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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).
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.
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A graph that has a topological ordering cannot have any cycles, because the edge into the earliest vertex of a cycle would have to be oriented the wrong way. Therefore, every graph with a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering.
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 ...
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.
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]
Cubesort is a parallel sorting algorithm that builds a self-balancing multi-dimensional array from the keys to be sorted. As the axes are of similar length the structure resembles a cube. After each key is inserted the cube can be rapidly converted to an array. [1]