Publication 2000 Issue No. 10 - October Abstract - Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed Systems
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Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed Systems
October 2000 (vol. 11 no. 10)
pp. 1090-1102
 ASCII Text x Piera Barcaccia, Maurizio A. Bonuccelli, Miriam Di Ianni, "Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 11, no. 10, pp. 1090-1102, October, 2000.
 BibTex x @article{ 10.1109/71.888647,author = {Piera Barcaccia and Maurizio A. Bonuccelli and Miriam Di Ianni},title = {Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed Systems},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {11},number = {10},issn = {1045-9219},year = {2000},pages = {1090-1102},doi = {http://doi.ieeecomputersociety.org/10.1109/71.888647},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed SystemsIS - 10SN - 1045-9219SP1090EP1102EPD - 1090-1102A1 - Piera Barcaccia, A1 - Maurizio A. Bonuccelli, A1 - Miriam Di Ianni, PY - 2000KW - Single hop distributed systemsKW - schedulingKW - NP-completenessKW - minimum schedule lengthKW - approximation algorithms.VL - 11JA - IEEE Transactions on Parallel and Distributed SystemsER -

Abstract—Switching networks are the core of many communication and multiprocessor systems. In these systems, a set of entities (communication equipment or processors) communicate through the switching network by exchanging messages. Simultaneous transmission or reception of two or more different messages through an input or output port results in the corruption of the messages (also called collision), which are useless and must be retransmitted later. This causes a performance degradation. Collisions can be avoided only by a proper scheduling of the messages. The same problem also arises in single-hop purely optical WDM systems, where simultaneous reception or transmission over the same wavelength channel results in a collision. In this paper, we study the problem of minimum length scheduling of a set of messages subject to precedence constraints. We show that the decision version of the problem is NP-complete even in very restricted cases. This means that the optimization problem cannot be solved in polynomial time, unless P=NP. Since the problem cannot be optimally solved by fast algorithms, we then investigate the existence of polynomial time approximation algorithms, by first proving that approximation algorithms cannot exist with performance ratio bounded by $4/3$ or smaller and successively presenting an $\epsilon$-approximation algorithm with $\epsilon <2$ for the case of two precedence classes of messages. Finally, we assess the existence of an asymptotically optimal schedule in the general case of an unrestricted number of precedence classes.

[1] A. Aggarwal, A. Bar-Noy, D. Coppersmith, R. Ramaswami, B. Schieber, and M. Sudan, “Efficient Routing and Scheduling Algorithms for Optical Networks,” Proc. Fifth ACM/SIAM Symp. Discrete Algorithms, pp. 412–423, Jan. 1994.
[2] D. Angluin and L.G. Valiant, “Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings,” J. Computer and System Sciences, vol. 18, no. 2, pp. 155–193, Apr. 1979.
[3] K. Ahuja, T.L. Magnati, and J.B. Orlin, Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
[4] P. Barcaccia and M.A. Bonuccelli, “Polynomial Time Optimal Algorithms for Time Slot Assignment of Variable Bandwidth Systems,” ACM/IEEE Trans. Networking, vol. 2, pp. 247–251, 1994.
[5] A.A. Bertossi, G. Bongiovanni, and M.A. Bonuccelli, “Time Slot Assignment in SS/TDMA Systems with Intersatellite Links,” IEEE Trans. Comm., vol. 35, pp. 602-608, 1987.
[6] M.S. Borella and B. Mukherjee, “Efficient Scheduling of Nonuniform Packet Traffic in a WDM/TDM Local Ligthwave Network with Arbitrary Transceiver Tuning Latencies,” IEEE J. Selected Areas on Comm., vol. 14, pp. 923–934, June 1996.
[7] D.P. Bovet and P. Crescenzi, Introduction to the Theory of Complexity. New York: Prentice Hall, 1993.
[8] G. Bongiovanni, D. Coppersmith, and C.K. Wong, “An Optimal Time Slot Assignment Algorithm for a SS/TDMA System with Variable Number of Transponders,” IEEE Trans. Comm., vol. 29, pp. 721–726, 1981.
[9] M.A. Bonuccelli, I.S. Gopal, and C.K. Wong, “Incremental Time Slot Assignment in SS/TDMA Satellite Systems,” IEEE Trans. Comm., vol. 39, pp. 1,147–1,156, 1991.
[10] R.E. Burkard, “Time-Slot Assignment for TDMA-Systems,” Computing, vol. 35, pp. 99–112, 1985.
[11] W.T. Chen, P.R. Sheu, and J.H. Yu, “Time Slot Assignment in TDM Multicast Switching Systems,” IEEE Trans. Comm., vol. 42, pp. 149–165, 1994.
[12] K.Y. Eng and A.S. Acampora, “Fundamental Conditions Governing TDM Switching Assignments in Terrestrial and Satellite Networks,” IEEE Trans. Comm., vol. 35, pp. 755–761, 1987.
[13] S. Even, A. Itai, and A. Shamir, “On the Complexity of Time Table and Multicommodity Flow Problems,” SIAM J. Computing, vol. 5, pp. 691–703, 1976.
[14] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[15] H. Gabow, T. Nishizeki, O. Kariv, D. Leven, and O. Terada, “Algorithms for Edge Coloring Graphs,” Technical Report 41/85, Univ. of Colorado, 1985.
[16] K. Hwang, Advanced Computer Architecture: Parallelism, Scalability, Programmability. McGraw-Hill, 1993.
[17] N. Huang, C. Wu, and G. Ma, “A Time-Wavelength Scheduling Algorithm for Interconnected WDM Star Networks,” Proc. IEEE INFOCOM '94, 1994.
[18] R. Jain, K. Somalwar, J. Werth, and J.C. Browne, “Heuristics for Scheduling I/O Operations,” IEEE Trans. Parallel and Distributed Systems, vol. 8, no. 3, pp. 310–320, Mar. 1997.
[19] F.T. Leighton, B.M. Maggs, and S.B. Rao, “Packet Routing and Job-Shop Scheduling in$O$(Congestion + Dilation) Steps,” Combinatorica, vol. 14, no. 2, pp. 167–186, 1994.
[20] F.T. Leighton, B.M. Maggs, and A. Richa, “Fast Algorithms for Finding$O$(Congestion + Dilation) Packet Routing Schedules,” Technical Report CMU-CS-96-152, School of Computer Science, Carnegie-Mellon Univ., 1996. To appear inCombinatorica.
[21] G.R. Pieris and G.H. Sasaki, “Scheduling Transmissions in WDM Broadcast-and-Select Networks,” IEEE/ACM Trans. Networking, vol. 2, pp. 105–110, Apr. 1994.
[22] J. Spencer, Ten Lectures on the Probabilistic Method. Philadelphia, Penn.: SIAM, 1987.
[23] M. Schwartz, Telecommunication Networks: Protocols, Modelling, and Analysis. Reading, Mass.: Addison Wesley, 1987.
[24] H.Y. Tyan, C.J. Hou, B. Wang, and C.C. Han, “On Supporting Time-Constrained Communications in WDMA-Based Star-Coupled Optical Networks,” Proc. IEEE Real Time Systems Symp., pp. 175–184, 1996.

Index Terms:
Single hop distributed systems, scheduling, NP-completeness, minimum schedule length, approximation algorithms.
Citation:
Piera Barcaccia, Maurizio A. Bonuccelli, Miriam Di Ianni, "Complexity of Minimum Length Scheduling for Precedence Constrained Messages in Distributed Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 11, no. 10, pp. 1090-1102, Oct. 2000, doi:10.1109/71.888647