2018-SPIRE-Block-Palindromes

We propose a new generalization of palindromes and gapped palindromes called block palindromes. A block palindrome is a string becomes a palindrome when identical substrings are replaced with a distinct character. We investigate several properties of block palindromes and in particular, study substrings of a string which are block palindromes.

SPIRE 2018: https://eventos.spc.org.pe/spire2018/wp/
arXiv: https://arxiv.org/abs/1806.00198

Transcript

1. Block Palindromes
   A New Generalization of Palindromes
   SPIRE 2018 in LIMA
   Keisuke Goto, Tomohiro I, Hideo Bannai, Shunsuke Inenaga
   

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2.  Standard Palindromes
   Palindromes
   a b c b a Same string
   a b c d e b a
   Same string
    Gapped Palindromes
   a b X b a
   X = cde
   gap
   SPIRE 2018 in LIMA 2/ 19
   

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3.  Palindromes represent characteristic structures of strings.
   There are several research about properties of palindromes
   
   maximal palindromes, palindrome factorization, ...
    Gapped palindromes model hairpin structures of DNA and
   RNA sequences
   Why Palindromes?
   where, G = C and U = A
   gap
   https://en.wikipedia.org/wiki/Stem-loop
   SPIRE 2018 in LIMA 3/ 19
   

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4.  A factorization f = f-n
   … f-1
   f0
   f1
   … fn
   of a string T is a
   block palindrome if f-i
   = fi
   for all 0 ≦ i ≦ n
   * f0
   may be empty string and f-i
   , fi
   for 0 < i ≦ n mustn’t
   Block Palindromes (BPs)
   f 2
   f 1
   f 0
   f 1
   f 2
   BPs are generalization of standard and gapped palindromes
   Same string
   LIMAisn‘tMALI
   f0
   f1
   f2
   f-1
   f-2
   We call a
   factor a block
   SPIRE 2018 in LIMA 4/ 19
   

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5.  We study basic properties of BPs, introducing
   representatives of BPs:
   
   Largest BPs (of a string)
   
   Maximal BPs (in a string)
    We propose an algorithm to enumerate all maximal BPs in
   a string T that runs in O(|T | + ||MBP(T )||) optimal time,
   where ||MBP(T )|| is the output size (i.e., the sum of # of
   factors in the outputs)
   Contributions
   SPIRE 2018 in LIMA 5/ 19
   

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6.  For a string T of length N, there are O(2N/2) BPs of T
    A unary string has 2N/2 BPs
   # of BPs of T
   a a a a a a a a a
   a a a a a a a a a
   a a a a a a a a a
   a a a a a a a a a
   a a a a a a a a a
   T =
   ・・・
   2N/2
   a a a a a a a a a
   SPIRE 2018 in LIMA 6/ 19
   

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7.  A string that is a (nonempty and proper)
   prefix and a suffix of T is called a border of T
    The outmost block of BPs of T is a border of T
    BPs can be obtained by stripping a border iteratively
   BPs and Borders
   o n i o n i n o n i o n
   T =
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   The BPs of T
   SPIRE 2018 in LIMA 7/ 19
   

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8. 8/ 19
   SPIRE 2018 in LIMA
   The largest BP of T
   o n i o n i n o n i o n
   T =
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   o n i o n i n o n i o n
   The BPs of T
   largest # of
   blocks
   Properties
    The largest BP is unique
   (obtained by stripping the shortest border iteratively)
    Each block is an unbordered string
    Any BP is represented by a factorization of the largest BP
   

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9. Factorization of Largest BPs
    Let f = f-n
   , ..., fn
   , g = g-m
   , ..., gm
   be BPs with f the largest
    In inductive steps on n > 0, we have 3 cases:
   
   (1) | fn
   | = |gm
   |
   
   (2) | fn
   | > |gm
   |
   
   (3) | fn
   | < |gm
   |
   SPIRE 2018 in LIMA 9/ 19
   

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10. g-m+1
    , ..., gm-1
    Factorization of Largest BPs
     Let f = f-n
    , ..., fn
    , g = g-m
    , ..., gm
    be BPs with f the largest
     In inductive steps on n > 0, we have 3 cases:
    
    (1) | fn
    | = |gm
    |
    
    (2) | fn
    | > |gm
    |
    
    (3) | fn
    | < |gm
    |
    f-n+1
    , ..., fn-1
    f-n
    fn
    gm
    g-m
    By the inductive hypothesis,
    f-n+1
    , ..., fn-1
    is finer than g-m+1
    , ..., gm-1
    ,
    which implies that f is finer than g
    SPIRE 2018 in LIMA 10/ 19
    

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11. Factorization of Largest BPs
     Let f = f-n
    , ..., fn
    , g = g-m
    , ..., gm
    be BPs with f the largest
     In inductive steps on n > 0, we have 3 cases:
    
    (1) | fn
    | = |gm
    |
    
    (2) | fn
    | > |gm
    | (cannot happen)
    
    (3) | fn
    | < |gm
    |
    f-n+1
    , ..., fn-1
    f-n
    gm
    g-m
    f-n
    g-m
    , s, gm
    , f-n+1
    , ..., fn-1
    , gm
    , s, gm
    has
    larger number of factors than f, a contradiction
    f-n+1
    , ..., fn-1
    s s gm
    g-m
    gm
    gm
    SPIRE 2018 in LIMA 11/ 19
    

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12. SPIRE 2018 in LIMA
    g-m+1
    , ..., gm-1
    Factorization of Largest BPs
     Let f = f-n
    , ..., fn
    , g = g-m
    , ..., gm
    be BPs with f the largest
     In inductive steps on n > 0, we have 3 cases:
    
    (1) | fn
    | = |gm
    |
    
    (2) | fn
    | > |gm
    | (cannot happen)
    
    (3) | fn
    | < |gm
    |
    f-n+1
    , ..., fn-1
    g-m
    gm
    f-n
    fn
    By the inductive hypothesis,
    f-n+1
    , ..., fn-1
    is finer than s, fn
    , g-m+1
    , ..., gm-1
    , fn
    , s,
    which implies that f is finer than g
    g-m+1
    , ..., gm-1
    s s
    fn
    fn
    12/ 19
    

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13.  We study largest BPs that occur as substrings in T
    Largest BPs in T
    When we fix a center block (that should be an unbordered
    string), there could be many largest BPs expanded from it
    T
    SPIRE 2018 in LIMA 13/ 19
    

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14.  We choose the maximal one as a representative of them
    Maximal BPs
    When we fix a center block (that should be an unbordered
    string), there could be many largest BPs expanded from it
    T
    maximal BP
    not maximal BP
    not maximal BP
    SPIRE 2018 in LIMA 14/ 19
    

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15.  We choose the maximal one as a representative of them
    Maximal BPs
    T
     Maximal BP whose f0
    occurs at T[b ... e] is unique
     Any largest BP is represented by a substring of the maximal BP
     ||MBP(T)|| ≦ N(2N-1)
    Properties
    The sum of # of factors of maximal BPs in T
    maximal BP
    not maximal BP
    not maximal BP
    SPIRE 2018 in LIMA 15/ 19
    

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16.  A naïve approach would be to compute the maximal BP for
    every center block
     Using a data structure for constant-time longest common
    extension queries, it can be done O(N3) time in total
    Enumeration of Maximal BPs
    We propose an algorithm running
    in O(N + ||MBP(T)||) time, which is optimal unless
    the maximal BPs can be represented more compactly
    SPIRE 2018 in LIMA 16/ 19
    

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17.  Our algorithm consists of two steps:
    
    Enumerate all pairs of occurrences of unbordered strings
    and sort them by the center position and beginning
    positions of the right arms
    Enumeration of Maximal BPs
    T
    SPIRE 2018 in LIMA 17/ 19
    

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18.  Our algorithm consists of two steps:
    
    Enumerate all pairs of occurrences of unbordered strings
    and sort them by the center position and beginning
    positions of the right arms
    
    For each center position, build maximal BPs by
    concatenating the enumerated pairs that are adjacent
    Enumeration of Maximal BPs
    T
    SPIRE 2018 in LIMA 18/ 19
    

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19.  We define block palindromes (BPs) which are new
    generalization of standard and gapped palindromes
     We introduce representatives of BPs
    
    Largest BPs (of strings)
    
    Maximal BPs (in strings)
     We study basic properties of these representatives and
    give efficient algorithms to compute/enumerate them
     Open problems:
    
    Is it possible to represent the maximal BPs compactly?
    (what if we only consider BPs with empty center blocks?)
    
    Can we utilize it to design faster enumeration algorithms?
    Conclusions and Future Work
    SPIRE 2018 in LIMA 19/ 19
    

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