Boyer–Moore string search algorithm

def alphabet_index(c):
    """
    Returns the index of the given character in the English alphabet, counting from 0.
    """
    return ord(c.lower()) - 97 # 'a' is ASCII character 97

def match_length(S, idx1, idx2):
    """
    Returns the length of the match of the substrings of S beginning at idx1 and idx2.
    """
    if idx1 == idx2:
        return len(S) - idx1
    match_count = 0
    while idx1 < len(S) and idx2 < len(S) and S[idx1] == S[idx2]:
        match_count += 1
        idx1 += 1
        idx2 += 1
    return match_count

def fundamental_preprocess(S):
    """
    Returns Z, the Fundamental Preprocessing of S. Z[i] is the length of the substring 
    beginning at i which is also a prefix of S. This pre-processing is done in O(n) time,
    where n is the length of S.
    """
    if len(S) == 0: # Handles case of empty string
        return []
    if len(S) == 1: # Handles case of single-character string
        return [1]
    z = [0 for x in S]
    z[0] = len(S)
    z[1] = match_length(S, 0, 1)
    for i in range(2, 1+z[1]): # Optimization from exercise 1-5
        z[i] = z[1]-i+1
    # Defines lower and upper limits of z-box
    l = 0
    r = 0
    for i in range(2+z[1], len(S)):
        if i <= r: # i falls within existing z-box
            k = i-l
            b = z[k]
            a = r-i+1
            if b < a: # b ends within existing z-box
                z[i] = b
            else: # b ends at or after the end of the z-box, we need to do an explicit match to the right of the z-box
                z[i] = a+match_length(S, a, r+1)
                l = i
                r = i+z[i]-1
        else: # i does not reside within existing z-box
            z[i] = match_length(S, 0, i)
            if z[i] > 0:
                l = i
                r = i+z[i]-1
    return z

def bad_character_table(S):
    """
    Generates R for S, which is an array indexed by the position of some character c in the 
    English alphabet. At that index in R is an array of length |S|+1, specifying for each
    index i in S (plus the index after S) the next location of character c encountered when
    traversing S from right to left starting at i. This is used for a constant-time lookup
    for the bad character rule in the Boyer-Moore string search algorithm, although it has
    a much larger size than non-constant-time solutions.
    """
    if len(S) == 0:
        return [[] for a in range(26)]
    R = [[-1] for a in range(26)]
    alpha = [-1 for a in range(26)]
    for i, c in enumerate(S):
        alpha[alphabet_index(c)] = i
        for j, a in enumerate(alpha):
            R[j].append(a)
    return R

def good_suffix_table(S):
    """
    Generates L for S, an array used in the implementation of the strong good suffix rule.
    L[i] = k, the largest position in S such that S[i:] (the suffix of S starting at i) matches
    a suffix of S[:k] (a substring in S ending at k). Used in Boyer-Moore, L gives an amount to
    shift P relative to T such that no instances of P in T are skipped and a suffix of P[:L[i]]
    matches the substring of T matched by a suffix of P in the previous match attempt.
    Specifically, if the mismatch took place at position i-1 in P, the shift magnitude is given
    by the equation len(P) - L[i]. In the case that L[i] = -1, the full shift table is used.
    Since only proper suffixes matter, L[0] = -1.
    """
    L = [-1 for c in S]
    N = fundamental_preprocess(S[::-1]) # S[::-1] reverses S
    N.reverse()
    for j in range(0, len(S)-1):
        i = len(S) - N[j]
        if i != len(S):
            L[i] = j
    return L

def full_shift_table(S):
    """
    Generates F for S, an array used in a special case of the good suffix rule in the Boyer-Moore
    string search algorithm. F[i] is the length of the longest suffix of S[i:] that is also a
    prefix of S. In the cases it is used, the shift magnitude of the pattern P relative to the
    text T is len(P) - F[i] for a mismatch occurring at i-1.
    """
    F = [0 for c in S]
    Z = fundamental_preprocess(S)
    longest = 0
    for i, zv in enumerate(reversed(Z)):
        longest = max(zv, longest) if zv == i+1 else longest
        F[-i-1] = longest
    return F

def string_search(P, T):
    """
    Implementation of the Boyer-Moore string search algorithm. This finds all occurrences of P
    in T, and incorporates numerous ways of pre-processing the pattern to determine the optimal 
    amount to shift the string and skip comparisons. In practice it runs in O(m) (and even 
    sublinear) time, where m is the length of T. This implementation performs a case-insensitive
    search on ASCII alphabetic characters, spaces not included.
    """
    if len(P) == 0 or len(T) == 0 or len(T) < len(P):
        return []

    matches = []

    # Preprocessing
    R = bad_character_table(P)
    L = good_suffix_table(P)
    F = full_shift_table(P)

    k = len(P) - 1      # Represents alignment of end of P relative to T
    previous_k = -1     # Represents alignment in previous phase (Galil's rule)
    while k < len(T):
        i = len(P) - 1  # Character to compare in P
        h = k           # Character to compare in T
        while i >= 0 and h > previous_k and P[i] == T[h]:   # Matches starting from end of P
            i -= 1
            h -= 1
        if i == -1 or h == previous_k:  # Match has been found (Galil's rule)
            matches.append(k - len(P) + 1)
            k += len(P)-F[1] if len(P) > 1 else 1
        else:   # No match, shift by max of bad character and good suffix rules
            char_shift = i - R[alphabet_index(T[h])][i]
            if i+1 == len(P):   # Mismatch happened on first attempt
                suffix_shift = 1
            elif L[i+1] == -1:   # Matched suffix does not appear anywhere in P
                suffix_shift = len(P) - F[i+1]
            else:               # Matched suffix appears in P
                suffix_shift = len(P) - L[i+1]
            shift = max(char_shift, suffix_shift)
            previous_k = k if shift >= i+1 else previous_k  # Galil's rule
            k += shift
    return matches

#include <stdint.h>
#include <stdlib.h>

#define ALPHABET_LEN 256
#define NOT_FOUND patlen
#define max(a, b) ((a < b) ? b : a)

// delta1 table: delta1[c] contains the distance between the last
// character of pat and the rightmost occurrence of c in pat.
// If c does not occur in pat, then delta1[c] = patlen.
// If c is at string[i] and c != pat[patlen-1], we can
// safely shift i over by delta1[c], which is the minimum distance
// needed to shift pat forward to get string[i] lined up 
// with some character in pat.
// this algorithm runs in alphabet_len+patlen time.
void make_delta1(int *delta1, uint8_t *pat, int32_t patlen) {
    int i;
    for (i=0; i < ALPHABET_LEN; i++) {
        delta1[i] = NOT_FOUND;
    }
    for (i=0; i < patlen-1; i++) {
        delta1[pat[i]] = patlen-1 - i;
    }
}

// true if the suffix of word starting from word[pos] is a prefix 
// of word
int is_prefix(uint8_t *word, int wordlen, int pos) {
    int i;
    int suffixlen = wordlen - pos;
    // could also use the strncmp() library function here
    for (i = 0; i < suffixlen; i++) {
        if (word[i] != word[pos+i]) {
            return 0;
        }
    }
    return 1;
}

// length of the longest suffix of word ending on word[pos].
// suffix_length("dddbcabc", 8, 4) = 2
int suffix_length(uint8_t *word, int wordlen, int pos) {
    int i;
    // increment suffix length i to the first mismatch or beginning
    // of the word
    for (i = 0; (word[pos-i] == word[wordlen-1-i]) && (i < pos); i++);
    return i;
}

// delta2 table: given a mismatch at pat[pos], we want to align 
// with the next possible full match could be based on what we
// know about pat[pos+1] to pat[patlen-1].
//
// In case 1:
// pat[pos+1] to pat[patlen-1] does not occur elsewhere in pat,
// the next plausible match starts at or after the mismatch.
// If, within the substring pat[pos+1 .. patlen-1], lies a prefix
// of pat, the next plausible match is here (if there are multiple
// prefixes in the substring, pick the longest). Otherwise, the
// next plausible match starts past the character aligned with 
// pat[patlen-1].
// 
// In case 2:
// pat[pos+1] to pat[patlen-1] does occur elsewhere in pat. The
// mismatch tells us that we are not looking at the end of a match.
// We may, however, be looking at the middle of a match.
// 
// The first loop, which takes care of case 1, is analogous to
// the KMP table, adapted for a 'backwards' scan order with the
// additional restriction that the substrings it considers as 
// potential prefixes are all suffixes. In the worst case scenario
// pat consists of the same letter repeated, so every suffix is
// a prefix. This loop alone is not sufficient, however:
// Suppose that pat is "ABYXCDBYX", and text is ".....ABYXCDEYX".
// We will match X, Y, and find B != E. There is no prefix of pat
// in the suffix "YX", so the first loop tells us to skip forward
// by 9 characters.
// Although superficially similar to the KMP table, the KMP table
// relies on information about the beginning of the partial match
// that the BM algorithm does not have.
//
// The second loop addresses case 2. Since suffix_length may not be
// unique, we want to take the minimum value, which will tell us
// how far away the closest potential match is.
void make_delta2(int *delta2, uint8_t *pat, int32_t patlen) {
    int p;
    int last_prefix_index = patlen-1;

    // first loop
    for (p=patlen-1; p>=0; p--) {
        if (is_prefix(pat, patlen, p+1)) {
            last_prefix_index = p+1;
        }
        delta2[p] = last_prefix_index + (patlen-1 - p);
    }

    // second loop
    for (p=0; p < patlen-1; p++) {
        int slen = suffix_length(pat, patlen, p);
        if (pat[p - slen] != pat[patlen-1 - slen]) {
            delta2[patlen-1 - slen] = patlen-1 - p + slen;
        }
    }
}

uint8_t* boyer_moore (uint8_t *string, uint32_t stringlen, uint8_t *pat, uint32_t patlen) {
    int i;
    int delta1[ALPHABET_LEN];
    int *delta2 = (int *)malloc(patlen * sizeof(int));
    make_delta1(delta1, pat, patlen);
    make_delta2(delta2, pat, patlen);

    // The empty pattern must be considered specially
    if (patlen == 0) {
        free(delta2);
        return string;
    }

    i = patlen-1;
    while (i < stringlen) {
        int j = patlen-1;
        while (j >= 0 && (string[i] == pat[j])) {
            --i;
            --j;
        }
        if (j < 0) {
            free(delta2);
            return (string + i+1);
        }

        i += max(delta1[string[i]], delta2[j]);
    }
    free(delta2);
    return NULL;
}