International audienceWe describe an algorithm, meant to be very general, to compute a presentation of the group of units of an order in a (semi-)simple algebra over QQ. Our method is based on a generalisation of Voronoï's algorithm for computing perfect forms, combined with Bass–Serre theory. It differs essentially from previously known methods to deal with such questions, e.g. for units in quaternion algebras. We illustrate this new algorithm by a series of examples where the computations are carried out completely
In this paper, we compare the makespan of preemptive and i-preemptive schedules where only a limited number i of preemptions is allowed. The problem is to schedule n independent jobs on m identical processors that operate in parallel. The objective is to minimize the makespan, i.e., the completion time of the last job that finishes. We show that the ratio of the optimal ipreemptive schedule length C ip * max versus the optimal preemptive schedule length C p * max is bounded from above by C ip * max ≤ (2 − 2/(m/(i + 1) + 1))C p * max. Furthermore, we show that the ratio of the length C LP T max of a nonpreemptive schedule following the longest processing time (LPT) rule versus the optimal preemptive schedule length C p * max is bounded from above by exactly the same bound when i = 0.
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