Efficient algorithms for assigning chain-like tasks on a chain-like network computer
The optimal allocation of a chain-like task system on a chain like network computers has been studied in a number of papers . The best known algorithm has a time complexity of O((m′-n)2n) if m′−n=O(√log n); otherwise O(m′ log m′ + m′(m′-n)) where m′ and n denote the number of un-mergeable modules and the number of processors respectively. The space complexity of Hsu's algorithm and Young and Chan's algorithm are ((m′−n)n) and O(m′(m′−n)) respectively. In this paper, we generalize the decision algorithm discussed in  and propose two algorithms for the optimization problem based on the generalized decision test. The first algorithm gives an optimal schedule in O(m′2) time. For the second one, the chain-like task system is divided into two categories, dominated and non-dominated task system. A dominated task system can be solved by an asymptotically optimal algorithm in O(m′) time. For a non-dominated task system, the worst case time complexity is O(m′n log(m′−n)). This is asymptotically the best algorithm if m′−n=ω(n). This algorithm has an efficient average case time complexity of O(m′ log m′) which is better than all existing algorithms except m′−n=o(√log n). The space complexity of our algorithm is O(m′) which is better than that of Young and Chan's or Hsu's algorithm. Furthermore, the optimal schedules obtained by using all these algorithms share a common property that the minimal number of processors is always used so that they are the optimally load balanced optimal schedules.
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