A new matching algorithm based on prime numbers

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A new matching algorithm based on prime numbers

• N. D. Atreas and C. Karanikas
• Department of Informatics
• Aristotle University of Thessaloniki

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Exact Matching find all the occurences of a
pattern within a text.

• 1. The Brute Force algorithm performs character
  by character comparison in O(N M) time
  complexity, where M is the length of the pattern
  and N is the length of the text.
• 2. The Knuth-Morris-Pratt algorithm Runs in
  O(NM) time, avoiding unecessary re-examinations
  of previously matched characters.

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• 3. The Boyer-Moore algorithm
• involves character by character comparison
  by using backwards checking. Best case execution
  O(N/M), worst time O(N).
• 4. The Karp Rabin algorithm
• It is a randomised algorithm that seeks a
  pattern within a text by using hashing. Expected
  running time O(NM).

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• A hash function must be
• efficiently computable
• highly discriminating for strings
• hash(x(j1 ... jM)) must be easily computable
  from hash(x(j jM-1)) and x(jM).
• not injective, i.e. the equality of two hash
  values suggests, but does not guarantee, equality
  of the inputs.

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Let x x(1),x(N) be a set of positive
integers and p(1)ltltp(N) be primes such that
p(1)gtMaxx(i), i1,..,N, we define the
transform
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Properties of T(x(1)x(N))

• T(x(1),x(N)) is one to one.
• x(1),,x(N) can be recovered from T(x) as the
  unique solution of a system of N linear
  Diophantine equations defined recursively
• (p(i1)p(N))x(i)p(i)c(i1) c(i)
• where c(1)T(x)p(1)P(N).

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Properties of T(x(1)x(N))

• T(x) can be used as a measure of similarity
  between two strings, since it can be used for
  counting the different elements between them.
• It provides a necessary and sufficient condition
  to detect whenever a binding operation on strings
  can be implemented.
• It is not a hash function.

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Modelling a hash function approximating T.
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Definition of the hash function

• We prove

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Final form of hash function

• Theorem

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Software implementation

• Let Xx(1),,x(N) be the text and
  Yy(1),,y(M) be the pattern.
• Compute T(y(1),,y(M)) and T(x(1),,x(M)) in O(M)
  time.
• Compute the hash values in O(N-M) time

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Software implementation

• for some i then x(i1),,x(iM-1) is a candidate
  for string matching.
• For all candidates perform at most p (p is the
  length of the alphabet) character comparisons to
  throw out false matches.
• The algorithm executes in O(N) time complexity.

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Conclusions

• We introduce the idea of a hash function
  approximation in order to reduce the
  computational complexity of an algorithm.
• Although the time bounds are the same or in some
  times inferiors compared to Boyer-Moore
  algorithm, our algorithm is superior for multiple
  matching problems.