s ( ) suppose P = { P, P,, P k } r 1 2 R ( L) row = 13

39 The two-line graph (TLG) refinement P= P P P an adequate system R ( L) row r, c, s ( ) suppose P P1 P2 = { P, P,, P k } r 1 2 Pi Pk = 13

40 The two-line graph (TLG) refinement P= P P P an adequate system R ( L) row r, c, s ( ) suppose P P1 P2 = { P, P,, P k } r 1 2 Pi Pk = r 1 r 2 r r ~ P r r ~ G P r r 1 ( ) the multisets agree in each part 13

41 The two-line graph (TLG) refinement P= P P P an adequate system R ( L) row r, c, s ( ) suppose P P1 P2 = { P, P,, P k } r 1 2 Pi Pk = r r r r ~ P r r ~ G P r r 1 ( ) the multisets agree in each part 13

42 The two-line graph (TLG) refinement P= P P P an adequate system R ( L) row r, c, s ( ) suppose P P1 P2 = { P, P,, P k } r 1 2 Pi Pk = r r r r ~ P r r ~ G P r r 1 ( ) 2 G( P ) ( ) analogous definitions for and ( P P P ) G P = G G G r, c, s ( ) ( ) ( ) ( ) c G P s 1 2 the multisets agree in each part 13

43 The TLG refinement 14

44 The TLG refinement G( ) P P 14

45 The TLG refinement G( P) P If is invariant so is P GP ( ) 14

46 The TLG refinement G( P) P If is invariant so is P GP ( ) If ' then P P G( P ') G( P) 14

47 The TLG refinement G( P) P If is invariant so is P GP ( ) If ' then P P G( P ') G( P) Can continue the process until it stops: [ ] ( ) ( ) ( ), GP G G P G P P 14

48 G vs. N Not much is known about the relation between the refinements G and N. However, it is easy to see 15

49 G vs. N Not much is known about the relation between the refinements G and N. However, it is easy to see G( P ) N( P ) S S 15

50 G vs. N Not much is known about the relation between the refinements G and N. However, it is easy to see G( P ) N( P ) S S G [ P ] N[ P ] S S 15

51 G vs. N Not much is known about the relation between the refinements G and N. However, it is easy to see G( P ) N( P ) S S G [ P ] N[ P ] S This does not mean that the natural refinement is redundant. It is possible to have S ( [ P] ) < G[ P] N G 15

52 Complexity The average complexity of the TLG refinement for 3 L PLR( r, s, n) is O( M log M ), where M = max( r, s, n) 16

53 Performance on random PLRs GP [ S ] N G N[ P S ] [ P ] S % optimal 1000 random PLRs in PLR(8,8,8;m) for each m Works well for dense PLRs and (full) Latin rectangles. 17

54 Problems for further study 18

55 Problems for further study What is the best combination of the operators N and G (in terms of best refinement and lowest complexity)? 18

56 Problems for further study What is the best combination of the operators N and G (in terms of best refinement and lowest complexity)? Find other refinement methods. 18

57 Problems for further study What is the best combination of the operators N and G (in terms of best refinement and lowest complexity)? Find other refinement methods. What is the most efficient way to conduct the subsequent search? 18

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59 19 K N A H T U O Y R O F R U O Y T N O I N E T A T