Download Algorithms on Strings by Maxime Crochemore, Christophe Hancart, Thierry Lecroq PDF

By Maxime Crochemore, Christophe Hancart, Thierry Lecroq

ISBN-10: 0511289324

ISBN-13: 9780511289323

This article and reference on string tactics and development matching provides examples relating to the automated processing of traditional language, to the research of molecular sequences and to the administration of textual databases. Algorithms are defined in a C-like language, with correctness proofs and complexity research, to cause them to able to enforce. The e-book can be a big source for college students and researchers in theoretical laptop technology, computational linguistics, computational biology, and software program engineering.

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When pref [i] = , the factor u = x[i . i + − 1] is a prefix of x but it is not necessarily the border of x[0 . i + − 1] because this border can be longer than u. In the same way, when border[j ] = , the factor v = x[j − + 1 . j ] is a prefix of x but it is not necessarily the longest prefix of x occurring at position j − + 1. The proposition that follows shows how the table border is expressed using the table pref . One can deduce from the statement an algorithm for computing the table border knowing the table pref .

Each negative comparison leads to the next step of the loop. Then there are at most m − 1 of them. Thus less than 2m comparisons on the overall. The previous argument also shows that the total time of all the executions of the loop of lines 7–8 is (m). The other instructions of the loop 3–9 take a constant time for each value of i giving again a global time (m) for their execution and that of the function. The bound of 2m on the number of comparisons performed by the function Prefixes is relatively tight.

We can note, moreover, that if g < i we have then g = i − 1, and that on the contrary, by definition of f , we have f < i ≤ g. The following lemma provides the justification for the correctness of the function Prefixes. 25 If i < g, we have the relation   pref [i − f ] pref [i] = g − i  g−i+ if pref [i − f ] < g − i, if pref [i − f ] > g − i, otherwise, where = |lcp(x[g − i . m − 1], x[g . m − 1])|. Proof Let us set u = x[f . g − 1]. The string u is a prefix of x by the definition of f and g.

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