1. The Rabin-Karp Algorithm
String Matching
Jonathan M. Elchison
19 November 2004
CS Algorithms
Dr. Shomper

2. Background String matching Naïve method We can do better
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 → = 14
CARD → = 22

5. Upper limits Problem Solution Example
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 = = 14
14 mod 13 = 1
BAN → 1
CARD = = 22
22 mod 13 = 9
CARD → 9

6. Searching

7. Spurious Hits Question Answer Possible cases
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 → = 14
CAM → = 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)
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

11. The Rabin-Karp Algorithm
Sources:
Cormen, Thomas S., et al. Introduction to Algorithms. 2nd ed. Boston: MIT Press, 2001.
Karp-Rabin algorithm. 15 Jan <
Shomper, Keith. “Rabin-Karp Animation.” to Jonathan Elchison. 12 Nov 2004.
The Rabin-Karp Algorithm
String Matching
Jonathan M. Elchison
19 November 2004
CS Algorithms
Dr. Shomper