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A subsequence which consists of a consecutive run of elements from the original sequence, such as ,, , from ,,,,, , is a substring. The substring is a refinement of the subsequence. The substring is a refinement of the subsequence.
A longest common subsequence (LCS) is the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest common substring : unlike substrings, subsequences are not required to occupy consecutive positions within the original sequences.
A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the ...
Given two strings, of length and of length , find a longest string which is substring of both and . A generalization is the k-common substring problem.Given the set of strings = {, …,}, where | | = and =.
The longest increasing subsequence problem is closely related to the longest common subsequence problem, which has a quadratic time dynamic programming solution: the longest increasing subsequence of a sequence is the longest common subsequence of and , where is the result of sorting.
In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S {\displaystyle S} is a substring of S {\displaystyle S} that occurs at the beginning of S {\displaystyle S} ; likewise, a suffix of a string S {\displaystyle S} is a ...
In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there ...