Two-way string-matching algorithm

In computer science, the two-way string-matching algorithm is a string-searching algorithm, discovered by Maxime Crochemore and Dominique Perrin in 1991. It takes a pattern of size m, called a “needle”, preprocesses it in linear time O(m), producing information that can then be used to search for the needle in any “haystack” string, taking only linear time O(n) with n being the haystack's length.

The two-way algorithm can be viewed as a combination of the forward-going Knuth–Morris–Pratt algorithm (KMP) and the backward-running Boyer–Moore string-search algorithm (BM). Like those two, the 2-way algorithm preprocesses the pattern to find partially repeating periods and computes “shifts” based on them, indicating what offset to “jump” to in the haystack when a given character is encountered.

Unlike BM and KMP, it uses only O(log m) additional space to store information about those partial repeats: the search pattern is split into two parts (its critical factorization), represented only by the position of that split. Being a number less than m, it can be represented in ⌈log₂ m⌉ bits. This is sometimes treated as "close enough to O(1) in practice", as the needle's size is limited by the size of addressable memory; the overhead is a number that can be stored in a single register, and treating it as O(1) is like treating the size of a loop counter as O(1) rather than log of the number of iterations. The actual matching operation performs at most 2n − m comparisons.

Breslauer later published two improved variants performing fewer comparisons, at the cost of storing additional data about the preprocessed needle:

• The first one performs at most n + ⌊(n − m)/2⌋ comparisons, ⌈(n − m)/2⌉ fewer than the original. It must however store ⌈log_{[[Golden ratio|Template:Nowrap]]} m⌉ additional offsets in the needle, using O(log² m) space.
• The second adapts it to only store a constant number of such offsets, denoted c, but must perform n + ⌊(Template:Frac + ε) * (n − m)⌋ comparisons, with ε = Template:Frac(F_c+2 − 1)⁻¹ = O(Template:Nowrap^−c) going to zero exponentially quickly as c increases.

The algorithm is considered fairly efficient in practice, being cache-friendly and using several operations that can be implemented in well-optimized subroutines. It is used by the C standard libraries glibc, newlib, and musl, to implement the memmem and strstr family of substring functions. As with most advanced string-search algorithms, the naïve implementation may be more efficient on small-enough instances; this is especially so if the needle isn't searched in multiple haystacks, which would amortize the preprocessing cost.

Critical factorization

Before we define critical factorization, we should define:

• A factorization is a partition Template:Tmath of a string Template:Mvar. For example, ("Wiki","pedia") is a factorization of "Wikipedia".
• A period of a string Template:Mvar is an integer Template:Mvar such that all characters Template:Mvar-distance apart are equal. More precisely, Template:Math holds for any integer Template:Math. This definition is allowed to be vacuously true, so that any word of length Template:Mvar has a period of Template:Mvar. To illustrate, the 8-letter word "educated" has period 6 in addition to the trivial periods of 8 and above. The minimum period of Template:Mvar is denoted as Template:Tmath.
• A repetition Template:Mvar in Template:Tmath is a non-empty string such that:
  • Template:Mvar is a suffix of Template:Mvar or Template:Mvar is a suffix of Template:Mvar;
  • Template:Mvar is a prefix of Template:Mvar or Template:Mvar is a prefix of Template:Mvar;

  In other words, Template:Mvar occurs on both sides of the cut with a possible overflow on either side. Examples include "an" for ("ban","ana") and "voca" for ("a","vocado"). Each factorization trivially has at least one repetition: the string Template:Mvar.

• A local period is the length of a repetition in Template:Tmath. The smallest local period in Template:Tmath is denoted as Template:Tmath. Because the trivial repetition Template:Mvar is guaranteed to exist and has the same length as Template:Mvar, we see that Template:Tmath.

Finally, a critical factorization is a factorization Template:Tmath of Template:Mvar such that Template:Tmath. The existence of a critical factorization is provably guaranteed. For a needle of length Template:Mvar in an ordered alphabet, it can be computed in Template:Math comparisons, by computing the lexicographically larger of two ordered maximal suffixes, defined for order ≤ and ≥.

The algorithm

The algorithm starts by computing a critical factorization of the needle n as the preprocessing step. This step produces the index (starting point) of the periodic right-half, and the period of this stretch. The suffix computation here follows the authors' formulation. It can alternatively be computed using the Duval's algorithm, which is simpler and still linear time but slower in practice.

Shorthand for inversion.
function cmp(a, b)
    if a > b return 1
    if a = b return 0
    if a < b return -1

function maxsuf(n, rev)
    length ← len(n)
    cur_period ← 1       currently known period.
    period_test_idx ← 1  index for period testing, 0 < period_test_idx <= cur_period.
    maxsuf_test_idx ← 0  index for maxsuf testing. greater than maxs.
    maxsuf_idx ← -1      the proposed starting index of maxsuf

    while maxsuf_test_idx + period_test_idx < length
        cmp_val ← cmp(
              n[maxsuf_test_idx + period_test_idx],
              n[maxsuf_idx      + period_test_idx]
        )
        if rev
            cmp_val *= -1
        if cmp_val < 0
            Suffix (maxsuf_test_idx + period_test_idx) is smaller. Period is the entire prefix so far.
            maxsuf_test_idx += period_test_idx
            period_test_idx ← 1
            cur_period ← maxsuf_test_idx - maxsuf_idx
        else if cmp_val == 0
            They are the same - we should go on.
            if period_test_idx == cur_period
                We are done checking this stretch of cur_period. reset period_test_idx.
                maxsuf_test_idx += cur_period
                period_test_idx ← 1
            else
                period_test_idx += 1
        else
            Suffix is larger. Start over from here.
            maxsuf_idx ← maxsuf_test_idx
            maxsuf_test_idx += 1
            cur_period ← 1
            period_test_idx ← 1
   return [maxsuf_idx, cur_period]

function crit_fact(n)
    [idx1, per1] ← maxsuf(n, false)
    [idx2, per2] ← maxsuf(n, true)
    if idx1 > idx2
        return [idx1, per1]
    else
        return [idx2, per2]

The comparison proceeds by first matching for the right-hand-side, and then for the left-hand-side if it matches. Linear-time skipping is done using the period.

function match(needle, haystack)
    needle_len   ← len(needle)
    haystack_len ← len(haystack)
    [length, cur_period] ← crit_fact(needle)
    Matches ← {}                             set of matches.

    Match the suffix.
    Use a library function like memcmp, or write your own loop.
    if needle[0] ... needle[length] == needle[length + 1] ... needle[length + cur_period]
        Matches ← {}
        pos ← 0
        s ← 0

    TODO. At least put the skip in.