The maximum flow problem: a real-time approach

The dynamic version of the maximum flow problem allows the graph underlying the flow network to change over time. The graph receives corrections to its structure or capacities and consequently the value of the maximum flow is modified. Several correction types are treated: edge capacity corrections and constant degree vertex additions/deletions. These corrections arrive in real time. In this paper, parallel and sequential solutions to the real-time maximum flow problem are developed on the reconfigurable multiple bus machine model and on the random access machine model, respectively. The parallel solution successfully meets the deadlines imposed in real time, while the sequential one fails to do so.The two solutions are then applied to a real-time process scheduler, an extension of Stone's static two-processor allocation problem. The scheduler allows processes to be created and destroyed, the amount of communication between two processes to change with time, and so on. The parallel algorithm is always able to compute the optimal schedule, while the solution obtained sequentially is only an approximation. The sequential solution gets worse with each new deadline to be met. In fact, after a sufficient number of steps, the quality improvement provided by the parallel approach over the sequential one is superlinear in the number of processors used by the parallel model.