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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.
The order extension principle is constructively provable for finite sets using topological sorting algorithms, where the partial order is represented by a directed acyclic graph with the set's elements as its vertices. Several algorithms can find an extension in linear time. [6]
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]
Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...
The importance of the fixed-point index is largely due to its role in the Lefschetz–Hopf theorem, which states: (,) =,where Fix(f) is the set of fixed points of f, and Λ f is the Lefschetz number of f.
A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems. Rubinstein and Thompson's 3-sphere recognition algorithm.
The archetypical example of a filter is the neighborhood filter at a point in a topological space (,), which is the family of sets consisting of all neighborhoods of . By definition, a neighborhood of some given point is any subset whose topological interior contains this point; that is, such that . Importantly, neighborhoods are not required to be open sets; those are called open ...