Knuth–Morris–Pratt algorithm

In computer science, the Knuth–Morris–Pratt string-searching algorithm (or KMP 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.I learned in 2012 that Yuri Matiyasevich had anticipated the linear-time pattern matching and pattern preprocessing algorithms of this paper, in the special case of a binary alphabet, already in 1969. He presented them as constructions for a Turing machine with a two-dimensional working memory. In computer science, the Knuth–Morris–Pratt string-searching algorithm (or KMP 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 conceived in 1970 by Donald Knuth and Vaughan Pratt, and independently by James H. Morris. This was the first linear-time algorithm for string matching. The three published it jointly in 1977. Independently, in 1969, Matiyasevich discovered a similar algorithm, coded by a two-dimensional Turing machine, while studying a string-pattern-matching recognition problem. A string-matching algorithm wants to find the starting index m in string S. The most straightforward algorithm is to look for a character match at successive values of the index m, the position in the string being searched, i.e. S. If the index m reaches the end of the string then there is no match, in which case the search is said to 'fail'. At each position m the algorithm first checks for equality of the first character in the word being searched, i.e. S =? W. If a match is found, the algorithm tests the other characters in the word being searched by checking successive values of the word position index, i. The algorithm retrieves the character W in the word being searched and checks for equality of the expression S =? W. If all successive characters match in W at position m, then a match is found at that position in the search string. Usually, the trial check will quickly reject the trial match. If the strings are uniformly distributed random letters, then the chance that characters match is 1 in 26. In most cases, the trial check will reject the match at the initial letter. The chance that the first two letters will match is 1 in 262 (1 in 676). So if the characters are random, then the expected complexity of searching string S is 1 million characters and W[] is 1000 characters, then the string search should complete after about 1.04 million character comparisons. That expected performance is not guaranteed. If the strings are not random, then checking a trial m may take many character comparisons. The worst case is if the two strings match in all but the last letter. Imagine that the string S is 999 A characters terminating in a final B character. The simple string-matching algorithm will now examine 1000 characters at each trial position before rejecting the match and advancing the trial position. The simple string search example would now take about 1000 character comparisons times 1 million positions for 1 billion character comparisons. If the length of W[] is n, then the worst-case performance is O(k⋅n). The KMP algorithm has a better worst-case performance than the straightforward algorithm. KMP spends a little time precomputing a table (on the order of the size of W[], O(n)), and then it uses that table to do an efficient search of the string in O(k). The difference is that KMP makes use of previous match information that the straightforward algorithm does not. In the example above, when KMP sees a trial match fail on the 1000th character (i = 999) because S ≠ W, it will increment m by 1, but it will know that the first 998 characters at the new position already match. KMP matched 999 A characters before discovering a mismatch at the 1000th character (position 999). Advancing the trial match position m by one throws away the first A, so KMP knows there are 998 A characters that match W[] and does not retest them; that is, KMP sets i to 998. KMP maintains its knowledge in the precomputed table and two state variables. When KMP discovers a mismatch, the table determines how much KMP will increase (variable m) and where it will resume testing (variable i). To illustrate the algorithm's details, consider a (relatively artificial) run of the algorithm, where W = 'ABCDABD' and S = 'ABC ABCDAB ABCDABCDABDE'. At any given time, the algorithm is in a state determined by two integers: