Presentation For VLSI Application of Suffix Trees

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Presentation For VLSIApplication of Suffix Trees

• Chunkai Yin
• Apr 2002

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Topics

• Approximate palindromes and repeats
• Multiple common substring problem

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Approximate palindromes and repeats

• Definitions
• Find all tandem repeats (naive method)
• Find all tandem repeats (Landau-Schmidt)
• Find all k-mismatch tandem repeats (L-S)

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Definitions

• Tandem repeat
• string written as AA, where A is a substring
• abcabc
• Specified by starting position and length of A
• K-mismatch palindrome/tandem
• a substring that becomes a palindrome/tandem
  after k or fewer characters are changed
• zzcbbcca 2-mismatch palindrome pzcb bcca
• axabaybb 2-mismatch tamdem axab aybb

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Find all tandem repeats (naive method)

• Guess a start position i , middle position j for
  the tandem
• Do a longest common extension query from i and j.
  If the extension from i reaches j, the tandem
  exists starting at position i.
• Time complexity O(n2)
• EX1
• fababd

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Problem Find all tandem repeats

• The Landau-Schmidt method divide the problem
• into four subproblems
• (Recursive, divide-and-conquer)
• Find all tandem repeats contained entirely in the
  first half of S (up to h n/2)
• Find all tandem repeats contained entirely in the
  second half of S (after h )
• Find all tandem repeats where the first copy
  contains position h of S.
• Find all tandem repeats where the second copy
  contains position h of S.

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How to handle this ?

• Subproblem A, B will be solved by recursively
  applying the method
• Subproblem C and D are symmetric, just consider
  subproblem C.
• Therefore, the core problem is subproblem C.

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Solve the subproblem 3

• Initialization
• h n/2 qhL L is a fixed number each time
• Compute the longest common extension (from left
  to right) from h and q.
• L1 is the length of extension.
• Compute the longest common extension (from right
  to left) from h-1 and q-1. L2

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Solve the subproblem 3

• If and only if L1L2gtL, a tandem repeat exists
  whose first copy contains position h with the
  length 2L, and it can begin at any position
  between h-L2 and hL1-L inclusive. The second
  copy begins L places to the right. Output the
  starting point.
• EX2

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L1, L2 and range for starting point
L
q
h
A
B

• ?1L-L2, L ?1 L2
• Ah-L2
• ?2L-L1, L ?2 L1
• Bh- ?2hL1-L

L1
L1
?2
?2
L2
L2
?1
?1
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An Example

• .b d c a b b d c a b b t t t.
• h q
• L5 L13 L23
• A1 B2
• Two forms available
• bdcab bdcab
• dcabb dcabb

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Time Complexity

• Time used for output O(Z)
• Z is the total number of tandem repeats in S
• Time used for the extension queries
• which is proportional to the number of
  extension queries ext.
• T(n)T(n/2) T(n/2) n n nn/2 (-gt)
  n/2(lt-)
• A B C D
• T(n) O(nlogn), totalO(nlogn)Z

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Find all k-mismatch tandem repeats (L-S)

• Immediate extension of the O(nlognZ) method for
  finding exact tandem repeats.
• Run k successive longest common extension queries
  forward from h and q and run k successive longest
  common extension queries backward from h-1 and
  q-1.
• Then try to find t, a midpoint of the tandem
  repeat.
• In the interval between h and q, the number
  of mismatches from h to t (found during the
  forward extension) plus the the number of
  mismatches from t1 to q-1(found during the
  backward ext) is ltk.

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Time complexity

• O(knlogn Z)

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Multiple common substring prob.

• Problem
• What substring are common to a large number of
  distinct strings?
• Definitions
• T a generalized suffix tree for K strings
• The identical suffixes appearing in more
• than one string end at distinct leaves.
  Each
• leaf has one of K unique string
  identifier.

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• C(v) the number of distinct leaf string
  identifiers in the subtree of v
• S(v) the total number of leaves in the subtree
  of v
• U(v) how many duplicate suffixes from the same
  string occur in vs subtree.
• C(v)S(v)-U(v)
• L(k) the length of the longest substring common
  to at least k of the strings.
• U(v)?ini(v)gt0(ni(v)-1)
• ni(v) the number of leaves with identifier i
• in the subtree rooted at v

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The key work we need do

• Our object get C(v)
• What is the algorithm?
• Define Li the list of leaves with identifier i,
  in increasing order of their DFS numbers

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Algorithm for the problem

• Build a generalized suffix tree T for the K
  strings.
• Number the leaves of T as they are encountered
  in a depth-first traversal of T
• For each string identifier i, extract the ordered
  list Li, of leaves with identifier i.
• For each identifier i, compute the lca of each
  consecutive pair of leaves in Li, and increment
  by one each time that w is the computed lca.

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Algorithm Cont.

• Lemma
• If we compute the lca for each consecutive
  pair of leaves in Li, then for any node v,
  exactly ni(v)-1 of the computed lcas will lie in
  the substring of v.
• With a bottom-up traversal of T, compute, for
  each node v, S(v), and U(v)?ini(v)gt0(ni(v)-1)
  ?h(w)w is in the subtree of v
• Set C(v)S(v)-U(v) for each v
• Accumulate the table of L(k) values.
• Time complexity O(n)

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End of the presentation

• Thank you
• Thank you
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