Knuth-Morris-Pratt Algorithm

1
Knuth-Morris-Pratt Algorithm

• left to right scan like the naïve algorithm
• one main improvement
• on a mismatch, calculate maximum possible shift
  to the right for the pattern

2
Basic Idea

• Definition
• For each position i in pattern P, define spi(P)
  to be the length of the longest proper suffix of
  P1..i that matches a prefix of P
• Define spi(P) to have the added condition that
  P(i1) is not equal to P(spi(P) 1)
• may denote as spi and spi when P is clear from
  context
• Usage
• mismatch occurs between P(i1) and T(k)
• Shift P to the right so that P(spi1) aligns
  with T(k)
• shift P i-spi spaces total
• If P is found, shift by n - spn places

3
Illustration of sp and sp

• 0 1
• 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
• a b c d a b c e a b c d a b c e f

spi 0 0 0 0 1 2 3 0 1 2 3 4 5 6 7 8 0
spi0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 8 0
4
Illustration 1 of KMP shift

• 0 1
• 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
• x y a b c a b d a b c f q f e a b

a b c a b d a b d
a b c a b d a b d
5
Illustration 2 of KMP shift

• 0 1
• 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
• x y a b x a b d a b c f q f e a b

a b x a b d a b d
a b x a b d a b d
6
spi and Z-boxes

• Definitions
• Position j gt 1 maps to i if i is the right end of
  a Z-box that starts at j
• Note, i j Zj-1 in this case
• Observation
• For any i gt 1, spi 0 if no j maps to i
• Otherwise, spi maxj maps to i Zj
• Choosing the smallest j that maps to i leads to
  the maximum possible Zj value

7
Z-based computation of spi

• for (i1iltni)
• spi 0
• for (jn jgt2 j--)
• i jZj-1
• spi Zj

8
Observations

• Original KMP defined in terms of failure
  functions F(i) and F(i)
• F(i) spi-1 and F(i) spi-1 for i 1 to n1
• 2m upper bound on number of comparisons
• once a position in T matches, it is never
  compared again to any position in P
• there may be cases where positions in T that
  mismatch are compared against multiple positions
  in P, but this can happen at most m times total
• Full implementation of KMP is on page 27

9
FSA KMP algorithm

• Definition
• For each position i in pattern P and each
  character x in S, define sp(i,x) (P) to be the
  length of the longest proper suffix of P1..i
  that matches a prefix of P and P(spi1) x
• Observation
• Now each position in T will be compared exactly
  once, even on a mismatch

10
Z-based computation of sp(i,x)

• for (i1iltni)
• for (all x in S)
• sp(i,x) 0
• for (jn jgt2 j--)
• i jZj-1
• x P(Zj1)
• sp(i,x) Zj