Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 46
WHILE (j > 0) & (i 0) & (i 0 demands that i > N; hence Q(i) implies Q(N),
signalling that no match exists. Of course we still have to convince ourselves that Q(i) and P(i, j) are indeed
invariants of the two repetitions. They are trivially satisfied when repetition starts, since Q(M) and P(x, M)
are always true.
Consider the first branch. Simultaneously decrementing k and j does not affect Q(i), and, since s k-
1 = p j-1 had been established, and P(i, j) holds prior to decrementing j, then P(i, j) holds after it as well.
In the second branch, it is sufficient to show that the statement i := i + d s[i-1] never falsifies the invariant
Q(i) because P(i, j) is satisfied automatically after the remaining assignments. Q(i) is satisfied after
incrementing i provided that before the assignment Q(i+d s[i-1] ) is guaranteed. Since we know that Q(i)
holds, it suffices to establish ~R(i+h) for h = 1 .. d s[i-1] -1. We now recall that d x is defined as the
distance of the rightmost occurrence of x in the pattern from the end. This is formally expressed as
Ak: M-d x ≤ k < M-1 : p k ≠ x
Substituting s i-1 for x, we obtain
Ak: M-d s[i-1] ≤ k < M-1 : s i-1 ≠ p k
= Ah: 1 ≤ h ≤ d s[i-1] -1 : si-1 ≠ p M-1-h
⇒ Ah: 1 ≤ h ≤ d s[i-1] -1 : ~R(i+h)
The following program includes the presented, simplified Boyer-Moore strategy in a setting similar to
that of the preceding KMP-search program.
PROCEDURE Search (VAR s, p: ARRAY OF CHAR; M, N: INTEGER; VAR r: INTEGER);
(* ADenS193_BM *)
(*search for pattern p of length M in text s of length N*)
(*if p is found, then r indicates the position in s, otherwise r = -1*)
VAR i, j, k: INTEGER;
d: ARRAY 128 OF INTEGER;
BEGIN
FOR i := 0 TO 127 DO d[i] := M END;
FOR j := 0 TO M-2 DO d[ORD(p[j])] := M-j-1 END;
i := M; j := M; k := i;
WHILE (j > 0) & (i 0) & (i