Prefix & Suffix Example W = ab is a prefix of X = abefac where Y = efac. Example W = cdaa is a suffix of X = acbecdaa where Y = acbe A string W is a prefix of a string X if X = WY for some string Y, denoted W X. A string W is a suffix of a string X if X = YW for some string Y, denoted W X. The empty string  is a prefix of any string.  is a suffix of any string.

Overlapping Suffix Lemma Suppose X Z and Y Z. a) if |X|  |Y|, then X Y ; b) if |X|  |Y|, then Y X ; c) if |X| = |Y|, then X = Y. X Z Y a) b) c)

The Knuth-Morris-Pratt Algorithm Use one auxiliary function (prefix function). Key idea on improvement: Achieve running time O(n+m)! Instead of precomputing the transition function in O(m |  |), efficiently compute it “on the fly” as needed. 3

Minimum Shifting Pattern P Text T P Question: What is the least shift s > s ? 1 q s+1 s+q shift s (minimum) shift s q matching chars s+1

The Prefix Function How much to shift depends on the pattern not the text. Prefix function measures length of the longest prefix of P[1..m] that is also a proper suffix of P[1..q].  [q] = max{ k: k < q and P[1..k] is a suffix of P[1..q] } b a c b a b a b a a b c b a b a b a b a c a T P[1..q] = ababa  [5] = 3 shift by 2 a b a b a c a P[1..k] = aba

Example  [ ] measures how well the pattern matches against a shift of itself. i P[1..i] a b a b a b a b c a  [i] P[1..8] a b a b a b a b c a P[1..6] a b a b a b a b c a  [8] = 6 P[1..4] a b a b a b a b c a  [6] = 4 P[1..2] a b a b a b a b c a  [4] = 2 P[] a b a b a b a b c a  [2] = 0 Ex. a b a b a b a b c a

Computing the Prefix Function Compute-Prefix-Function(P) m  length[P]  [1]  0 k  0 for q  2 to m // invariant k =  [q  1] do while k > 0 and P[k+1]  P[q] do k   [k] if P[k+1] = P[q] then k  k+1  [q]  k return  k+1 q  [k]+1 Ex. q = 9 and k = 6 p[k+1] = a  c = p[q]  [9] =  [  [  [6]]] = 0 k+1 a b a b a b a b c a k a b a b a b a b c a 6 q a b a b a b a b c a 4 a b a b a b a b c a 2 a b a b a b a b c a 0

Running-time Analysis 1 Compute-Prefix-Function(P) 2 m  length[P] 3  [1]  0 4 k  0 5 for q = 2 to m 6do while k > 0 and P[k+1]  P[q] 7do k   [k] // decrease k by at least 1 8 if P[k+1] = P[q] 9 then k  k+1 //  m  1 increments, each by 1 10  [q]  k 11 return  # decrements  # increments, thus line 7 is executed at most m  1 times in total. Total time  (m).

KMP Algorithm KMP-Matcher(T, P) // n = |T| and m = |P|   Compute-Prefix-Function(P) //  (m) time. q  0 for i  1 to n do while q > 0 and P[q+1]  T[i] do q   [q] if P[q+1] = T[i] then q  q+1 //  n total increments if q = m then print “Pattern occurs with shift” i  m q   [q] //  (n) time Total time  (m+n).

A KMP Example i  [i] abababbababbaababbababaa ababbababaa abababbababbaababbababaa ababbababaa abababbababbaababbababaa ababbababaa abababbababbaababbababaa ababbababaa abababbababbaababbababaa ababbababaa shift by q   [q] = 4  2 shift by 9  4 = 5 shift by 6  1 = 5 shift by 1  0 = 1 P[1..i] a b a b b a b a b a a