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The traditional ld (Unix linker) requires that its library inputs be sorted in topological order, since it processes files in a single pass. This applies both to static libraries ( *.a ) and dynamic libraries ( *.so ), and in the case of static libraries preferably for the individual object files contained within.
The canonical application of topological sorting is in scheduling a sequence of jobs or tasks based on their dependencies.The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started (for example, when washing clothes, the washing machine must finish before we put the clothes in the dryer).
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 ...
The green and blue boxes combine to form the entire sorting network. For any arbitrary sequence of inputs, it will sort them correctly, with the largest at the bottom. The output of each green or blue box will be a sorted sequence, so the output of each pair of adjacent lists will be bitonic, because the top one is blue and the bottom one is green.
Topological space; Topological property; Open set, closed set. Clopen set; Closure (topology) Boundary (topology) Dense (topology) G-delta set, F-sigma set; closeness (mathematics) neighbourhood (mathematics) Continuity (topology) Homeomorphism; Local homeomorphism; Open and closed maps; Germ (mathematics) Base (topology), subbase; Open cover ...
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]
The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort , merge sort ), multiplying large numbers (e.g., the Karatsuba algorithm ), finding the closest pair of points , syntactic ...
In the field of computer science, a pre-topological order or pre-topological ordering of a directed graph is a linear ordering of its vertices such that if there is a directed path from vertex u to vertex v and v comes before u in the ordering, then there is also a directed path from vertex v to vertex u. [1] [2]