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

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

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,...

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

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

Filter-based algorithms (1) Apply the filter function to partial rows of the text matrix There can only be an occurrence where the pattern’s column vector occurs in the reduced text Use 1D pattern matching to find those occurrences

Filter-based algorithms: a simple filter function A simple example of a filter function: f(x) = x[0] Pattern: Text: aaa bab aaa a b a aaa bab aaa baa baa aba abb bbb bab bb ab bb baa abb aba aa ab aa aaa bab aaa baa baa aba abb bbb bab baa abb aba

Filter-based algorithms: Takaoka-Zhu Filter function: hash function from the (1D) Karp-Rabin algorithm aaa bab aaa baa baa aba abb bbb bab bb ab bb baa abb aba aa ab aa

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 pattern’s rows

Filter-based algorithms: Baker-Bird (1) Pattern: Trie based on pattern rows {aaa, bab}: q0 q1q2q3 q4q5q6 a aa a b b aaa bab aaa

Filter-based algorithms: Baker-Bird (2) Pattern: Aho-Corasick automaton based on pattern rows {aaa, bab}: q0 q1q2q3 q4q5q6 a aa a b b b b b b a b a a aaa bab aaa q3 q6 q3 b

Filter-based algorithms: Baker-Bird (3) Pattern: Text: aaa bab aaa q3 q6 q3 aaa bab aaa baa baa aba abb bbb bab bb ab bb baa abb aba aa ab aa q4q5 q6q4q5 q3 q4 q2q4 q2q3q4 q5q6q4 q5 q4q5q6 q5q6 q2q3 q4q5q6 q5q2q3

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 0 m-1 2*m-1 3*m-1

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 aaa bab aaa aaa bab aaa baa baa aba abb bbb bab bb ab bb baa abb aba aa ab aa

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

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:

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:

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:

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:

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

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 Polcar’s algorithm, sets of matrices can be represented by their maximal elements

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 –...

Questions