/* Fill the table.
For 0 < i < m:
0 < table[i] <= i is defined such that
- rhaystack[0..i-1] == needle[0..i-1] and rhaystack[i] != needle[i]
- implies
- forall 0 <= x < table[i]: rhaystack[x..x+m-1] != needle[0..m-1],
+ forall 0 < x < table[i]: needle[x..i-1] != needle[0..i-1-x],
and table[i] is as large as possible with this property.
+ This implies:
+ 1) For 0 < i < m:
+ If table[i] < i,
+ needle[table[i]..i-1] = needle[0..i-1-table[i]].
+ 2) For 0 < i < m:
+ rhaystack[0..i-1] == needle[0..i-1]
+ and exists h, i <= h < m: rhaystack[h] != needle[h]
+ implies
+ forall 0 <= x < table[i]: rhaystack[x..x+m-1] != needle[0..m-1].
table[0] remains uninitialized. */
{
size_t i, j;
+ /* i = 1: Nothing to verify for x = 0. */
table[1] = 1;
j = 0;
+
for (i = 2; i < m; i++)
{
+ /* Here: j = i-1 - table[i-1].
+ The inequality needle[x..i-1] != needle[0..i-1-x] is known to hold
+ for x < table[i-1], by induction.
+ Furthermore, if j>0: needle[i-1-j..i-2] = needle[0..j-1]. */
unsigned char b = (unsigned char) needle[i - 1];
for (;;)
{
+ /* Invariants: The inequality needle[x..i-1] != needle[0..i-1-x]
+ is known to hold for x < i-1-j.
+ Furthermore, if j>0: needle[i-1-j..i-2] = needle[0..j-1]. */
if (b == (unsigned char) needle[j])
{
+ /* Set table[i] := i-1-j. */
table[i] = i - ++j;
break;
}
+ /* The inequality needle[x..i-1] != needle[0..i-1-x] also holds
+ for x = i-1-j, because
+ needle[i-1] != needle[j] = needle[i-1-x]. */
if (j == 0)
{
+ /* The inequality holds for all possible x. */
table[i] = i;
break;
}
+ /* The inequality needle[x..i-1] != needle[0..i-1-x] also holds
+ for i-1-j < x < i-1-j+table[j], because for these x:
+ needle[x..i-2]
+ = needle[x-(i-1-j)..j-1]
+ != needle[0..j-1-(x-(i-1-j))] (by definition of table[j])
+ = needle[0..i-2-x],
+ hence needle[x..i-1] != needle[0..i-1-x].
+ Furthermore
+ needle[i-1-j+table[j]..i-2]
+ = needle[table[j]..j-1]
+ = needle[0..j-1-table[j]] (by definition of table[j]). */
j = j - table[j];
}
+ /* Here: j = i - table[i]. */
}
}