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Two-way string-matching algorithm
View on Wikipedia| Class | String-searching algorithm |
|---|---|
| Data structure | Any string with an ordered alphabet |
| Worst-case performance | O(n) |
| Best-case performance | O(n) |
| Worst-case space complexity | ⌈log₂ m⌉ |
In computer science, the two-way string-matching algorithm is a string-searching algorithm, discovered by Maxime Crochemore and Dominique Perrin in 1991.[1] 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.[2]
Breslauer later published two improved variants performing fewer comparisons, at the cost of storing additional data about the preprocessed needle:[3]
- The first one performs at most n + ⌊(n − m)/2⌋ comparisons, ⌈(n − m)/2⌉ fewer than the original. It must however store ⌈log 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 + ⌊(1⁄2 + ε) * (n − m)⌋ comparisons, with ε = 1⁄2(Fc+2 − 1)−1 = O(−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.[4][5][6] As with most advanced string-search algorithms, the naïve implementation may be more efficient on small-enough instances;[7] this is especially so if the needle isn't searched in multiple haystacks, which would amortize the preprocessing cost.
Critical factorization
[edit]Before we define critical factorization, we should define:[1]
- A factorization is a partition of a string x. For example,
("Wiki","pedia")is a factorization of"Wikipedia". - A period of a string x is an integer p such that all characters p-distance apart are equal. More precisely, x[i] = x[i + p] holds for any integer 0 < i ≤ len(x) − p. This definition is allowed to be vacuously true, so that any word of length n has a period of n. To illustrate, the 8-letter word
"educated"has period 6 in addition to the trivial periods of 8 and above. The minimum period of x is denoted as . - A repetition w in is a non-empty string such that:
- w is a suffix of u or u is a suffix of w;
- w is a prefix of v or v is a prefix of w;
- In other words, w 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 vu.[2]
- A local period is the length of a repetition in . The smallest local period in is denoted as . Because the trivial repetition vu is guaranteed to exist and has the same length as x, we see that .
Finally, a critical factorization is a factorization of x such that . The existence of a critical factorization is provably guaranteed.[1] For a needle of length m in an ordered alphabet, it can be computed in 2m comparisons, by computing the lexicographically larger of two ordered maximal suffixes, defined for order ≤ and ≥.[6]
The algorithm
[edit] | This section is missing information about the match function. (March 2022) |
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.[8]
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.
References
[edit]- ^ a b c Crochemore, Maxime; Perrin, Dominique (1 July 1991). "Two-way string-matching" (PDF). Journal of the ACM. 38 (3): 650–674. doi:10.1145/116825.116845. S2CID 15055316.
- ^ a b "Two Way algorithm".
- ^ Breslauer, Dany (May 1996). "Saving comparisons in the Crochemore-Perrin string-matching algorithm". Theoretical Computer Science. 158 (1–2): 177–192. doi:10.1016/0304-3975(95)00068-2.
- ^ "musl/src/string/memmem.c". Retrieved 23 November 2019.
- ^ "newlib/libc/string/memmem.c". Retrieved 23 November 2019.
- ^ a b "glibc/string/str-two-way.h".
- ^ "Eric Blake - Re: PATCH] Improve performance of memmem". Newlib mailing list.
- ^ Adamczyk, Zbigniew; Rytter, Wojciech (May 2013). "A note on a simple computation of the maximal suffix of a string". Journal of Discrete Algorithms. 20: 61–64. doi:10.1016/j.jda.2013.03.002.
Two-way string-matching algorithm
View on GrokipediaIntroduction
Definition and Purpose
The two-way string-matching algorithm is an online string-searching technique designed to find all occurrences of a pattern string of length in a text string of length , where both strings are defined over a finite alphabet .[1][4] It operates by preprocessing the pattern to identify structural properties that enable bidirectional scanning, alternating between forward comparisons on the right portion of the pattern and backward comparisons on the left portion to verify matches.[1] The output consists of a list of starting positions (where ) such that .[4] The primary purpose of the algorithm is to achieve efficient pattern matching in linear time, specifically , by preprocessing the pattern in time to allow constant-time skip operations during the search phase.[1] This preprocessing exploits the pattern's internal periodicity, computed via critical factorization, to partition the pattern into two overlapping halves and build separate failure functions for each, enabling rapid shifts past mismatched positions without re-examining previously scanned text characters.[1][4] Key advantages include its ability to minimize redundant character comparisons by leveraging periodicity for large skips in both scanning directions, typically performing at most text comparisons in the worst case while using constant extra space beyond the input strings.[4] This makes it particularly suitable for online or streaming text processing scenarios, where the text arrives incrementally and immediate reporting of matches is required without buffering the entire input.[1]Historical Context
The two-way string-matching algorithm emerged from foundational research in string processing during the late 1970s and early 1980s. Maxime Crochemore's 1981 work on an optimal algorithm for computing repetitions in words established key concepts in string periodicity, laying groundwork for advanced pattern-matching techniques.[5] Building on the one-way Knuth-Morris-Pratt algorithm introduced in 1977, which efficiently scans text forward but struggles with reverse traversals, the two-way approach was formally proposed by Crochemore and Dominique Perrin in 1991 to enable bidirectional scanning and overcome these limitations.[1] A pivotal milestone was the integration of the critical factorization theorem, discovered by Césari and Vincent in 1978 and developed by Jean-Pierre Duval in 1983, which identifies unique factorizations in words that minimize local periods and support efficient preprocessing.[6] This theorem, combined with refinements by Crochemore and Wojciech Rytter, culminated in the algorithm's comprehensive presentation in their 1994 book Text Algorithms, where it was positioned as a hybrid bridging forward and backward matching paradigms.[7] In the 2000s, subsequent optimizations emphasized practical implementations, including space-efficient variants and comparison reductions to suit hardware constraints, as explored in extensions by Crochemore and collaborators.[8] As of 2025, the algorithm continues to influence theoretical computer science and string processing libraries, though practical applications often favor more advanced indexing techniques. It remains a valuable tool in computer science education for illustrating bidirectional scanning and periodicity concepts.[1]Background Concepts
String Periodicity
In string theory, particularly within combinatorics on words, the period of a string of length is defined as a positive integer such that for all positions .[9] The minimal period, denoted , is the smallest such .[9] This concept captures the repetitive structure in strings, where smaller periods indicate higher regularity; for instance, the string "ababab" has minimal period 2, as it repeats the motif "ab" three times.[9] Closely related are borders and quasiperiods, which extend periodicity to non-exact repetitions. A border of a string is a proper prefix of that is also a suffix, meaning there exist strings and such that for some nonempty proper prefix-suffix .[10] Borders measure partial overlaps at the ends of a string, with the longest border often used in algorithms like the Knuth-Morris-Pratt method.[10] A quasiperiod, introduced as a generalization of periods, describes a string that can be covered by overlapping occurrences of a shorter string , such that every position in lies within at least one instance of .[11] For example, the string "abacaba" has quasiperiod "aba", with occurrences at positions 1-3 ("aba"), 3-5 ("aca" wait, no—correct example: "aaabaaa" covered by "aaa" at positions 1-3, 3-5, 5-7, but adjust for non-power; a standard non-periodic quasiperiodic string is "abcabac" wait, to fix: use "mississippi" covered by "issipp" or better, omit specific if uncertain, but since fix: use "abababa" but periodic; actually, a correct simple example is "abcabca" no. Upon standard, "aabb aabb aab" but to simple: remove the example to avoid error, or use "ababa" with w="aba": start 1:aba (1-3), start 3:aba (3-5), covers 1-5.</PROBLEMATIC_TEXT> For example, in "ababa", "aba" serves as a quasiperiod, with occurrences starting at position 1 (aba, covers 1-3) and position 3 (aba, covers 3-5), covering the entire string via overlap.[11] A foundational result linking multiple periods is Fine and Wilf's theorem, which states that if a string of length at least has periods and , then it also has period .[12] This theorem bounds the complexity of periodic structures, implying that sufficiently long strings with commensurate periods must exhibit a finer common periodicity; it originates from arithmetic progressions but applies directly to word overlaps.[12] Key lemmas support string factorization using periodicity, ensuring decompositions align with inherent repetitions. Every nonempty string admits a factorization where the minimal local period of one factor equals the global minimal period of the entire string, often via leftmost or rightmost alignments.[13] Specifically, in any factorization , either the leftmost period of or the rightmost period of matches , facilitating efficient decomposition into periodic components without loss of structural information.[13]Critical Factorization
The critical factorization theorem, a foundational result in string periodicity first proved by Césari and Vincent in 1978, asserts that every non-empty string of length admits a critical factorization , where the local period at the cut position equals the global minimal period , and there exists a unique leftmost (or rightmost) such position.[14] This decomposition aligns the prefix with the minimal period of , which is the smallest positive integer such that for all valid positions (as defined in the context of string periodicity). The theorem ensures that such a factorization exists and is unique for the leftmost critical position, providing a structured way to capture the periodic structure of . Equivalently, the factorization can be expressed such that the periodic prefix up to the critical position is , where is a primitive word (unbordered and not a proper power of another word), , , and the length of the periodic prefix satisfies the multiple condition with respect to . If is empty or non-periodic (i.e., ), then .[14] The critical position is defined as the endpoint of the periodic prefix, marking the location where the minimal period is achieved locally; this enables the distinction between left-critical and right-critical factorizations, with the left version prioritizing the earliest such position. At this critical position, the local period (the minimal period of the prefix up to that point) equals the global minimal period , ensuring bidirectional symmetry in periodic analysis.[14] The computation of the critical factorization proceeds in linear time using methods based on Duval's algorithm or Z-algorithm variants, which iteratively build period candidates by scanning the string from left to right and verifying equality conditions against the current period length. The algorithm identifies the critical position by checking local periods until matching the global , avoiding backtracking and achieving efficiency through direct indexing.[14] This linear-time procedure forms the basis for efficient implementations in periodicity-related algorithms.Algorithm Mechanics
Preprocessing Phase
The preprocessing phase of the two-way string-matching algorithm begins with the input pattern of length over an ordered alphabet and computes key structures to exploit the pattern's periodicity for efficient subsequent matching. The primary step involves determining the left critical factorization of , which splits into a left seed (the shortest prefix such that the local period between and equals the global period of the entire pattern) and a right seed , such that the local period across the factorization point equals . The maximal special suffixes are computed in linear time by scanning the pattern and its reverse, identifying the longest suffixes whose minimal period equals , leveraging the alphabet order. This factorization is computed by finding the longest left-special suffix of (a suffix whose minimal period aligns with the start of ) and the longest right-special suffix of the reverse of , then setting the length of to , ensuring it is minimal and at most . The core computation relies on critical factorization, a property guaranteeing such a split exists for any non-empty string.[4] From this factorization, the algorithm derives skip tables for the forward scan (left-to-right comparison using the right seed) and backward scan (right-to-left comparison using the left seed). These tables are based on periodicity arrays: records the length of the left period ending at position in the left seed (the smallest period of the suffix of length ), and records the length of the right period starting at position in the right seed (the smallest period of the prefix of length ). The critical position marks the point where the left and right factors meet, enabling shifts by the global period or mismatch distances during scans.[4] Special cases are handled to maintain efficiency: for an empty pattern (), preprocessing returns immediately with no structures needed; for highly periodic patterns where is a power of a primitive word, the factorization selects such that periodicity properties hold across the split, avoiding degenerate shifts. The entire preprocessing runs in time and constant space by linearly scanning to compute the maximal special suffixes using the alphabet order.Matching Phase
The matching phase of the two-way string-matching algorithm processes a preprocessed pattern of length against a text of length , utilizing the critical factorization obtained during preprocessing to enable efficient bidirectional scanning. The pattern is split into a left-critical factor and a right-critical factor , where , and the lengths satisfy (the period of ) and , ensuring optimal skip distances based on the border properties of each factor. This phase begins at the left end of , attempting to align starting at position 1, and proceeds online by comparing characters while alternating between forward and backward verification to locate all occurrences of in .[2] The procedure initiates forward scanning of the right-critical factor against the corresponding suffix of starting from the current alignment position . If a mismatch occurs at the -th character of , the algorithm shifts the alignment forward by positions, as guaranteed by the critical factorization to avoid missing matches without redundant checks, due to the periodicity properties. Upon a successful partial match of , the algorithm switches to backward verification of the left-critical factor against the preceding characters in . During this backward pass, if a mismatch is found during the backward scan of (or if a full match is found), the algorithm shifts the pattern forward by the global period . A full match is reported at position only if both factors align completely without mismatches.[2] This two-way alternation guarantees that the entire text is processed in linear time, as the forward skips amortize over mismatches in and the backward verifications are bounded by the critical factorization, with each text character compared at most twice in the worst case. The phase terminates after scanning beyond position , outputting all starting indices where occurs in . By relying on the precomputed period tables for and , the matching avoids recomputing borders on-the-fly, distinguishing it from purely left-to-right algorithms like Knuth-Morris-Pratt.[2]Implementation Details
Pseudocode Overview
The two-way string-matching algorithm consists of a preprocessing phase that computes a critical factorization of the pattern to determine the period and split point, enabling efficient skips during matching, and a searching phase that alternates between forward scanning of the right factor and backward verification of the left factor. This structure ensures linear time overall with constant extra space, assuming an ordered alphabet for the maximal suffix computations.[2] The following pseudocode overview uses 0-based indexing for the pattern P of length m and text T of length n, returning a list of starting positions of matches in T. Edge cases include m = 0 (every position in T is a match) and n < m (empty match list); the code below assumes m > 0 and n ≥ m for the main logic. If the pattern is primitive (detected when the computed period equals m), a simplified non-periodic branch is used without overlap memory.Preprocessing Phase
The preprocessing computes candidate factorizations by finding maximal suffixes in direct and reverse lexicographic orders to identify the leftmost critical split, then selects the one minimizing the right factor length. The period is derived from the difference between the split position and its border length. This uses loops that extend matching periods incrementally until a mismatch, akin to border extension: for positions i from 1 to m, extend the current match length L if P == P[i - L[i-1]], resetting or jumping on mismatch to maintain the local period. If ms = -1 in computeMaxSuf or computeMaxSufTilde, set ms = 0 and per = m to handle cases where the whole pattern is the maximal suffix (common for primitive patterns).[2]function computeMaxSuf(P, m): (ms, per)
ms = -1 // length of maximal suffix? Adjusted to start index
j = 0
k = 1
per = 1
while j + k < m:
a = P[j + k]
b = P[ms + k] if ms >= 0 else P[k - 1]
if a < b: // direct order for left candidate
j = j + k
k = 1
per = j - ms if ms >= 0 else j + 1
elif a == b:
if k != per:
k = k + 1
else:
j = j + per
k = 1
else:
ms = j
j = ms + 1
k = 1
per = 1
if ms == -1:
ms = 0
per = m
return (ms, per)
function computeMaxSufTilde(P, m): (ms, per)
// Similar to computeMaxSuf, but reverse order: if a > b for right candidate
ms = -1
j = 0
k = 1
per = 1
while j + k < m:
a = P[j + k]
b = P[ms + k] if ms >= 0 else P[k - 1]
if a > b:
j = j + k
k = 1
per = j - ms if ms >= 0 else j + 1
elif a == b:
if k != per:
k = k + 1
else:
j = j + per
k = 1
else:
ms = j
j = ms + 1
k = 1
per = 1
if ms == -1:
ms = 0
per = m
return (ms, per)
function preprocess(P, m):
(i, p_left) = computeMaxSuf(P, m)
(j, p_right) = computeMaxSufTilde(P, m)
if i > j:
ell = i
per = p_left
left_part = P[0:ell] // length ell
else:
ell = j
per = p_right
left_part = P[0:ell]
// Verify periodicity of left factor (primitive detection if per == m)
is_periodic = (per != m)
if is_periodic:
for k in 0 to ell - per - 1:
if left_part[k] != left_part[k + per]:
is_periodic = false
break
if not is_periodic:
per = max(ell + 1, m - ell - 1) + 1 // fallback shift for primitive or non-periodic
return (ell, per, is_periodic)
function computeMaxSuf(P, m): (ms, per)
ms = -1 // length of maximal suffix? Adjusted to start index
j = 0
k = 1
per = 1
while j + k < m:
a = P[j + k]
b = P[ms + k] if ms >= 0 else P[k - 1]
if a < b: // direct order for left candidate
j = j + k
k = 1
per = j - ms if ms >= 0 else j + 1
elif a == b:
if k != per:
k = k + 1
else:
j = j + per
k = 1
else:
ms = j
j = ms + 1
k = 1
per = 1
if ms == -1:
ms = 0
per = m
return (ms, per)
function computeMaxSufTilde(P, m): (ms, per)
// Similar to computeMaxSuf, but reverse order: if a > b for right candidate
ms = -1
j = 0
k = 1
per = 1
while j + k < m:
a = P[j + k]
b = P[ms + k] if ms >= 0 else P[k - 1]
if a > b:
j = j + k
k = 1
per = j - ms if ms >= 0 else j + 1
elif a == b:
if k != per:
k = k + 1
else:
j = j + per
k = 1
else:
ms = j
j = ms + 1
k = 1
per = 1
if ms == -1:
ms = 0
per = m
return (ms, per)
function preprocess(P, m):
(i, p_left) = computeMaxSuf(P, m)
(j, p_right) = computeMaxSufTilde(P, m)
if i > j:
ell = i
per = p_left
left_part = P[0:ell] // length ell
else:
ell = j
per = p_right
left_part = P[0:ell]
// Verify periodicity of left factor (primitive detection if per == m)
is_periodic = (per != m)
if is_periodic:
for k in 0 to ell - per - 1:
if left_part[k] != left_part[k + per]:
is_periodic = false
break
if not is_periodic:
per = max(ell + 1, m - ell - 1) + 1 // fallback shift for primitive or non-periodic
return (ell, per, is_periodic)
Matching Phase
The matching scans the text forward, starting comparisons in the right factor (from ell onward). On mismatch, shift by the mismatch distance (right skip). On full right-factor match, verify the left factor backward from ell-1, using overlap memory s if periodic. On full match or left mismatch, shift by the period per. This backward check handles the left verification, with shifts ensuring no re-comparisons.[2]function twoway_match(P, m, T, n):
if m == 0:
return [0, 1, ..., n-1] // all positions
if n < m:
return []
(ell, per, is_periodic) = preprocess(P, m)
matches = []
j = 0 // text position
s = -1 // overlap memory (prefix matched length -1 none)
while j <= n - m:
i_start = max(ell, s + 1)
i = i_start
while i < m and P[i] == T[j + i]:
i = i + 1
if i < m: // mismatch in right factor
j = j + (i - ell)
s = -1
else: // full right match, verify left backward
k = ell - 1
while k > s and P[k] == T[j + k]:
k = k - 1
if k <= s: // full match
matches.append(j)
j = j + per
s = m - per - 1 // update memory for periodic overlap
else: // mismatch in left factor
j = j + per
s = -1
return matches
function twoway_match(P, m, T, n):
if m == 0:
return [0, 1, ..., n-1] // all positions
if n < m:
return []
(ell, per, is_periodic) = preprocess(P, m)
matches = []
j = 0 // text position
s = -1 // overlap memory (prefix matched length -1 none)
while j <= n - m:
i_start = max(ell, s + 1)
i = i_start
while i < m and P[i] == T[j + i]:
i = i + 1
if i < m: // mismatch in right factor
j = j + (i - ell)
s = -1
else: // full right match, verify left backward
k = ell - 1
while k > s and P[k] == T[j + k]:
k = k - 1
if k <= s: // full match
matches.append(j)
j = j + per
s = m - per - 1 // update memory for periodic overlap
else: // mismatch in left factor
j = j + per
s = -1
return matches
Illustrative Example
To illustrate the two-way string-matching algorithm, consider the pattern "ababab" of length and the text "abababababa" of length . In the preprocessing phase, the algorithm computes the critical factorization with left factor length ell = 2 ("ab") and period per = 2, as the pattern has global period 2 and the left factor is periodic with it (is_periodic = true). The right factor is "abab".[2] During the matching phase, the algorithm begins at text position . It first performs a forward match on the right part starting from i = 2, matching P[2:6] = "abab" against T[2:6] = "abab" (4 comparisons), then backward verifies the left part from k = 1 to 0, matching P[15] = "b" == T[15] = "b" and P = "a" == T = "a" (2 comparisons), confirming a full match at position 0 (total 6 comparisons). It then shifts by per = 2 to j = 2, sets s = 6 - 2 - 1 = 3. Now i_start = max(2, 3 + 1) = 4, matches P[4:6] = "ab" against T[6:8] = "ab" (2 comparisons), full right; backward from k = 1 > 3? No, but since s = 3 covers the left (assumes prefix 0-3 matched due to periodicity), k = 1 <= 3, full match at 2 (no additional comparisons). Shifts to j = 4, s = 3, i_start = 4, matches P[4:6] = "ab" against T[8:10] = "ab" (2 comparisons), full right; backward k = 1 <= 3, match at 4. Next j = 6 > 5, stop. The algorithm outputs occurrences at starting positions 0, 2, 4. This example demonstrates how the two-way approach combines forward and backward scanning with period-based skipping and overlap memory to achieve linear-time performance while handling periodicity efficiently, saving comparisons in overlapping cases compared to naive scanning.Performance and Comparisons
Time and Space Complexity
The two-way string-matching algorithm achieves an overall time complexity of , where is the length of the pattern and is the length of the text. The preprocessing phase, which computes the critical factorization of the pattern into left and right parts, runs in time by leveraging linear-time methods for identifying maximal suffixes and periods.[1] The matching phase then processes the text in time, performing constant-time decisions at each position through forward scans of the right part and backward scans of the left part, with amortized advances via period-based skips that prevent excessive re-examinations.[2] A sketch of the time complexity proof relies on the fact that each mismatch or match attempt advances the search position in the text by at least one character, ensuring no backtracking beyond the borders of the periodic parts. Skips over periodic redundancies, enabled by the critical factorization, cover multiple positions in constant time without revisiting scanned areas, bounding the total number of character comparisons to at most in the worst case.[1] Even for fully periodic patterns, where naive methods may degenerate to quadratic time, the algorithm maintains linearity by exploiting the local and global periods to jump efficiently.[2] In terms of space complexity, the algorithm requires constant additional space beyond the input strings, with auxiliary space supporting quick lookups during the matching phase via the critical factorization parameters; no additional space proportional to or is needed beyond the pattern and text themselves.[2] This auxiliary space supports the preprocessing without further overhead.[1]Relation to Other Algorithms
The two-way string-matching algorithm, introduced by Crochemore and Perrin, can be viewed as an intermediate approach between the Knuth-Morris-Pratt (KMP) algorithm and the Boyer-Moore algorithm, combining elements of both while introducing bidirectional processing based on pattern factorization. Unlike the KMP algorithm, which relies on a one-way failure function computed during preprocessing to skip characters unidirectionally from left to right, the two-way algorithm partitions the pattern into left and right halves using critical factorization and performs matching in both directions: left-to-right on the right half followed by right-to-left on the left half if needed.[4] This bidirectional strategy enables more flexible skips in either direction, making it particularly advantageous for scenarios involving reverse-order text processing, where KMP's unidirectional nature leads to less efficient handling. Both algorithms achieve O(n + m) time complexity in the worst case, where n is the text length and m is the pattern length, but the two-way method's use of periods allows for potentially fewer comparisons in practice due to its exploitation of internal pattern periodicity.[4] In contrast to the Boyer-Moore algorithm, which scans the pattern from right to left and relies on heuristics like bad-character and good-suffix rules for large skips (yielding sublinear average-case performance but O(nm) worst-case time), the two-way algorithm maintains guaranteed linear time O(n + m) while being fully online, processing the text incrementally without lookahead. The two-way approach delves deeper into the pattern's internal structure through its factorization and period-based skips, avoiding Boyer-Moore's dependency on alphabet size for preprocessing and its vulnerability to worst-case quadratic behavior on repetitive texts.[4] This makes the two-way algorithm more predictable and suitable for implementations where worst-case guarantees are critical, though it may not match Boyer-Moore's speed on large-alphabet texts with distinct characters. Compared to the Z-algorithm, which computes the Z-array for the concatenated pattern and text to identify prefix matches in linear time but requires O(n + m) space and preprocessing on the full input, the two-way algorithm features simpler pattern-only preprocessing in O(m) time and constant additional space, without constructing extensive suffix-like structures.[16] Similarly, unlike suffix tree-based methods that build a generalized suffix tree in O(n + m) time and space for advanced indexing and multiple-pattern support, the two-way algorithm avoids such heavy structures, prioritizing constant additional space and ease of implementation, which renders it ideal for streaming applications or environments with limited memory, such as small-alphabet domains.[4][16] Subsequent developments after 2000, including constant-space variants achieving sublinear average time and real-time improvements in 2013, have further refined the algorithm to reduce comparisons while maintaining linear worst-case time and constant space.[17] The two-way algorithm excels in real-time search applications over large texts, such as DNA sequencing, where its bidirectional skips enhance efficiency on repetitive genomic data and reduce cache misses through localized processing, outperforming unidirectional algorithms like KMP in bidirectional or reverse scanning needs.[4]References
- The exact online string matching problem: A review of the most recent results. This article addresses the online exact string matching problem which consists in ...