Greedy Õ(C +D) Hot-Potato Routing on Trees

In hot-potato (deflection) routing, nodes in the network have no buffers for packets in transit. A hotpotato routing algorithm is greedy if packets are advanced from their sources toward their destinations whenever possible. The dilation D is the longest distance a packet has to travel; the congestion C is the maximum number of packets that traverse any edge. The routing time of a routing-algorithm is the time for the last packet to reach its destination. A well known lower bound on the routing time is Ω(C +D). When is it possible to design routing algorithms whose routing times match this lower bound? Here, we address this fundamental question within the context of hot-potato routing for the specific case of a tree with n nodes. In particular, we present two greedy, hot-potato routing algorithms: i. A deterministic algorithm, which has a routing time of O((δ ·C+D) lg n), where δ is the maximum node degree; thus, for bounded degree trees, the routing time becomes O((C +D) lg n). ii. A randomized algorithm which has a routing time of O((C + D) lg n) with high probability, for any node degree. Randomization is used for adjusting packet priorities. Both algorithms are online and simple yet efficient. They are built upon the idea of using safe deflections, which are deflections whose net effect is to “recycle” edges from one path to another. These are the first known hot-potato routing algorithms (whether greedy or not) for trees whose routing time is within logarithmic factors of the Ω(C +D) lower bound. Computer Science Department, RPI, 110 8th Street, Troy, NY 12180, USA. Email: buschc@cs.rpi.edu Computer Science Department, RPI, 110 8th Street, Troy, NY 12180, USA. Email: magdon@cs.rpi.edu Computer Science Department, University of Cyprus, P. O. Box 20537, Nicosia CY-1678, Cyprus. Email: mavronic@ucy.ac.cy Computer Science Department, ETH Zurich, 8092 Zurich, Switzerland. Email: wattenhofer@inf.ethz.ch

[1]  Albert G. Greenberg,et al.  Sharp approximate models of deflection routing in mesh networks , 1993, IEEE Trans. Commun..

[2]  W. Daniel Hillis,et al.  The connection machine , 1985 .

[3]  Costas Busch.  Õ(congestion + dilation) hot-potato routing on leveled networks , 2002, SPAA '02.

[4]  Friedhelm Meyer auf der Heide,et al.  Routing with Bounded Buffers and Hot-Potato Routing in Vertex-Symmetric Networks , 1995, ESA.

[5]  Ted H. Szymanski.  An analysis of 'hot-potato' routing in a fiber optic packet switched hypercube , 1990, Proceedings. IEEE INFOCOM '90: Ninth Annual Joint Conference of the IEEE Computer and Communications Societies@m_The Multiple Facets of Integration.

[6]  Burton J. Smith.  Architecture And Applications Of The HEP Multiprocessor Computer System , 1982, Optics & Photonics.

[7]  Shai Halevi,et al.  Potential Function Analysis of Greedy Hot-Potato Routing , 1998, Theory of Computing Systems.

[8]  Pierre Fraigniaud,et al.  Routing in Trees , 2001, ICALP.

[9]  Bruce M. Maggs,et al.  Fast Algorithms for Finding O(Congestion + Dilation) Packet Routing Schedules , 1999, Comb..

[10]  Costas Busch,et al.  Õ(Congestion + Dilation) Hot-Potato Routing on Leveled Networks , 2002, SPAA '02.

[11]  Zhensheng Zhang,et al.  Performance analysis of multihop lightwave networks with hot potato routing and distance-age-priorities , 1994, IEEE Trans. Commun..

[12]  Louxin Zhang,et al.  Optimal bounds for matching routing on trees , 1997, SODA '97.

[13]  Michael Mitzenmacher,et al.  Constant time per edge is optimal on rooted tree networks , 1996, SPAA '96.

[14]  Antonios Symvonis,et al.  Lower bounds for hot-potato permutation routing on trees , 2000, SIROCCO.

[15]  Antonios Symvonis,et al.  Many-to-Many Routings on Trees via Matchings , 1997, Theor. Comput. Sci..

[16]  Friedhelm Meyer auf der Heide,et al.  Shortest-Path Routing in Arbitrary Networks , 1999, J. Algorithms.

[17]  Maurice Herlihy,et al.  Hard-Potato routing , 2000, STOC '00.

[18]  Noga Alon,et al.  Routing permutations on graphs via matchings , 1993, SIAM J. Discret. Math..

[19]  Noga Alon,et al.  Routing Permutations on Graphs Via Matchings , 1994, SIAM J. Discret. Math..

[20]  Anthony S. Acampora,et al.  Multihop lightwave networks: a comparison of store-and-forward and hot-potato routing , 1991, IEEE INFCOM '91. The conference on Computer Communications. Tenth Annual Joint Comference of the IEEE Computer and Communications Societies Proceedings.

[21]  Christian Scheideler,et al.  Locally efficient on-line strategies for routing packets along fixed paths , 1999, SODA '99.

[22]  Bruce M. Maggs,et al.  Packet routing and job-shop scheduling inO(congestion+dilation) steps , 1994, Comb..

[23]  Charles L. Seitz,et al.  Mosaic C: An Experimental Fine-Grain Multicomputer , 1992, 25th Anniversary of INRIA.

[24]  Ioannis Caragiannis,et al.  Experimental Evaluation of Hot-Potato Routing Algorithms on 2-Dimensional Processor Arrays (Research Note) , 2000, Euro-Par.

[25]  Rafail Ostrovsky,et al.  Universal O(congestion + dilation + log1+εN) local control packet switching algorithms , 1997, STOC '97.

[26]  Yuval Rabani,et al.  Distributed packet switching in arbitrary networks , 1996, STOC '96.

[27]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[28]  Uriel Feige,et al.  Exact analysis of hot-potato routing , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[29]  Allan Borodin,et al.  Deterministic Many-to-Many Hot Potato Routing , 1997, IEEE Trans. Parallel Distributed Syst..

[30]  Mikkel Thorup,et al.  Direct routing on trees , 1998, SODA '98.

[31]  P. Baran,et al.  On Distributed Communications Networks , 1964 .

[32]  Rene L. Cruz,et al.  Bounds on Maximum Delay in Networks with Deflection Routing , 1995, IEEE Trans. Parallel Distributed Syst..

[33]  Arnold L. Rosenberg,et al.  On Bufferless Routing of Variable Length Messages in Leveled Networks , 1996, IEEE Trans. Computers.

[34]  Nicholas F. Maxemchuk,et al.  Comparison of deflection and store-and-forward techniques in the Manhattan Street and Shuffle-Exchange Networks , 1989, IEEE INFOCOM '89, Proceedings of the Eighth Annual Joint Conference of the IEEE Computer and Communications Societies.

[35]  Satish Rao,et al.  Hot-potato routing on processor arrays , 1993, ACM Symposium on Parallelism in Algorithms and Architectures.