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Computing autotopism groups of partial Latin rectangles: a pilot study Ra ul M. Falc on (U. Seville ); Daniel Kotlar (Tel-Hai College ); Rebecca J. Stones (Nankai U. ) 19 December 2016 1 2 3 2 4

  1. Computing autotopism groups of partial Latin rectangles: a pilot study Ra´ ul M. Falc´ on (U. Seville ); Daniel Kotlar (Tel-Hai College ); Rebecca J. Stones (Nankai U. ) 19 December 2016 · · · · · · 1 2 3 · · · 2 4 1 5 6 7 · · · · · 1 5 3 4 · 2 · 5 · 3 · 4 · · · · · 4 3 5 1 2 · · · · · · 2 1 3

  2. Partial Latin rectangles An r × s partial Latin rectangle is an r × s matrix containing symbols from [ n ] ∪ {·} such that each row and each column contains at most one copy of any symbol in [ n ]. · 5 4 3 2 · · · 5 1 · · · 4 1

  3. Partial Latin rectangles An r × s partial Latin rectangle is an r × s matrix containing symbols from [ n ] ∪ {·} such that each row and each column contains at most one copy of any symbol in [ n ]. · 5 4 3 2 · · · 5 1 · · · 4 1 Every partial Latin rectangle L ∈ PLR ( r , s , n ) is uniquely determined by its entry set: entry � �� � Ent ( L ) := { ( i , j , L [ i , j ]) : i ∈ [ r ] , j ∈ [ l ] , and L [ i , j ] ∈ [ n ] } . � � (2 , 5 , 1) , (1 , 3 , 4) , (1 , 2 , 5) , Ent (above) = (1 , 5 , 2) , (1 , 4 , 3) , (3 , 4 , 1) , (3 , 1 , 4) , (2 , 1 , 5)

  4. Isotopisms and autotopisms � r × s partial Latin rectangles � on symbol set [ n ] The isotopism θ := ( α, β, γ ) ∈ S r × S s × S n acts on PLR ( r , s , n ). · · · · · 1 2 3 4 3 4 2 · · · · · 3 4 2 1 2 3 4 swap first two rows α = (12) · · · · · · · · · · 5 5 swap last two columns β = (56) · · · · · · · · 6 5 6 5 do nothing to symbols γ = id − − − − − − − − − − − − − − − − − → · · · · · · · · 6 5 5 6 · · · · · · · · · · 6 6

  5. Isotopisms and autotopisms � r × s partial Latin rectangles � on symbol set [ n ] The isotopism θ := ( α, β, γ ) ∈ S r × S s × S n acts on PLR ( r , s , n ). · · · · · 1 2 3 4 3 4 2 · · · · · 3 4 2 1 2 3 4 swap first two rows α = (12) · · · · · · · · · · 5 5 swap last two columns β = (56) · · · · · · · · 6 5 6 5 do nothing to symbols γ = id − − − − − − − − − − − − − − − − − → · · · · · · · · 6 5 5 6 · · · · · · · · · · 6 6 And, in some cases, we can apply an isotopism θ and end up back where we started = ⇒ θ is an autotopism.

  6. Isotopisms and autotopisms � r × s partial Latin rectangles � on symbol set [ n ] The isotopism θ := ( α, β, γ ) ∈ S r × S s × S n acts on PLR ( r , s , n ). · · · · · 1 2 3 4 3 4 2 · · · · · 3 4 2 1 2 3 4 swap first two rows α = (12) · · · · · · · · · · 5 5 swap last two columns β = (56) · · · · · · · · 6 5 6 5 do nothing to symbols γ = id − − − − − − − − − − − − − − − − − → · · · · · · · · 6 5 5 6 · · · · · · · · · · 6 6 And, in some cases, we can apply an isotopism θ and end up back where we started = ⇒ θ is an autotopism. The set of autotopisms form a group, named the autotopism group.

  7. 1 2 3 4 This member of PLR (2 , 2 , 4) has 4 autotopisms. ( id , id , id ), ((12) , id , (13)(24)), ( id , (12) , (12)(34)), ((12) , (12) , (14)(23)),

  8. 1 2 3 4 This member of PLR (2 , 2 , 4) has 4 autotopisms. ( id , id , id ), ((12) , id , (13)(24)), ( id , (12) , (12)(34)), ((12) , (12) , (14)(23)), ...forming a group isomorphic to C 2 × C 2 .

  9. 1 2 3 4 This member of PLR (2 , 2 , 4) has 4 autotopisms. ( id , id , id ), ((12) , id , (13)(24)), ( id , (12) , (12)(34)), ((12) , (12) , (14)(23)), ...forming a group isomorphic to C 2 × C 2 . Note : The row and column permutations determine the autotopism.

  10. How to efficiently compute the autotopism group? Input : partial Latin rectangle. Output : its autotopism group.

  11. How to efficiently compute the autotopism group? Input : partial Latin rectangle. Output : its autotopism group. By the looks of things, the answer is...

  12. How to efficiently compute the autotopism group? Input : partial Latin rectangle. Output : its autotopism group. By the looks of things, the answer is... Basically, the answer depends on the partial Latin rectangle.

  13. How to efficiently compute the autotopism group? Input : partial Latin rectangle. Output : its autotopism group. By the looks of things, the answer is... Basically, the answer depends on the partial Latin rectangle. This work is a “pilot study” to (a) identify design goals of future software for computing the autotopism group, and (b) eliminate unpromising methods.

  14. How to efficiently compute the autotopism group? Input : partial Latin rectangle. Output : its autotopism group. By the looks of things, the answer is... Basically, the answer depends on the partial Latin rectangle. This work is a “pilot study” to (a) identify design goals of future software for computing the autotopism group, and (b) eliminate unpromising methods. We experimentally compare 6 families of methods...

  15. Backtracking methods... Family 1: Alpha-beta backtracking.

  16. Backtracking methods... Family 1: Alpha-beta backtracking. At each level of the α search tree, we designate 3 1 4 · · a 1 α row i �− → row a α · · i 2 provided it doesn’t clash.

  17. Backtracking methods... Family 1: Alpha-beta backtracking. At each level of the α search tree, we designate 3 1 4 · · a 1 α row i �− → row a α · · i 2 provided it doesn’t clash. Once α is determined... β At each level of the β search tree, we designate j b β 3 1 4 column j �− → column b · · 1 provided it doesn’t clash. · · 2

  18. Backtracking methods... Family 1: Alpha-beta backtracking. At each level of the α search tree, we designate 3 1 4 · · a 1 α row i �− → row a α · · i 2 provided it doesn’t clash. Once α is determined... β At each level of the β search tree, we designate j b β 3 1 4 column j �− → column b · · 1 provided it doesn’t clash. · · 2 Then we check if ( α, β, ??) is an autotopism.

  19. Backtracking methods... Family 2: Entrywise backtracking.

  20. Backtracking methods... Family 2: Entrywise backtracking. α ( i ) = a At each level of the search tree, β ( j ) = b we designate γ ( L [ i, j ]) = L [ a, b ] j b θ entry ( i , j , L [ i , j ]) → �− entry ( a , b , L [ a , b ]) a 3 1 4 · · 1 i provided it doesn’t clash. · · 2

  21. Graph methods... Family 3: McKay, Meynert, and Myrvold method.

  22. Graph methods... Family 3: McKay, Meynert, and Myrvold method. Vertex set : Ent ( L ) ∪ { R i : i ∈ [ r ] and row i of L is non-empty } ∪ { S j : j ∈ [ s ] and column j of L is non-empty } ∪ { N k : k ∈ [ n ] and symbol k occurs in L } where each of the four subsets, Ent ( L ), { R i } , { S j } , and { N k } , are assigned a distinct color. Edge set : { R i L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { S j L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { N L [ i , j ] L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } .

  23. Graph methods... Family 3: McKay, Meynert, and Myrvold method. Vertex set : Ent ( L ) ∪ { R i : i ∈ [ r ] and row i of L is non-empty } ∪ { S j : j ∈ [ s ] and column j of L is non-empty } ∪ { N k : k ∈ [ n ] and symbol k occurs in L } where each of the four subsets, Ent ( L ), { R i } , { S j } , and { N k } , are assigned a distinct color. Edge set : { R i L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { S j L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { N L [ i , j ] L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } . The automorphism group of this graph is isomorphic to the autotopism group of the partial Latin rectangle.

  24. Graph methods... Family 3: McKay, Meynert, and Myrvold method. Vertex set : Ent ( L ) ∪ { R i : i ∈ [ r ] and row i of L is non-empty } ∪ { S j : j ∈ [ s ] and column j of L is non-empty } ∪ { N k : k ∈ [ n ] and symbol k occurs in L } where each of the four subsets, Ent ( L ), { R i } , { S j } , and { N k } , are assigned a distinct color. Edge set : { R i L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { S j L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } ∪ { N L [ i , j ] L [ i , j ]: ( i , j , L [ i , j ]) ∈ Ent ( L ) } . The automorphism group of this graph is isomorphic to the autotopism group of the partial Latin rectangle. We compute this using Nauty .

  25. Graph methods... Family 4: Bipartite graph method.

  26. Graph methods... Family 4: Bipartite graph method. column index 1 row index · 5 4 3 2 1 2 transform into bipartite graph · · · 5 1 − − − − − − − − − → 2 3 · · · 4 1 3 4 5 (Edges are colored to illustrate construction.)

  27. Graph methods... Family 4: Bipartite graph method. column index 1 row index · 5 4 3 2 1 2 transform into bipartite graph · · · 5 1 − − − − − − − − − → 2 3 · · · 4 1 3 4 5 (Edges are colored to illustrate construction.) Then compute the automorphism group of the bipartite graph using Nauty .

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