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An alternative algorithm for topological sorting is based on depth-first search.The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort or the node has no outgoing edges (i.e., a leaf node):
Randomized algorithm similar to depth-first search used in generating a maze. Algorithms that use depth-first search as a building block include: Finding connected components. Topological sorting. Finding 2-(edge or vertex)-connected components. Finding 3-(edge or vertex)-connected components. Finding the bridges of a graph.
Therefore, the order in which the strongly connected components are identified constitutes a reverse topological sort of the DAG formed by the strongly connected components. [7] Donald Knuth described Tarjan's SCC algorithm as one of his favorite implementations in the book The Stanford GraphBase. [8] He also wrote: [9]
It is also possible to check whether a given directed graph is a DAG in linear time, either by attempting to find a topological ordering and then testing for each edge whether the resulting ordering is valid [18] or alternatively, for some topological sorting algorithms, by verifying that the algorithm successfully orders all the vertices ...
Many topological sorting algorithms will detect cycles too, since those are obstacles for topological order to exist. Also, if a directed graph has been divided into strongly connected components , cycles only exist within the components and not between them, since cycles are strongly connected.
A depth-first search (DFS) is an algorithm for traversing a finite graph. DFS visits the child vertices before visiting the sibling vertices; that is, it traverses the depth of any particular path before exploring its breadth. A stack (often the program's call stack via recursion) is generally used when implementing the algorithm.
The pairwise sorting network is very similar to the Batcher odd-even mergesort, but differs in the structure of operations. While Batcher repeatedly divides, sorts and merges increasingly longer subsequences, the pairwise method does all the subdivision first, then does all the merging at the end in the reverse sequence.
I emphasize that (un)directional edges should be moved to a separate article and that both DFS article and the topological sorting article point to it, hence bridging the 'gap' in understanding how topological sort can be used – together with the principles of the four edge classifications – to find cycles in a depth-first search tree in ...