Faster Algorithms for Online Topological Ordering

We present two algorithms for maintaining the topological order of a directed acyclic graph with n vertices, under an online edge insertion sequence of m edges. Efficient algorithms for online topological ordering have many applications, including online cycle detection, which is to discover the first edge that introduces a cycle under an arbitrary sequence of edge insertions in a directed graph. The current fastest algorithms for the online topological ordering problem run in time O(min(m3/2 logn,m3/2 + n2 logn)) and O(n2.75) (the latter algorithm is faster for dense graphs, i.e., when m > n11/6). In this paper we present faster algorithms for this problem. We first present a simple algorithm with running time O(n5/2) for the online topological ordering problem. This is the current fastest algorithm for this problem on dense graphs, i.e., when m > n5/3. We then present an algorithm with running time O((m+n logn) √ m), which is an improvement over the O(min(m3/2 logn,m3/2 +n2 logn)) algorithm it is a strict improvement when m is sandwiched between ω(n) and O(n4/3). Our results yield an improved upper bound of O(min(n5/2,(m+n logn) √ m)) for the online topological ordering problem.