Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 43
Evidently the condition P(i-D, D) & Q(i-D) must hold before assigning j := D, if the invariant P(ij,
j) & Q(i-j) is to hold thereafter. This precondition is therefore our guideline for finding an appropriate
expression for D along with the condition P(i-j, j) & Q(i-j), which is assumed to hold prior to the
assignment (all subsequent reasoning concerns this point of the program).
The key observation is that thanks to P(i-j, j) we know that
p 0 ... p j-1 = s i-j ... s i-1
(we had just scanned the first j characters of the pattern and found them to match). Therefore the condition
P(i-D, D) with D < j, i.e.,
p 0 ... p D-1 = s i-D ... s i-1
translates into an equation for D:
p 0 ... p D-1 = p j-D ... p j-1
As to Q(i-D), this condition follows from Q(i-j) provided ~R(i-k) for k = D+1 ... j. The validity of
~R(i-k) for j = k is guaranteed by the inequality s[i] # p[j]. Although the conditions ~R(i-k) ≡ ~P(ik,
M) for k = D+1 ... j-1 cannot be evaluated from the scanned text fragment only, one can evaluate the
sufficient conditions ~P(i-k,k). Expanding them and taking into account the already found matches
between the elements of s and p, we obtain the following condition:
p 0 ... p k-1 ≠ p j-k ... p j-1 for all k = D+1 ... j-1
that is, D must be the maximal solution of the above equation. Fig. 1.12 illustrates the function of D.
examples
i,j
string
pattern
shifted pattern
A A A A A C
A A A A A B
A A A A A B
j=5, D=4, (max. shift =j-D=1)
p 0… p 3 = p 1… p 4
i,j
A B C A B D
A B C A B C
A B C A B C
j=5, D=2, (max. shift =j-D=3)
p 0… p 1 = p 3… p 4
p 0… p 2 # p 2… p 4
p 0… p 3 # p 1… p 4
i,j
A B C D E A
A B C D E F
A B C
j=5, D=0, (max. shift =j-D=5)
p 0… p 0 # p 4… p 4
p 0… p 1 # p 3… p 4
p 0… p 2 # p 2… p 4
p 0… p 3 # p 1… p 4
Fig. 1.12. Partial pattern matches and computation of D.
If there is no solution for D, then there is no match in positions below i+1. Then we set D := -1. Such a
situation is shown in the upper part in Fig. 1.13.
The last example in Fig. 1.12 suggests that we can do even slightly better; had the character p j been an
A instead of an F, we would know that the corresponding string character could not possibly be an A, and
the shift of the pattern with D = -1 had to be performed in the next loop iteration (see Fig. 1.13, the lower