Solving Di erence Constraints

A system of diierence constraints consists of a set of inequalities of the form x i ? x j b i;j. Such systems occur in many applications, e.g., temporal reasoning. The best known algorithm to determine if a system of diierence constraints is feasible (i.e., if it has a solution) and to compute a solution if the system is feasible runs in (mn) time, where n is the number of variables and m is the number of constraints. In this paper, we explore the problem of maintaining a solution to a system of diierence constraints as the system undergoes changes such as the addition or removal of constraints. The problem arises in the context of an interactive system that allows users to model the temporal behavior of a multimedia application as a system of diierence constraints. We present an incremental algorithm for the problem (which enables immediate feedback to the user as and when a constraint added or modiied by the user creates an infeasible system.) Our algorithm processes the addition of a constraint in time O(m + n log n) and the removal of a constraint in constant time, as long as the original system is feasible. (The time taken to process the addition of a constraint can be more precisely described as O(kk + jj log jj), where jj denotes the number of variables whose values are changed to compute the new feasible solution, and kk denotes the number of constraints involving the variables whose values are changed.) If the original system is infeasible, the algorithm processes any change in O(m + n log n) amortized time. The algorithm we present can also be used to check for the existence of negative cycles in a dynamic (i.e., changing) graph.