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In an empty graph, each vertex forms a component with one vertex and zero edges. [3] More generally, a component of this type is formed for every isolated vertex in any graph. [4] In a connected graph, there is exactly one component: the whole graph. [4] In a forest, every component is a tree. [5] In a cluster graph, every component is a ...
In formal language theory and pattern matching (including regular expressions), the concatenation operation on strings is generalised to an operation on sets of strings as follows: For two sets of strings S 1 and S 2, the concatenation S 1 S 2 consists of all strings of the form vw where v is a string from S 1 and w is a string from S 2, or ...
Every m-edge string graph can be partitioned into two subsets, each a constant fraction the size of the whole graph, by the removal of O(m 3/4 log 1/2 m) vertices. It follows that the biclique-free string graphs, string graphs containing no K t,t subgraph for some constant t, have O(n) edges and more strongly have polynomial expansion. [5]
Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal.
In information theory, linguistics, and computer science, the Levenshtein distance is a string metric for measuring the difference between two sequences. The Levenshtein distance between two words is the minimum number of single-character edits (insertions, deletions or substitutions) required to change one word into the other.
For LCS(R 3, C 1), C and A do not match, so LCS(R 3, C 1) gets the longest of the two sequences, (A). For LCS(R 3, C 2), C and G do not match. Both LCS(R 3, C 1) and LCS(R 2, C 2) have one element. The result is that LCS(R 3, C 2) contains the two subsequences, (A) and (G). For LCS(R 3, C 3), C and C match, so C is appended to LCS(R 2, C 2 ...
A string-searching algorithm, sometimes called string-matching algorithm, is an algorithm that searches a body of text for portions that match by pattern. A basic example of string searching is when the pattern and the searched text are arrays of elements of an alphabet ( finite set ) Σ.
The function used to determine the distance between two clusters, known as the linkage function, is what differentiates the agglomerative clustering methods. In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other.