Sparse ( 0 , 1 ) array and perfect phylogeny Yanzhen Xiong Shanghai - PowerPoint PPT Presentation

Sparse ( 0 , 1 ) array and perfect phylogeny Yanzhen Xiong Shanghai Jiao Tong University Joint work with Yaokun Wu Yichang, August 23, 2019 1/34

Outline Sparse ( 0 , 1 ) array 1 Perfect phylogeny 2 Problems 3 2/34 2 / 34

For any positive integer n , let [ n ] denote the set { 1 , . . . , n } . Given positive integers a 1 , . . . , a n , a map M ∈ { 0 , 1 } [ a 1 ] ×···× [ a n ] is called an n -dimensional ( 0 , 1 ) array (or tensor) of size a 1 × · · · × a n . For every nonempty subsets S i ⊆ [ a i ] , i ∈ [ n ] , the restriction of M to S 1 × · · · × S n is a subarray of M. 3/34 3 / 34

Taking Boolean sum to get the projection Let M be an n -dim’l ( 0 , 1 ) array of size a 1 × · · · × a n . For � [ n ] � { i 1 , . . . , i k } ∈ , let M i 1 ,..., i k be the k -dim’l ( 0 , 1 ) array of size k a i 1 × · · · × a i k such that M i 1 ,..., i k ( t i 1 , . . . , t i k ) = 0 if and only if � M ( t 1 , . . . , t n ) = 0 t j ∈ [ a j ] j / ∈ { i 1 , . . . , i k } We call M i 1 ,..., i k the k -dim’l projection of M to { i 1 , . . . , i k } . 4/34 4 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Taking Boolean sum to get the projection 0 0 1 0 0 0 0 0   0 0 1 1 0 0 0 1 2-dim’l proj. to { x , y } 1 0 0 0 0 0 1 0     0 0 0 1   0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 2-dim’l proj. to { y , z }  0 0 1 1  0 0 0 0 1 0 0 1 1 0 0 0   0 0 0 1 1 0 0 1 1 0 0 1 1-dim’l proj. to { y } y � � 1 0 1 1 x z 5/34 5 / 34

Sparsity, k -dimensional margins Let M be an n -dim’l ( 0 , 1 ) array of size a 1 × · · · × a n . We say that M has the sparse property Q n if for all nonempty subsets S 1 ⊆ [ a 1 ] , . . . , S n ⊆ [ a n ] , it holds n � � M ( t 1 , . . . , t n ) ≤ ( | S j | − 1 ) + 1 . j = 1 ( t 1 ,..., t n ) ∈ S 1 ×···× S n We say that M has the sparse property Q k if every k -dim’l projection of M has the sparse property Q k for positive integer k ≤ n . 6/34 6 / 34

Q 2 : For every subarray of size a × b , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + 1 . � 1 � 1    1 0 0 0  0 1 0 1 1 1 1 0 1 0 1 1 0 0       0 1 0 1 0 1 1 1   0 0 0 1 not Q 2 sparse Q 2 not Q 2 7/34 7 / 34

Q 2 : For every subarray of size a × b , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + 1 . � 1 � 1    1 0 0 0  0 1 0 1 1 1 1 0 1 0 1 1 0 0       0 1 0 1 0 1 1 1   0 0 0 1 not Q 2 sparse Q 2 not Q 2 7/34 7 / 34

Q 2 : For every subarray of size a × b , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + 1 . � 1 � 1    1 0 0 0  0 1 0 1 1 1 1 0 1 0 1 1 0 0       0 1 0 1 0 1 1 1   0 0 0 1 not Q 2 sparse Q 2 not Q 2 7/34 7 / 34

Q 2 : For every subarray of size a × b , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + 1 . � 1 � 1    1 0 0 0  0 1 0 1 1 1 1 0 1 0 1 1 0 0       0 1 0 1 0 1 1 1   0 0 0 1 not Q 2 sparse Q 2 not Q 2 7/34 7 / 34

Q 2 : For every subarray of size a × b , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + 1 . � 1 � 1    1 0 0 0  0 1 0 1 1 1 1 0 1 0 1 1 0 0       0 1 0 1 0 1 1 1   0 0 0 1 not Q 2 sparse Q 2 not Q 2 7/34 7 / 34

Q 3 : For every subarray of size a × b × c , ♯ 1 ≤ ( a − 1 ) + ( b − 1 ) + ( c − 1 ) + 1 . 1 0 0 1 0 1 1 0 Sparse Q 3 8/34 8 / 34

Theorem (2019+ Wu-X) Let k and n be integers with 3 ≤ k ≤ n and let M be an n-dim’l ( 0 , 1 ) array. If M has the property Q k − 1 , then M has the property Q k . 1 It can happen that M has the property Q k but does not have the 2 property Q k − 1 . 9/34 9 / 34

1 0 0 1 0 1 1 0 It has the property Q 3 but does not have the property Q 2 . 10/34 10 / 34

Let M be an n -dim’l ( 0 , 1 ) array of size a 1 × · · · × a n . We say that M has the sparse property Q n if n � � M ( t 1 , . . . , t n ) = ( a j − 1 ) + 1 . ( t 1 ,..., t n ) ∈ [ a 1 ] ×···× [ a n ] j = 1 We say that M has the sparse property Q k if every k -dim’l projection of M has the sparse property Q k . 11/34 11 / 34

Sparse completion Theorem (2019+ Wu-X) Let n be an integer with n ≥ 3 and let M be an n-dimensional ( 0 , 1 ) array satisfying the property Q n . Then, exact one of the following holds: M has the properties Q k and Q k for all k ∈ [ n ] . 1 M has the property Q k for no k ∈ { 2 , . . . , n − 1 } . 2 Let M and M ′ be two n -dim’l ( 0 , 1 ) arrays of equal size. If M ≤ M ′ (entry-wise) and M ′ has the sparse properties Q 2 and Q n , then we call M ′ a sparse completion of M. 12/34 12 / 34

From array to partition system    ∅ ∅ ∅  1 0 0 0 a 1 1 0 0 b c ∅ ∅         0 1 1 1 ∅ d e f     0 0 0 1 ∅ ∅ ∅ g X = { a , b , c , d , e , f , g } {{ a | bc | def | g , ab | cd | e | fg }} 13/34 13 / 34

From partition system to array X = { a , b , c , d , e , f , g } C = {{ abc | defg , bde | acfg , ef | bd | acg }} ∅ ∅ g a , c e ∅ ∅ f b d ∅ ∅ 14/34 14 / 34

From partition system to array X = { a , b , c , d , e , f , g } C = {{ abc | defg , bde | acfg , ef | bd | acg }} ∅ ∅ g a , c e ∅ ∅ f b d ∅ ∅ 14/34 14 / 34

From partition system to array X = { a , b , c , d , e , f , g } C = {{ abc | defg , bde | acfg , ef | bd | acg }} ∅ ∅ g a , c e ∅ ∅ f b d ∅ ∅ 14/34 14 / 34

From partition system to array X = { a , b , c , d , e , f , g } C = {{ abc | defg , bde | acfg , ef | bd | acg }} ∅ ∅ g a , c e ∅ ∅ f b d ∅ ∅ 14/34 14 / 34

From partition system to array X = { a , b , c , d , e , f , g } C = {{ abc | defg , bde | acfg , ef | bd | acg }} 0 0 ∅ ∅ 1 1 a , c g 0 1 ∅ e 0 1 ∅ f 1 1 b d 0 0 ∅ ∅ M C : Intersection array of C 14/34 14 / 34

Perfect phylogeny We call a partition system compatible if it can be displayed on a labeled tree. a f d b , c g e X = { a , b , c , d , e , f , g } {{ abc | de | fg , a | bc | dfg | e }} 15/34 15 / 34

Perfect phylogeny We call a partition system compatible if it can be displayed on a labeled tree. a f d b , c g e X = { a , b , c , d , e , f , g } {{ abc | de | fg , a | bc | dfg | e }} 15/34 15 / 34