# Efficient algorithms for assigning chain-like tasks on a chain-like network computer

## Abstract

The optimal allocation of a chain-like task system on a chain like network computers has been studied in a number of papers [1][2][3][4]. The best known algorithm has a time complexity of *O*((*m′*-*n*)^{2}n) 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[4]. 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 [4] 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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## References

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*IEEE Transactions on Computers*, vol 37:pp. 48–57, January 1988.Google Scholar - 2.J.P. Sheu and Z.F. Chaing. “Efficient allocation of chain-like task on chain-like network computers”.
*Information Processing Letters*, vol 36:pp. 241–245, 1990.MathSciNetGoogle Scholar - 3.C.C. Hsu. “A Two-phase Approach for the Optimal Assignment of a Chain-like Task on a Chain-like Network Computer”. Technical report, National Taiwan Institute of Technology, 1993.Google Scholar
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