On competitive on-line algorithms for the dynamic priority-ordering problem

Abstract

The vertices of a directed acyclic graph (DAG) are correctly prioritized if every vertex v in the graph is assigned a priority, denoted by priority( v), such that if there is an edge in the DAG from vertex v to vertex w then priority( v)< priority( w). The dynamic priority-ordering problem is to maintain a correct prioritization of the graph as the DAG is modified. We show that the Alpern et al. algorithm for this problem does not have a constant competitive ratio, where the cost of the algorithm is measured in terms of the number of primitive priority-manipulation operations. The proof shows that there exists no algorithm for the problem that has a constant competitive ratio, as long as the allowed primitive priority-manipulation operations satisfy a simple property. The proof also shows that there exists no algorithm for the problem of maintaining a topological-sort ordering that has a constant competitive ratio.