BitParallelism in Approximate Matching Agrep

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Bit-Parallelism in Approximate Matching Agrep
Dynamic Programming
Ni Bing Nov. 30th, 2006
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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Introduction

• Approximate matching
• Given a text Tt0t1...tn-1, a pattern
  Pp0p1...pm-1, (m
• Find all the substrings tste of T such that,
• Transformation of P to tste can be done with
  less than k errors, where
• Errors can be insertion, deletion and
  substitution

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Introduction
w number of bits in one machine word, for
example w32 for 32-bit machine
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Introduction

• Reasons of capability of Agrep against
  traditional DP
• Bit-parallelism permits executing several
  operations simultaneously over a set of bits or
  numbers stored in a single computer word.

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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Equivalence between Agrep DP
Table DP is the combination of the table Rd (0satisfy cell Rdi,j1
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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Exact matching Cell Dependencies
Definition R0i,j 1, if p0 pi match up to
tj 0, or else.
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Exact matching Column-wise Bit-parallelism (
Agrep )
R0i,j R0i-1,j-1 (pi tj)
, where

• Denote R0j as all the bits in the j-th column
• Assume p0 pm-1 is in the top-down direction

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Approximate matching - Cell Dependencies
Definition R1i,j 1, if p0 pi match up to
tj with not larger than 1 error 0, or else.
R1i,j (R0i-1,j-1 1) (R0i,j-1 1)
(R0i-1,j 1) R1i-1,j-1 (pi
tj)
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Approximate matching Column-wise
Bit-parallelism ( Agrep )
R1i,j R0i-1,j-1 R0i,j-1
R0i-1,j R1i-1,j-1 (pi tj)
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One more word

• The Column-wise Bit-parallelism in Agrep is
  called later as
• NFA (non-deterministic Finite Automaton) based
  Bit-parallelism

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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Dynamic Programming- Cell Dependencies
Definition Local alignment of DP DPi,j d,
such that p0 pi match up to tj with minimum d
errors
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Dynamic Programming

• Differential DPi,j
• Myer99
• Hyy Nava03

Bit-Parallelism possible ?
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Dynamic Programming- Cell Structure

• The difference in the
• horizontal direction,
• vertical direction, and
• diagonal direction

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Dynamic Programming- Cell representation (Myer99)

• To represent the difference
• Vertically,
• Hirozontally,
• And diagonally,

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Dynamic Programming- input/output function
(Myer99)

• 1 Conceptually think of as a function of
• modulated by a boolean value Xv,
• Xv Eq or
• Brute-force enumeration
• Symmetrically,

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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Comparisons Observations
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Outline

• Introduction
• Equivalence between Agrep DP
• Agrep
• Dynamic Programming
• Comparisons Observations
• Foresee

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Foresee

• Bit-parallelism done on structure instead of text