高效两向字符串匹配算法：介于KMP与Boyer-Moore间的简化解决方案
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更新于2024-07-14
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"Two-Way String Matching算法是1991年由Maxime Crochemore和Dominique Perrin提出的一种创新的字符串匹配方法，它位于经典的Knuth-Morris-Pratt (KMP)算法和Boyer-Moore算法之间，具有独特的性能特点。这个算法的时间复杂度是线性的，与Galil和Seiferas的算法一样，使用了常数空间，这意味着它在处理大量数据时效率极高。
该算法的关键在于其设计思路巧妙地结合了全局性和局部信息。它利用了组合学中的一个已有结果——Critical Factorization Theorem（关键分解定理），该定理将单词的全局周期与其局部重复块关联起来。通过这种方法，算法能够高效地搜索目标模式在输入字符串中的位置，即使模式包含回文或者有重复子串也能快速定位。
在技术领域，Two-Way String Matching算法适用于编程技术中的模式识别和文本处理场景。它在非数值算法分析和问题复杂性方面具有价值，特别是在图论和模式识别的方法论中，因为它提供了一种既简单又高效的解决方案。算法的直观性和易分析性使得它在教学和实际开发中非常受欢迎，对于理解和优化字符串搜索问题具有重要意义。
Two-Way String Matching算法作为一种高效且结构清晰的工具，不仅提升了字符串匹配任务的性能，还为理解字符串处理的底层原理提供了新的视角。对于IT专业人士和研究人员来说，掌握这种算法不仅有助于提高代码执行效率，还能推动算法理论的进一步发展。"
Two- Way String-Matching
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called a period of x if
x[i] =x[i+ p],
whenever both sides are defined. In other terms, p is a period of x if two
letters of xat distance p always coincide. The smallest period of x is called
the period of x and is denoted by p(x). One has the inequalities
O<p(x)sl xl.
Let x be a word and 1 be an integer such that O s 1< I x 1. An integer
r > 1 is called a local period of x at position 1 if one has
x[i] =x[i+r]
for all indices i such that 1 – r + 1< i <1 and such that both sides of the
equality are defined (see Figure 3). The local period of x at position 1 is the
smallest local period at position 1. It is denoted r( x, 1). It is an easy conse-
quence of the definitions that
1 sr(x, l) <p(x).
It is convenient to reformulate the definition of a local period as follows: An
integer r is a local period of x at position 1 if and only if there exists a word w
of length r such that one of the four conditions is satisfied:
(i) x = x’ wwx” with \x’wl =1,
(ii) x = x’ wu
with I X’W I = 1 and u prefix of w,
(iii) x = vwx” with I v I = 1 and v suffix of w,
(iv) x = vu
with I v I = 1, v suffix of w and u prefix of w.
The case (i) is represented on Figure 3. The other cases correspond to a
position close to the ends of x or equivalently to an overflow of w. The
relation between both definitions is given by
w[i] =x[l+i] or
w[i] =x[l–r+i]
(for 1 s i < r), according to which expression is defined on the right-hand
side.
Our algorithm relies on the following result due to Cesari, Vincent, and
Duval (see [22] for precise references).
THEOREM (CRITICAL FACTORIZATION THEOREM). For each word x, there
exists at least one position 1 with O s 1< P(x) such that
r(x, l) =p(x).
A position 1 such that r( x, 1) = p(x) is called a critical position of x. For
instance, the word
x = abaabaa
has period 3. It has three critical positions, namely 2, 4, 5. We shall prove the
Critical Factorization Theorem in Section 3 and give an algorithm allowing to
compute a critical position in Section 4. Before coming to this description, we
present our application of this theorem to string-matching. We begin with an
informal description.
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