Knuth–Morris–Pratt algorithm

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The Knuth–Morris–Pratt string searching algorithm searches for occurrences of a "word" W within a main "text string" S by employing the observation that when a mismatch occurs, the word itself embodies sufficient information to determine where the next match could begin, thus bypassing re-examination of previously matched characters.

The algorithm was discovered by Donald Knuth and Vaughan Pratt and independently by J. H. Morris in 1977, but the three published it jointly.

In this article, we will be using zero-based arrays for our strings; thus the 'C' in W (shown below) will be denoted W[2]. In general, notation follows C syntax, when it is not too illogical.

Contents

[edit] The KMP algorithm

[edit] A worked example of the search algorithm

To illustrate the algorithm's details, we work through a (relatively artificial) run of the algorithm. At any given time, the algorithm is in a state determined by two integers, m and i, which denote respectively the position within S which is the beginning of a prospective match for W, and the index in W denoting the character currently under consideration. This is depicted, at the start of the run, like:

m: 01234567890123456789012
S: ABC ABCDAB ABCDABCDABDE
W: ABCDABD
i: 0123456

We proceed by comparing successive characters of W to "parallel" characters of S, moving from one to the next if they match. However, in the fourth step, we get S[3] is a space and W[3] = 'D', a mismatch. Rather than beginning to search again at S[1], we note that no 'A' occurs between positions 0 and 3 in S except at 0; hence, having checked all those characters previously, we know there is no chance of finding the beginning of a match if we check them again. Therefore we move on to the next character, setting m = 4 and i = 0.

m: 01234567890123456789012
S: ABC ABCDAB ABCDABCDABDE
W:     ABCDABD
i:     0123456

We quickly obtain a nearly complete match "ABCDAB" when, at W[6] (S[10]), we again have a discrepancy. However, just prior to the end of the current partial match, we passed an "AB" which could be the beginning of a new match, so we must take this into consideration. As we already know that these characters match the two characters prior to the current position, we need not check them again; we simply reset m = 8, i = 2 and continue matching the current character. Thus, not only do we omit previously matched characters of S, but also previously matched characters of W.

m: 01234567890123456789012
S: ABC ABCDAB ABCDABCDABDE
W:         ABCDABD
i:         0123456

This search fails immediately, however, as the pattern still does not contain a space, so as in the first trial, we return to the beginning of W and begin searching at the next character of S: m = 11, reset i = 0.

m: 01234567890123456789012
S: ABC ABCDAB ABCDABCDABDE
W:            ABCDABD
i:            0123456

Once again we immediately hit upon a match "ABCDAB" but the next character, 'C', does not match the final character 'D' of the word W. Reasoning as before, we set m = 15, to start at the two-character string "AB" leading up to the current position, set i = 2, and continue matching from the current position.

m: 01234567890123456789012
S: ABC ABCDAB ABCDABCDABDE
W:                ABCDABD
i:                0123456

This time we are able to complete the match, whose first character is S[15].

[edit] Description of and pseudocode for the search algorithm

The above example contains all the elements of the algorithm. For the moment, we assume the existence of a "partial match" table T, described below, which indicates where we need to look for the start of a new match in the event that the current one ends in a mismatch. The entries of T are constructed so that if we have a match starting at S[m] that fails when comparing S[m + i] to W[i], then the next possible match will start at index m + i - T[i] in S (that is, T[i] is the amount of "backtracking" we need to do after a mismatch). This has two implications: first, T[0] = -1, which indicates that if W[0] is a mismatch, we cannot backtrack and must simply check the next character; and second, although the next possible match will begin at index m + i - T[i], as in the example above, we need not actually check any of the T[i] characters after that, so that we continue searching from W[T[i]]. The following is a sample pseudocode implementation of the KMP search algorithm:

algorithm kmp_search:
    input:
        an array of characters, S (the text to be searched)
        an array of characters, W (the word sought)
    output:
        an integer (the zero-based position in S at which W is found)

    define variables:
        an integer, m ← 0 (the beginning of the current match in S)
        an integer, i ← 0 (the position of the current character in W)
        an array of integers, T (the table, computed elsewhere)

    while m + i is less than the length of S, do:
        if W[i] = S[m + i],
            let i ← i + 1
            if i equals the length of W,
                return m
        otherwise,
            let m ← m + i - T[i],
            if i is greater than 0,
                let i ← T[i]

    (if we reach here, we have searched all of S unsuccessfully)
    return the length of S

[edit] The efficiency of the search algorithm

Assuming the prior existence of the table T, the search portion of the Knuth-Morris-Pratt algorithm has complexity O(k), where k is the length of S and the O is big-O notation. As except for the fixed overhead incurred in entering and exiting the function, all the computations are performed in the while loop, we will calculate a bound on the number of iterations of this loop; in order to do this we first make a key observation about the nature of T. By definition it is constructed so that if a match which had begun at S[m] fails while comparing S[m + i] to W[i], then the next possible match must begin at S[m + (i - T[i])]. In particular the next possible match must occur at a higher index than m, so that T[i] < i.

Using this fact, we will show that the loop can execute at most 2k times. For in each iteration, it executes one of the two branches in the loop. The first branch invariably increases i and does not change m, so that the index m + i of the currently scrutinized character of S is increased. The second branch adds i - T[i] to m, and as we have seen, this is always a positive number. Thus the location m of the beginning of the current potential match is increased. Now, the loop ends if m + i = k; therefore each branch of the loop can be reached at most k times, since they respectively increase either m + i or m, and m ≤ m + i: if m = k, then certainly m + i ≥ k, so that since it increases by unit increments at most, we must have had m + i = k at some point in the past, and therefore either way we would be done.

Thus the loop executes at most 2k times, showing that the time complexity of the search algorithm is O(k).

[edit] The "partial match" table (also known as "failure function")

The goal of the table is to allow the algorithm not to match any character of S more than once. The key observation about the nature of a linear search that allows this to happen is that in having checked some segment of the main string against an initial segment of the pattern, we know exactly at which places a new potential match which could continue to the current position could begin prior to the current position. In other words, we "pre-search" the pattern itself and compile a list of all possible fallback positions that bypass a maximum of hopeless characters while not sacrificing any potential matches in doing so.

We want to be able to look up, for each position in W, the length of the longest possible initial segment of W leading up to (but not including) that position, other than the full segment starting at W[0] that just failed to match; this is how far we have to backtrack in finding the next match. Hence T[i] is exactly the length of the longest possible proper initial segment of W which is also a segment of the substring ending at W[i - 1]. We use the convention that the empty string has length 0. Since a mismatch at the very start of the pattern is a special case (there is no possibility of backtracking), we set T[0] = -1, as discussed above.

[edit] A worked example of the table-building algorithm

We consider the example of W = "ABCDABD" first. We will see that it follows much the same pattern as the main search, and is efficient for similar reasons. We set T[0] = -1. To find T[1], we must discover a proper suffix of "A" which is also a prefix of W. But there are no proper suffixes of "A", so we set T[1] = 0. Likewise, T[2] = 0.

Continuing to T[3], we note that there is a shortcut to checking all suffixes: let us say that we discovered a proper prefix ending at W[2] with length 2 (the maximum possible); then its first character is a proper prefix of a proper prefix of W, hence a proper prefix itself, and it ends at W[1], which we already determined cannot occur. Therefore we need not even concern ourselves with substrings having length 2, and as in the previous case the sole one with length 1 fails, so T[3] = 0.

We pass to the subsequent W[4], 'A'. The same logic shows that the longest substring we need consider has length 1, and although in this case 'A' does work, recall that we are looking for segments ending before the current character; hence T[4] = 0 as well.

Considering now the next character, W[5], which is 'B', we exercise the following logic: if we were to find a subpattern beginning before the previous character W[4], yet continuing to the current one W[5], then in particular it would itself have a proper initial segment ending at W[4] yet beginning before it, which contradicts the fact that we already found that 'A' itself is the earliest occurrence of a proper segment ending at W[4]. Therefore we need not look before W[4] to find a terminal string for W[5]. Therefore T[5] = 1.

Finally, we see that the next character in the ongoing segment starting at W[4] = 'A' would be 'B', and indeed this is also W[5]. Furthermore, the same argument as above shows that we need not look before W[4] to find a segment for W[6], so that this is it, and we take T[6] = 2.

Therefore we compile the following table:

i 0 1 2 3 4 5 6
W[i] A B C D A B D
T[i] -1 0 0 0 0 1 2

[edit] Description of and pseudocode for the table-building algorithm

The example above illustrates the general technique for assembling the table with a minimum of fuss. The principle is that of the overall search: most of the work was already done in getting to the current position, so very little needs to be done in leaving it. The only minor complication is that the logic which is correct late in the string erroneously gives non-proper substrings at the beginning. This necessitates some initialization code.

algorithm kmp_table:
    input:
        an array of characters, W (the word to be analyzed)
        an array of integers, T (the table to be filled)
    output:
        nothing (but during operation, it populates the table)

    define variables:
        an integer, pos ← 2 (the current position we are computing in T)
        an integer, cnd ← 0 (the zero-based index in W of the next character of the current candidate substring)

    (the first few values are fixed but different from what the algorithm might suggest)
    let T[0] ← -1, T[1] ← 0

    while pos is less than the length of W, do:
        (first case: the substring continues)
        if W[pos - 1] = W[cnd], let T[pos] ← cnd + 1, pos ← pos + 1, cnd ← cnd + 1

        (second case: it doesn't, but we can fall back)
        otherwise, if cnd > 0, let cnd ← T[cnd]

        (third case: we have run out of candidates.  Note cnd = 0)
        otherwise, let T[pos] ← 0, pos ← pos + 1

[edit] The efficiency of the table-building algorithm

The complexity of the table algorithm is O(n), where n is the length of W. As except for some initialization all the work is done in the while loop, it is sufficient to show that this loop executes in O(n) time, which will be done by simultaneously examining the quantities i and i - j. In the first branch, i - j is preserved, as both i and j are incremented simultaneously, but naturally, i is increased. In the second branch, j is replaced by T[j], which we saw above is always strictly less than j, thus increasing i - j. In the third branch, i is incremented and j is not, so both i and i - j increase. Since i ≥ i - j, this means that at each stage either i or a lower bound for i increases; therefore since the algorithm terminates once i = n, it must terminate after at most 2n iterations of the loop, since i - j begins at 1. Therefore the complexity of the table algorithm is O(n).

[edit] The efficiency of the KMP algorithm

Since the two portions of the algorithm have, respectively, complexities of O(k) and O(n), the complexity of the overall algorithm is O(n + k).

As is apparent from the worked example, the algorithm makes its greatest gains over a naive string-searching algorithm when it can skip the most characters at a time. That is, the less it has to backtrack, the faster it goes, which is reflected in the table T by the presence of zeroes. A word such as "ABCDEFG" works well with this algorithm because it has no repetitions of its beginning, so that its table is simply all zeroes with a -1 at the beginning. On the opposite end of the spectrum, W = "AAAAAAA" works terribly, because its table is

i 0 1 2 3 4 5 6
W[i] A A A A A A A
T[i] -1 0 1 2 3 4 5

This is the worst possible pattern for T, and it can be exploited by a word such as S = "AAAAAABAAAAAABAAAAAAA", in which the algorithm tries to match each 'A' against each 'B' before giving up; this results in the maximum possible number of iterations, verging on twice the number of characters of S as the number of repetitions of "AAAAAAB" increases. Although the table-building routine is rapid for this word (since no backtracking ever occurs), this routine is run only once for a given W, while the search routine is potentially run many times. If each time, this W is searched for in strings like S, the overall efficiency will be as poor as possible. By way of comparison, this particular combination is a best case for the Boyer-Moore string search algorithm.

Note that in practice, the KMP algorithm is not good at searching in natural language texts because it can only skip characters when the first part of the pattern actually matches a part of the text. In practice this only happens seldomly in natural language texts. For instance, consider how many times a prefix of the pattern "text" matches something in this paragraph.

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