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A General Theory for Deadlock Avoidance in Wormhole-Routed Networks
July 1998 (vol. 9 no. 7)
pp. 626-638

Abstract—Most machines of the last generation of distributed memory parallel computers possess specific routers which are used to exchange messages between nonneighboring nodes in the network. Among the several technologies, wormhole routing is usually preferred because it allows low channel-setup time and reduces the dependency between latency and internode distance. However, wormhole routing is very susceptible to deadlock because messages are allowed to hold many resources while requesting others. Therefore, designing deadlock-free routing algorithms using few hardware facilities is a major problem for wormhole-routed networks. In this paper, we describe a general theoretical framework for the study of deadlock-free routing functions. We give a general definition of what can be a routing function. This definition captures many specific definitions of the literature (e.g., vertex-dependent, input-dependent, source-dependent, path-dependent, etc.). Using our definition, we give a necessary and sufficient condition which characterizes deadlock-free routing functions. Our theory embraces, at a high level, most of the theories related to deadlock avoidance in wormhole-routed networks previously derived in the literature. In particular, it applies not only to one-to-one routing, but also to one-to-many routing. The latter paradigm is used to solve the multicast problem with the path-based or tree-based facility.

[1] V.S. Adve and M.K. Vernon, “Performance Analysis of Mesh Interconnection Networks with Deterministic Routing,” IEEE Trans. Parallel and Distributed Systems, vol. 5, no. 3, pp. 225-246, Mar. 1994.
[2] K.V. Anjan and T.M. Pinkston, “An Efficient, Fully Adaptive Deadlock Recovery Scheme: Disha,” Proc. 22nd Ann. Int'l Symp. Computer Architecture, pp. 201-210, June 1995.
[3] K.V. Anjan, T.M. Pinkston, and J. Duato, Generalized Theory for Deadlock-Free Adaptive Routing and Its Application to Disha Concurrent Proc. 10th Int'l Parallel Processing Symp., Apr. 1996.
[4] K. Aoyama and A.A. Chien, "The Cost of Adaptativity and Virtual Lanes," J. VLSI Design, 1993.
[5] A. Arruabarrena, R. Beivide, C. Izu, and J. Miguel, "A Performance Evaluation of Adaptive Routing in Bidimensional Cut-Through Networks," Parallel Processing Letters, vol. 3, pp. 469-484, 1993.
[6] B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, "Improved Routing Strategies with Succinct Tables," J. Algorithms, vol. 11, pp. 307-341, 1990.
[7] A. Ben-Dor, S. Halevi, and A. Schuster, "Potential Function Analysis of Greedy Hot-Potato Routing," Proc. 13th ACM Symp. Principles Distributed Computing, pp. 225-234, 1993.
[8] P. Berman, L. Gravano, J. Sanz, and G. Pifarre, "Adaptive Deadlock- and Livelock-Free Routing with All Minimal Paths in Torus Networks," Proc. Fourth ACM Symp. Parallel Algorithms and Architectures, June 1992.
[9] J.-C. Bermond and C. Peyrat, "de Bruijn and Kautz Networks: A Competitor for the Hypercube?," Hypercube and Distributed Computers, F. Andre and J. Verjus, eds., pp. 279-294. NorthHolland, 1989.
[10] J.-C. Bermond and M. Syska, "Routage Wormhole et Canaux Virtuels," Algorithmique Parallèle, M.N.M. Cosnard and Y. Robert, eds., pp. 149-158.Masson, 1992.
[11] N. Biggs, Algebraic Graph Theory.London: Cambridge Univ. Press, 1974.
[12] R. Boppana and S. Chalasani, "New Wormhole Routing Algorithms for Multicomputers," Proc. Seventh Int'l Parallel Processing Symp., pp. 419-423,Newport Beach, Calif., Apr. 1993.
[13] C.M. Chiang and L. Ni, "Deadlock-Free Multi-Head Wormhole Routing," Proc. First High Performance Computing Asia Conf., 1995.
[14] A. Chien, "A Cost and Speed Model for k-ary n-Cube Wormhole routers," Proc. Hot Interconnects '93,Palo Alto, Calif., Aug. 1993.
[15] F. Chung, E. Coffman, M. Reiman, and B. Simon, "The Forwarding Indices of Communication Networks," IEEE Trans. Information Theory, vol. 33, pp. 224-232, 1987.
[16] R. Cypher and L. Gravano, "Adaptive, Deadlock-Free Packet Routing in Torus Networks with Minimal Storage," Proc. 1992 Int'l Conf. Parallel Processing, vol. III, pp. 204-211, Aug. 1992.
[17] R. Cypher and L. Gravano, "Requirements for Deadlock-Free, Adaptive Packet Routing," Proc. 11th ACM Symp. Principles Distributed Computing, 1992.
[18] W.J. Dally, "Virtual-Channel Flow Control," IEEE Trans. Parallel and Distributed Systems, vol. 3, no. 2, pp. 194-205, Mar. 1992.
[19] W.J. Dally and H. Aoki, "Deadlock-Free Adaptive Routing in Multicomputer Networks Using Virtual Channels," IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 4, pp. 466-475, Apr. 1993.
[20] W. Dally and C. Seitz, "The Torus Routing Chip," Distributed Computing, vol. 1, pp. 187-196, 1986.
[21] W.J. Dally and C.L. Seitz, “Deadlock-Free Message Routing in Multiprocessor Interconnection Networks,” IEEE Trans. Computers, Vol. C-36, No. 5, May 1987, pp. 547-553.
[22] M. Di Ianni, "Wormhole Deadlock Prediction," Proc. Euro-Par '97, LNCS 1300, 1997.
[23] J. Duato, “On the Design of Deadlock-Free Adaptive Routing Algorithms for Multicomputers: Design Methodologies,” Proc. Parallel Architectures and Languages Europe 91, June 1991.
[24] J. Duato, "On the Design of Deadlock Free Adaptative Routing Algorithms for Multicomputers: Theoretical Aspects," Lecture Notes in Computer Science 487, Distributed Memory Computing, A Bode, ed., pp. 234-243. Springer Verlag, 1991.
[25] J. Duato, " Channel Classes: A New Concept for Deadlock Avoidance in Wormhole Networks," Parallel Processing Letters, vol. 2, pp. 347-354, 1992.
[26] J. Duato, "A New Theory of Deadlock-Free Adaptive Routing in Wormhole Networks," IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 12, pp. 1,320-1,331, Dec. 1993.
[27] J. Duato, "On the Design of Deadlock-Free Adaptive Multicast Routing," Parallel Processing Letters, vol. 3, pp. 321-334, 1993.
[28] J. Duato, "A Theory to Increase the Effective Redundancy in Wormhole Networks," Parallel Processing Letters, vols. 1and 2, pp. 125-138, 1994.
[29] J. Duato, “A Necessary and Sufficient Condition for Deadlock-Free Adaptive Routing in Wormhole Networks,” IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 10, pp. 1,055–1,067, Oct. 1995.
[30] J. Duato,"A Theory of Deadlock-Free Adaptive Multicast Routing in Wormhole Networks," Trans. Parallel and Distributed Systems, vol. 6, no. 9, Sept. 1995, pp. 976-987.
[31] U. Feige and P. Raghavan, "Exact Analysis of Hot Potato Routing. 33rd Ann. Symp. Foundations of Computer Science, pp. 553-562, Oct. 1992.
[32] S. Felperin, P. Raghavan, and E. Upfal, "A Theory of Wormhole Routing in Parallel Computers," Proc. 33rd Ann. IEEE Conf. Foundations of Computer Science, pp. 563-572, 1992.
[33] E. Fleury and P. Fraigniaud, "Deadlocks in Adaptive Wormhole Routing," Research Report 94-09, Laboratoire de l'Informatique du Parallélisme, ENS-Lyon, France, 1994.
[34] P. Fraigniaud, "Asymptotically Optimal Broadcasting and Gossiping in Faulty Hypercube Multicomputers," IEEE Trans. Computers, vol. 41, no. 11, pp. 1,410-1,419, Nov. 1992.
[35] P. Fraigniaud and C. Gavoille, "Memory Requirement for Universal Routing Schemes," Proc. 14th Ann. ACM Symp. Principles of Distributed Computing (PODC), pp. 223-230, 1995.
[36] P. Fraigniaud and E. Lazard, "Methods and Problems of Communication in Usual Networks," Discrete Applied Math., vol. 53, pp. 79-133, 1994.
[37] D. Gelernter, "A DAG-Based Algorithm for Prevention of Store-And-Forward Deadlock in Packet Networks," IEEE Trans. Computers, vol. 30, pp. 709-715, 1981.
[38] C. Germain, "Etude des Mécanismes de Communication Pour une Machine Massivement Parallèle: MEGA," PhD thesis, UniversitéParis Sud, Orsay, France, 1989.
[39] C.J. Glass and L.M. Ni, “Adaptive Routing in Mesh-Connected Networks,” Proc. 1992 Int'l Conf. on Distributed Computing Systems, pp. 12-19, May 1992.
[40] C. Glass and L. Ni, "Maximally Fully Adaptive Routing in 2D Meshes," Proc. 1992 Int'l Conf. Parallel Processing, vol. I, pp. 101-104, Aug. 1992.
[41] C.J. Glass and L.M. Ni, "The Turn Model for Adaptive Routing," Proc. 19th Int'l Symp. Computer Architecture, vol. 20, no. 2, pp. 278-287, May 1992.
[42] C.J. Glass and L.M. Ni, "Fault-Tolerant Wormhole Routing in Meshes," Proc. 23rd Int'l Symp. Fault-Tolerant Computing, pp. 240-249, 1993.
[43] C.J. Glass and L.M. Ni, “The Turn Model for Adaptive Routing,” J. ACM, vol. 41, no. 5, pp. 874-902, Sept. 1994.
[44] D. Bäumer et al., "Framework Development for Large Systems," Comm. ACM, vol. 40, no. 10, Oct. 1997, pp. 52-59.
[45] K.D. Gunther, "Prevention of Deadlocks in Packet-Switched Data Transport Systems," IEEE Trans. Commun., vol. 29, pp. 512-524, Apr. 1981.
[46] B. Hajek, "Bound on Evacuation Time for Deflection Routing," Distributed Computing, vol. 5, pp. 1-6, 1991.
[47] S. Hedetniemi, S. Hedetniemi, and A. Liestman, "A Survey of Gossiping and Broadcasting in Communication Networks," Networks, vol. 18, pp. 319-349, 1986.
[48] M.-C. Heydemann, J.-C. Meyer, and D. Sotteau, "On Forwarding Indices of Networks," Discrete Applied Mathematics, vol. 23, pp. 103-123, 1989.
[49] D.N. Jayasimha, D. Manivannan, J.A. May, L. Schwiebert, and S.L. Hary, “A Foundation for Designing Deadlock-Free Routing Algorithms in Wormhole Networks,” Proc. Symp. Parallel and Distributed Processing, pp. 190-197, Oct. 1996.
[50] P. Kermani and L. Kleinrock, "Virtual Cut-Through: A New Computer Communication Switching Technique," Computers Networks, vol. 3, pp. 267-286, 1979.
[51] F.T. Leighton,"Average Case Analysis of Greedy Routing Algorithms on Arrays," Proc. Second Ann. ACM Symp. Parallel Algorithms and Architectures, SPAA '90, Crete, July2-6, 1990.
[52] X. Lin, P.K. McKinley, L.M. Ni, “The Message Flow Model for Routing in Wormhole-Routed Networks,” IEEE Trans. Parallel and Distributed Systems, vol. 6, no. 7, pp. 755–760, July 1995.
[53] X. Lin and L. Ni, "Deadlock-Free Multicast Wormhole Routing in Multicomputer Networks," Proc. Int'l Symp. Computer Architecture, June 1991.
[54] D.H. Linder and J.C. Harden, "An Adaptive and Fault Tolerant Wormhole Routing Strategy for k-Ary n-Cubes," IEEE Trans. Computers, vol. 40, no. 1, pp. 2-12, Jan. 1991.
[55] The T9000 Products-Overview Manual, P.T.M. May and P. Welch, eds. SGS-Thomson-Inmos, 1991.
[56] P.M. Merlin and P.J. Schweitzer, "Deadlock Avoidance in Store-and-Forward Networks-I: Store-and-Forward Deadlock," IEEE Trans. Commun., vol. 28, no. 3, pp. 345-354, Mar. 1980.
[57] J. Misic and Z. Jovanovic, "Routing Function and Deadlock Avoidance in a Star Graph Interconnection Network," J. Parallel and Distributed Computing, vol. 22, pp. 216-228, 1994.
[58] L.M. Ni and P.K. McKinley, "A Survey of Wormhole Routing Techniques in Direct Networks," Computer, vol. 26, no. 2, pp. 62-76, Feb. 1993.
[59] P.S. Pacheco, Parallel Programming with MPI. Morgan Kaufmann, 1997.
[60] D.K. Panda, S. Singal, and P. Prabhakaran, “Multidestination Message Passing Mechanism Conforming to Base Wormhole Routing Scheme,” Proc. Parallel Computer Routing and Comm. Workshop, pp. 131–145, 1994.
[61] H. Park and D.P. Agrawal, “Generic Methodologies for Deadlock-Free Routing,” Proc. Int'l Parallel Processing Symp., pp. 638-643, Apr. 1996.
[62] D. Peleg and E. Upfal, "A Trade-Off Between Space and Efficiency for Routing Tables," J. ACM, vol. 36, pp. 510-530, 1989.
[63] J. Peters and M. Syska, "Circuit-Switched Broadcasting in Torus Networks," IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 3, pp. 246-255, 1996.
[64] G.D. Pifarré, L. Gravano, S.A. Felperin, and J.L.C. Sanz, "Fully Adaptive Minimal Deadlock-Free Packet Routing in Hypercubes, Meshes and Other Networks: Algorithms and Simulations," IEEE Trans. Parallel Distributed Systems, vol. 5, no. 3, pp. 247-263, Mar. 1994.
[65] A. Reddy, "Fault Tolerance of Adaptive Routing Algorithms in Multicomputers," technical report, IBM Almaden Research Center, San Jose, Calif., 1992.
[66] J.D. Rumeur, "Communcation dans les Réseaux de Processeurs," Collection Etudes et Recherches en Informatique.Masson, 1994.
[67] M.R. Samatham and D.K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. Computers, vol. 38, no. 4, pp. 567-581, Apr. 1989.
[68] L. Schwiebert, "Deadlock-Free Oblivious Wormhole Routing with Cyclic Dependencies," Proc. Ninth Ann. ACM Symp. Parallel Algorithms and Architectures (SPAA '97), pp. 149-158, 1997.
[69] L. Schwiebert and D.N. Jayasimha, “A Universal Proof Technique for Deadlock-Free Routing in Interconnection Networks,” Proc. Symp. Parallel Algorithms and Architectures, pp. 175–184, July 1995.
[70] L. Schwiebert and D.N. Jayasimha, "A Necessary and Sufficient Condition for Deadlock-Free Wormhole Routing," J. Parallel and Distributed Computing, vol. 32, no. 1, pp. 103-117, Jan. 1996.
[71] A. Tanenbaum, Computer Networks pp. 15-21. Prentice Hall, 1981.
[72] C. Trefftz and P. McKinley, "Performance Evaluation of Wormhole Routing in Octagonal Mesh Direct Network," Proc. Int'l Conf. Parallel and Distributed Systems, pp. 25-32, 1992.
[73] E. Upfal, "On the Theory of Interconnection Networks for Parallel Computer," Proc. Int'l Colloquium on Automata, Languages, and Programming (ICALP '94), 1994.
[74] L. Valiant, "A Scheme for Fast Parallel Communication," SIAM J. Computing, vol. 11, pp. 350-361, 1982.
[75] V. Varavithya and P. Mohapatra, "Tree-Based Multicasting on Wormhole Routed Multistage Interconnection Networks," Proc. 1997 Int'l Conf. Parallel Processingi, 1997.
[76] C. Yang and Y. Tsai, "Fault-Tolerant and Deadlock-Free Wormhole Routing in the Hybercube Network," Proc. Int'l Conf. Parallel and Distributed Systems, pp. 9-16, 1992.

Index Terms:
Routing, wormhole, deadlock, multicast.
Citation:
Eric Fleury, Pierre Fraigniaud, "A General Theory for Deadlock Avoidance in Wormhole-Routed Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 9, no. 7, pp. 626-638, July 1998, doi:10.1109/71.707539
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