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The Rabin-Karp Algorithm

The Rabin-Karp Algorithm. String Matching. Jonathan M. Elchison 19 November 2004 CS-3410 Algorithms Dr. Shomper. Background. String matching Naïve method n ≡ size of input string m ≡ size of pattern to be matched O( (n-m+1)m ) Θ ( n 2 ) if m = floor( n/2 ) We can do better.

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The Rabin-Karp Algorithm

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  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-m+1)m ) • Θ( n2 ) if m = floor( n/2 ) • We can do better

  3. How it works • Consider a hashing scheme • Each symbol in alphabet Σ can be represented by an ordinal value { 0, 1, 2, ..., d } • |Σ| = d • “Radix-d digits”

  4. How it works • Hash pattern P into a numeric value • Let a string be represented by the sum of these digits • Horner’s rule (§ 30.1) • Example • { A, B, C, ..., Z } → { 0, 1, 2, ..., 26 } • BAN → 1 + 0 + 13 = 14 • CARD → 2 + 0 + 17 + 3 = 22

  5. 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 < 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

  6. Searching

  7. 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

  8. 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 ← ( d*p + P[ i ] ) mod q t0 ← ( d*t0 + T[ i ] ) mod q for s ← 0 to n – m ► Matching do if p = ts then if P[ 1..m ] = T[ s+1 .. s+m ] then print “Pattern occurs with shift” s if s < n – m then ts+1 ← ( d * ( ts – T[ s + 1 ] * h ) + T[ s + m + 1 ] ) mod q

  9. Performance • Preprocessing (determining each pattern hash) • Θ( m ) • Worst case running time • Θ( (n-m+1)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

  10. Demonstration • http://www-igm.univ-mlv.fr/~lecroq/string/node5.html

  11. Sources: • Cormen, Thomas S., et al. Introduction to Algorithms. 2nd ed. Boston: MIT Press, 2001. • Karp-Rabin algorithm. 15 Jan 1997. <http://www-igm.univ-mlv.fr/~lecroq/string/node5.html>. • Shomper, Keith. “Rabin-Karp Animation.” E-mail to Jonathan Elchison. 12 Nov 2004. The Rabin-Karp Algorithm String Matching Jonathan M. Elchison 19 November 2004 CS-3410 Algorithms Dr. Shomper

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