2 Dimensional Parameterized Matching

1
2 Dimensional Parameterized Matching

• Carmit Hazay
• Moshe Lewenstein
• Dekel Tsur

2
Outline

• Definitions
• HistoryMotivation
• Text Preprocessing
• Algorithm Outline
• Pattern Preprocessing

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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
s
(a)x
x
x
y
y
y
t
(b)y
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1D Parameterized Matching

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

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2D Parameterized Matching

• Input Text T and pattern P
  Tnn, Pmm.
• Output All text locations (i,j),
• such that (P)Ti,j Tim-1,jm-1.
• Example-

T
a b c a a b b b b
(x)a (y)b (z)c
P
x y z x x y y y y
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Parameterized Matching History

• Introduced by Brenda Baker Baker93.
• Two Dimensions AACLP03This work.
• Used in scaled matching ABL99.
• Periodicity of parameterized matching
  ApostolicoGiancarlo.
• Approximate parameterized matching AEL,
  HLS04.
• Others AFM94, Bak95, Bak97.

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Mismatch pairs

• Pair of locations such that the characters
  disagree parameterized.
• Example,

a a b a a a
x x y x z y
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1D Encoding

• Encode every text location by its predecessor
  location.

First a to its left
a b a d d a b d b c b d a a b d a a a a b b b
T
1 3 6 13 14 15
16 17 18
Encoded T
0 1 3 6 13 14 15 16 17
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1D Encoding

• Two p-matching strings have the same encoded
  texts.

S
a b b c b a a c b b c b a
0 0 2 0 3 1 6 4 5 9 8 10 7
Encoded S
x y y z y x x z y y z y x
T
0 0 2 0 3 1 6 4 5 9 8 10 7
Encoded T
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1D Encoding

• Two strings p-match iff encoded strings match.
• Reduction to exact matching problem.

S
a b b c b b a c b b c b a
0 0 2 0 3 5 6 4 5 9 8 10 7
Encoded S
x y y z y x x z y y z y x
T
0 0 2 0 3 1 6 4 5 9 8 10 7
Encoded T
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2D Mismatch Pairs

• Same as 1D mismatch pairs, but with 2D strings.
• Example

a b a b a b b a b
x y x y y y y y y
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2D Encoding

• First idea,
• Encode the linearization of text and pattern.
• Overflow problem!!

b
a
Different character than b
b
a
b
Different character than a
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2D Encoding

• Second idea, use strips.
• Strip Substring of T of size nm.
• i-th strip of T, is nm substring
  T1n,iim-1.

i
Encode the linearization of strips. Why overflow
problem solved? Every predecessor within mm
window.
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Text Preprocessing

• For first strip, compute predecessors on its
  linearization.
• How to compute predecessors for rest of strips?
• First solution Do same as above.
• Time O(n2m).
• Can we do better?

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Update strips

• Yes, exploit information from previous strips.
• When moving from strip i to i1, update only O(n)
  pointers of first and last column.

i
i1
Time O(n2)
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Check Predecessors

• Are we done?
• No!!
• Need to check every predecessor against every
  text location contains it.
• Worst case O(n2m2).
• How to improve?

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Algorithm Outline

• Use Duel and Sweep paradigm
• Find candidates - Dueling
• Divide candidates by strips
• Update predecessors of every new strip
• Check new predecessors - Sweep
• Assume pattern witness table given.

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Witness

• Witness Mismatch pair between P and its
  alignment to location (a,b).

a
b
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Set Candidates

• Using duel-
• Two text locations with witness one can be
  eliminated.
• Apply algorithm of ABF94 and return list of
  candidates.
• Time O(n2).

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Sweep Technique

• Observation,
• All candidates agree with each other.
• Hence,
• Mismatch pair eliminates all candidates
  containing it.
• Therefore,
• For every predecessor, enough to find one
  candidate that contains it.

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Sweep Technique

• How to find?
• Create new nn array A such that,
• Ai,j largest row among candidates that
  starts at column j and overlap with row i.

x
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Sweep Technique

• For every predecessor (i,j), (x,y), use range
  minima query to find highest candidate contain it.

In case of a mismatch pair, eliminate all
candidates containing it. How?
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Sweep Technique

• Use mismatch vector.
• Every mismatch pair translates into range.
• For new predecessors, add mistake predecessors,
  and delete old ones.

All candidates within this range are
eliminated. O(n2) time.
m
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Sweep Technique

• Observation-
• T p-matches P
• Every text location and its predecessor are not
  mismatch pair
• of distinct characters in P and T equal
• Left to do?
• Count distinct characters for every candidate.
• Use algorithm of Amir Cole Dar Church, time O(n2).

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Overview
Checking all predecessors takes linear
time. Total time O(n2).
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Pattern Preprocessing

• Find witness table for P in time O(m2.5
  polylogm).
• For every pattern location (i,j), create list of
  size O( ) pointers.
• Pointer i is predecessor of lines above
  (i,j).
• Reduce to exact matching with dont cares.

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Pattern Preprocessing

• End cases, multiple cases.

Less than
B1
A1
A2
B2
B3
A3
A4
B4
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Open Questions

• Can the algorithm time complexity be reduced into
  O(n2m2)?