Twodimensional pattern matching

1
Two-dimensional pattern matching
M.G.W.H. van de Rijdt
23 August 2005
2
Introduction

• Problem description
• Naive algorithm
• Filter-based algorithms
• A simple filter function
• Takaoka-Zhu
• Baker-Bird
• Baeza-Yates Régnier
• Polcar
• Conclusions
• Future work
• Questions

3
Problem description

• One-dimensional pattern matching finding all
  occurrences of a pattern string in a text string
• Two-dimensional pattern matching finding all
  occurrences of a 2D pattern matrix in a 2D text
  matrix
• Applications image processing, ...

4
Naive algorithm

• Simply check for each position in the text
  whether there is a match there
• Most straightforward, but inefficient, solution
• Better algorithms
• use gathered information to disregard a larger
  area of the text at onces
• and/or
• precompute information to determine more quickly
  whether a match exists on a position in the text

5
Filter-based algorithms (0)

• Define a filter function, which transforms each
  row of the pattern matrix to a single value
• Using this function, reduce the pattern matrix to
  a single (column) vector

6
Filter-based algorithms (1)

• Apply the filter function to partial rows of the
  text matrix
• There can only be an occurrence where the
  patterns column vector occurs in the reduced
  text
• Use 1D pattern matching to find those occurrences

7
Filter-based algorithms a simple filter function

• A simple example of a filter function f(x)
  x0
• Pattern
• Text

8
Filter-based algorithms Takaoka-Zhu

• Filter function hash function from the (1D)
  Karp-Rabin algorithm

9
Filter-based algorithms Baker-Bird (0)

• Based on Aho-Corasick automaton
• Aho-Corasick is an algorithm for (1D)
  multipattern matching
• It uses a special automaton, based on the pattern
  strings
• Filter function for Baker-Bird state in the
  Aho-Corasick automaton, based on the patterns
  rows

10
Filter-based algorithms Baker-Bird (1)

• Pattern
• Trie based on pattern rows aaa, bab

a
a
q1
q2
q3
a
q0
b
a
b
q4
q5
q6
11
Filter-based algorithms Baker-Bird (2)

• Pattern
• Aho-Corasick automaton based on pattern rows
  aaa, bab

a
a
q1
q2
q3
a
b
b
a
b
q0
a
b
a
b
q4
q5
q6
a
b
b
b
12
Filter-based algorithms Baker-Bird (3)

• Pattern
• Text

13
Baeza-Yates Régnier (0)

• Say our pattern has m rows
• In the text, each occurrence of the pattern
  intersects with exactly one row of the form i m
  1

14
Baeza-Yates Régnier (1)

• Algorithm idea
• use 1D multipattern matching to search for
  occurrences of any pattern row in these rows of
  the text
• where such a match occurs, check if there is a
  match with the entire pattern in the surrounding
  area

15
Polcar (0)

• In some 1D pattern matching algorithms, we view
  an occurrence of the pattern as a suffix of a
  prefix of the text
• For Polcar, we do the same in two dimensions

16
Polcar (1)

• For each prefix of the text A, we compute the set
  of suffixes of A that are also a prefix of the
  pattern

17
Polcar (1)

• For each prefix of the text A, we compute the set
  of suffixes of A that are also a prefix of the
  pattern

18
Polcar (1)

• For each prefix of the text A, we compute the set
  of suffixes of A that are also a prefix of the
  pattern

19
Polcar (1)

• For each prefix of the text A, we compute the set
  of suffixes of A that are also a prefix of the
  pattern

20
Polcar (2)

• In derivations of the corresponding 1D pattern
  matching algorithms, sets of prefixes of the
  pattern are represented by their element of
  maximum length
• In 2D there is not always one unique maximum
• But these sets of matrices can be represented by
  their maximal elements

21
Conclusions

• Presentation of several 2D pattern matching
  algorithms
• All of them have been formally derived
• derivation is a formal proof
• derivations show the major design decisions
• Similarities between the filter-based algorithms
• Several improvements to existing algorithms
• most notably in Polcars algorithm, sets of
  matrices can be represented by their maximal
  elements

22
Future work

• Derive other existing algorithms
• Construct a taxonomy
• Find new algorithms
• Expand existing pattern matching toolkits (SPARE
  Time / SPARE Parts) or create a new 2D pattern
  matching toolkit
• Thorough performance analysis
• Further generalisations of the 2D pattern
  matching problem
• Multipattern matching
• More than two dimensions
• Approximate 2D pattern matching
• Patterns of non-rectangular shapes
• ...

23
Questions
24
(No Transcript)