236605 Advanced Topics Spring Semester, 2000 1.1 Shortest Remaining Processing Time (srpt) Rule

1 Algorithms for a Single Machine We show that simple greedy algorithms are optimal for some scheduling problems on a single machine with release dates and preemptions. SRPT Rule: At each moment, process the job with the shortest remaining processing time. Note that the implementation of SRPT is possible only when preemptions are allowed. Theorem 1.1 ((1]) SRPT rule is optimal for 1jr j ; pmtnj P j C j. Proof: Consider a schedule, S 1 , which is not according to SRPT. Assume that on time t, a job k with remaining processing time p 0 k is processed, while some other available job, j, with shorter remaining processing time, p 0 j , is waiting. Consider all the time intervals from t and ahead, in which j or k are processed. We reallocate these intervals to j and k in the following way: rst allocate intervals to j until it is completed, then allocate the remaining intervals to k. Since both jobs are completed in the schedule S 1 , the total length of these intervals is p 0 k + p 0 j , therefore, both jobs are completed also in the new schedule. We do not change the schedule of jobs other than j; k, thus, Fact 1.2 For any job i 2 J n fj; kg, the completion time of i is not aaected from the reallocation of the intervals. Claim 1.3 The value of C j + C k is reduced. Proof: Let c be the completion time of j; k in the schedule S 1. In the new schedule, the completion time of k is C k = c. Since p 0 j < p 0 k and in the schedule S 1 the job k is processed at time t, the completion time of the rst job out of j; k in S 1 is larger than the completion time of j in the new schedule (which is now the completion time of intervals of total length p 0 j). The reallocation of the intervals follows the SRPT rule. We conclude that any schedule that does not follow the SRPT rule can be converted into a one that follows SRPT in which P j C j is smaller. Note that we need to check which job achieves the SRPT only when the processing of the current job is completed or when a …