+ /* short-circuit obvious comparisons */
+ if (xvec_length == 0 || yvec_length == 0) /* Prob: 1% */
+ return (xvec_length == 0 && yvec_length == 0 ? 1.0 : 0.0);
+
+ if (lower_bound > 0)
+ {
+ /* Compute a quick upper bound.
+ Each edit is an insertion or deletion of an element, hence modifies
+ the length of the sequence by at most 1.
+ Therefore, when starting from a sequence X and ending at a sequence Y,
+ with N edits, | yvec_length - xvec_length | <= N. (Proof by
+ induction over N.)
+ So, at the end, we will have
+ edit_count >= | xvec_length - yvec_length |.
+ and hence
+ result
+ = (xvec_length + yvec_length - edit_count)
+ / (xvec_length + yvec_length)
+ <= (xvec_length + yvec_length - | yvec_length - xvec_length |)
+ / (xvec_length + yvec_length)
+ = 2 * min (xvec_length, yvec_length) / (xvec_length + yvec_length).
+ */
+ volatile double upper_bound =
+ (double) (2 * MIN (xvec_length, yvec_length))
+ / (xvec_length + yvec_length);
+
+ if (upper_bound < lower_bound) /* Prob: 74% */
+ /* Return an arbitrary value < LOWER_BOUND. */
+ return 0.0;
+
+#if CHAR_BIT <= 8
+ /* When X and Y are both small, avoid the overhead of setting up an
+ array of size 256. */
+ if (xvec_length + yvec_length >= 20) /* Prob: 99% */
+ {
+ /* Compute a less quick upper bound.
+ Each edit is an insertion or deletion of a character, hence
+ modifies the occurrence count of a character by 1 and leaves the
+ other occurrence counts unchanged.
+ Therefore, when starting from a sequence X and ending at a
+ sequence Y, and denoting the occurrence count of C in X with
+ OCC (X, C), with N edits,
+ sum_C | OCC (X, C) - OCC (Y, C) | <= N.
+ (Proof by induction over N.)
+ So, at the end, we will have
+ edit_count >= sum_C | OCC (X, C) - OCC (Y, C) |,
+ and hence
+ result
+ = (xvec_length + yvec_length - edit_count)
+ / (xvec_length + yvec_length)
+ <= (xvec_length + yvec_length - sum_C | OCC(X,C) - OCC(Y,C) |)
+ / (xvec_length + yvec_length).
+ */
+ int occ_diff[UCHAR_MAX + 1]; /* array C -> OCC(X,C) - OCC(Y,C) */
+ int sum;
+
+ /* Determine the occurrence counts in X. */
+ memset (occ_diff, 0, sizeof (occ_diff));
+ for (i = xvec_length - 1; i >= 0; i--)
+ occ_diff[(unsigned char) string1[i]]++;
+ /* Subtract the occurrence counts in Y. */
+ for (i = yvec_length - 1; i >= 0; i--)
+ occ_diff[(unsigned char) string2[i]]--;
+ /* Sum up the absolute values. */
+ sum = 0;
+ for (i = 0; i <= UCHAR_MAX; i++)
+ {
+ int d = occ_diff[i];
+ sum += (d >= 0 ? d : -d);
+ }
+
+ upper_bound = 1.0 - (double) sum / (xvec_length + yvec_length);
+
+ if (upper_bound < lower_bound) /* Prob: 66% */
+ /* Return an arbitrary value < LOWER_BOUND. */
+ return 0.0;
+ }
+#endif
+ }
+