Parameterized Pattern Matching

1
Parameterized Pattern Matching

• Amihood Amir
• Martin Farach
• V. Muthukrishnan

2
Parameterized Matching

• Input two strings s and t st, over
  alphabets ?s and ?t.
• s parameterize matches t if bijection
  ?s ?t , such that (s) t.

Example
a
a
b
b
b
(a)x
x
x
y
y
y
(b)y
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Parameterized Matching

• Input Two strings T, P Tn, Pm.
• Output All text locations i,
• such that (P)Ti Tim-1.

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Parameterized Matching History

• Introduced by Brenda Baker Baker93.
• Others AFM94, Bak95, Bak97.
• Two Dimensions AACLP03.
• Used in scaled matching ABL99.
• Periodicity of parameterized matching
  ApostolicoGiancarlo.
• Approximate parameterized matching HLS04.

5
Alternate Definition

• Notice
• Alphabet bijection between S and T means
• Si Ti for all I
• Where Si Ti if
• Si ? Sk, k1,,i-1
  and
• Ti ? Tk, k1,,i-1
• or for all k1,,i-1
• SiSk iff TiTk

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Parameterized Matching Algorithm

• Run KMP with the following modifications
• Construct table A1,,Am where
• largest k, 1klti, s.t. PiPk
• Ai
• i , if no such k exists

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• 2. Replace equality checks as follows
• Instead of PiTj? do
• Compare (Pi,Tj)
• If Aii and Tj?Tk, kj-i1,,j
• then return equal
• If Ai?i and TjTj-iAi
• then return equal
• return not equal
• End

8

• Instead of PiPj? do
• Compare (Pi,Pj)
• If (Aii or i-Aij) and Pj?Pk,
  k1,,j
• then return equal
• If i-Ailtj and PjPj-iAi
• then return equal
• return not equal
• End

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Correctness

• Automaton construction guarantees that failure
  arrow points to
• largest prefix that parameter matches the
  suffix.

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TIME

• KMP is linear time, but we have a new Compare
  subroutine.
• Take text size to be 2m, and Compare takes time
  O(log s), where smin(S,m).
• This is the time to search if Tj or Pj
  appears in a balanced tree.

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TIME

• Automaton Construction O(m log s) .
• Text Scanning O(n log s) .
• Can we do better?

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Alphabet S1,,n

• Can be done in linear time.
• How?
• Construct array
• 1 list of indices of symbol 1
• 2 list of indices of symbol 2
• .
• .
• m list of indices of symbol m.

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To check if Tj?Tk, kj-i1,,j

• Assume the symbol in Tj is a.
• Check if
• previous index to j in as list lt j-i1

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LOWER BOUNDS

• What about general alphabets?
• Element distincness Problem (EDP)
• Input Array A1,,An of natural numbers.
• Decide If all elements of A are distinct (i.e.
  no i?j where Aiaj)

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TIME FOR EDP

• In comparison model
• General alphabets ?(n log n)
• Alphabet S1,,n linear time.
• (construct array of indices)

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Linear Reduction

• Claim EDP is linearly reducible to Parameterized
  Matching.
• Proof Let A1,,An be an array of numbers.
• In linear time, check if A1 is unique.
• If so, construct SA2,A3,,An,A1

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Linear Reduction (cont.)

• A S iff all elements of A are distinct.
• trivial
• By induction on the prefixes of A.
• A1 is unique we checked.
• Assume A1,,Ak are distinct.
• In particular, Ak is unique.

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Linear Reduction (cont.)

• But Ak was parameter-matched to Sk, so Sk
  only appears once in S.
• But SkAk1.
• This means that Ak1 is unique.