The Rabin-Karp Algorithm

1
The Rabin-Karp Algorithm

• String Matching

Jonathan M. Elchison 19 November 2004 CS-3410
Algorithms Dr. Shomper
2
Background

• String matching
• Naïve method
• n size of input string
• m size of pattern to be matched
• O( (n-m1)m )
• T( n2 ) if m floor( n/2 )
• We can do better

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How it works

• Consider a hashing scheme
• Each symbol in alphabet S can be represented by
  an ordinal value 0, 1, 2, ..., d
• S d
• Radix-d digits

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How it works

• Hash pattern P into a numeric value
• Let a string be represented by the sum of these
  digits
• Horners rule ( 30.1)
• Example
• A, B, C, ..., Z ? 0, 1, 2, ..., 26
• BAN ? 1 0 13 14
• CARD ? 2 0 17 3 22

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Upper limits

• Problem
• For long patterns, or for large alphabets, the
  number representing a given string may be too
  large to be practical
• Solution
• Use MOD operation
• When MOD q, values will be lt q
• Example
• BAN 1 0 13 14
• 14 mod 13 1
• BAN ? 1
• CARD 2 0 17 3 22
• 22 mod 13 9
• CARD ? 9

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Searching
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Spurious Hits

• Question
• Does a hash value match mean that the patterns
  match?
• Answer
• No these are called spurious hits
• Possible cases
• MOD operation interfered with uniqueness of hash
  values
• 14 mod 13 1
• 27 mod 13 1
• MOD value q is usually chosen as a prime such
  that 10q just fits within 1 computer word
• Information is lost in generalization (addition)
• BAN ? 1 0 13 14
• CAM ? 2 0 12 14

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Code

• RABIN-KARP-MATCHER( T, P, d, q )
• n ? length T
• m ? length P
• h ? dm-1 mod q
• p ? 0
• t0 ? 0
• for i ? 1 to m ? Preprocessing
• do p ? ( dp P i ) mod q
• t0 ? ( dt0 T i ) mod q
• for s ? 0 to n m ? Matching
• do if p ts
• then if P 1..m T s1 .. sm
• then print Pattern occurs with shift s
• if s lt n m
• then ts1 ? ( d ( ts T s 1 h )
  T s m 1 ) mod q

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Performance

• Preprocessing (determining each pattern hash)
• T( m )
• Worst case running time
• T( (n-m1)m )
• No better than naïve method
• Expected case
• If we assume the number of hits is constant
  compared to n, we expect O( n )
• Only pattern-match hits not all shifts

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Demonstration

• http//www-igm.univ-mlv.fr/lecroq/string/node5.ht
  ml

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The Rabin-Karp Algorithm

• Sources
• Cormen, Thomas S., et al. Introduction to
  Algorithms. 2nd ed. Boston MIT Press, 2001.
• Karp-Rabin algorithm. 15 Jan 1997.
  lthttp//www-igm.univ-mlv.fr/lecroq/string/node5.h
  tmlgt.
• Shomper, Keith. Rabin-Karp Animation. E-mail
  to Jonathan Elchison. 12 Nov 2004.

• String Matching

Jonathan M. Elchison 19 November 2004 CS-3410
Algorithms Dr. Shomper