Maintaining a Topological Order Under Edge Insertions

Abstract A topological order of the vertices of a directed acyclic graph G = ( V , E ) is any total order ord such that if ( x , y ) ϵ E , then x precedes y in ord . In this paper we consider the dynamic version of this problem, and provide simple algorithms and data structures achieving O( n ) amortized time per edge insertion starting from an empty graph, which favorably compares to the trivial O( m + n ) time bound per operation obtained applying the off-line algorithm. The additional space requirement, beside the representation of the graph itself, is O( n ). Experimental results show that our algorithm performs in practice orders of magnitude faster than the off-line algorithm.