toporder

Toporder is a term sometimes used in computer science and mathematics as shorthand for a topological order, or for the operation that produces such an order from a directed acyclic graph (DAG). In this context, a topological order is a linear arrangement of the graph’s vertices in which every directed edge u → v satisfies u before v. This ordering reflects dependency relationships and is possible only when the graph is acyclic.

To compute a toporder, several algorithms are commonly employed. Kahn’s algorithm uses the in-degrees of vertices, repeatedly removing nodes with zero in-degree and updating the in-degrees of their neighbors. A depth-first search approach yields a valid topological order by recording a reverse postorder of finished vertices. Both methods run in O(V + E) time and require O(V) extra space, where V is the number of vertices and E the number of edges.

Applications of toporders include task scheduling, where some tasks must precede others, build systems that resolve compile-time dependencies, database and workflow planning, and compiler optimizations such as instruction scheduling. Because a DAG can have multiple valid topological orders, a toporder is not necessarily unique.

In literature, the term toporder may refer to either the result (the topological order itself) or the process (the topological sorting operation). It is closely related to concepts such as linear extensions of partial orders and is often used interchangeably with topological sort in informal contexts. See also: topological sort, DAG, partial order, linear extension.