Routing equal-size messages on a slotted ring

We deal with the problem of routing messages on a slotted ring network in this paper. We study the computational complexity and algorithms for this routing by means of the results known in the literature for the multi-slot just-in-time scheduling problem. We consider two criteria for the routing problem: makespan, or minimum routing time, and diagonal makespan. A diagonal is simply a schedule of ring links i = 0, . . . , q − 1 in q consecutive time slots respectively. The number of diagonals between the earliest and the latest diagonals with non-empty packets is referred to as the diagonal makespan. For the former, we show that the optimal routing of messages of size k, is NP-hard in the strong sense, while an optimal routing when k = q can be computed in O(n log n) time. We also give an O(n logn)-time constant factor approximation algorithm for unit size messages. For the latter, we prove that the optimal routing of messages of size k, where k divides the size of the ring q, is NP-hard in the strong sense even for any fixed k ≥ 1, while an optimal routing when k = q can be computed in O(n logn) time. We also give an O(n logn)-time approximation algorithm with an absolute error 2q − k.