Two-Hop Virtual Path Layout in Tori S´ebastien Choplin, Lata Narayanan, Jaroslav Opatrny Department of Computer Science, Concordia University, Montreal, Quebec, Canada, H3G 1M8, email : lata,opatrny @cs.concordia.ca,
[email protected]
Abstract We consider a problem motivated by the design of ATM (Asynchronous Transfer Mode) networks. Given a physical network and an All-to-All traffic, the problem consists in designing a virtual network with a given diameter, which can be embedded in the physical one with a minimum congestion (the congestion is the maximum load of a physical link). Here we propose a method to solve this problem when the diameter is 2 and we use this method to find the optimal solution when the physical network is a mesh or a torus. Keywords : Virtual Path Layout, Projective Plane, Diameter, Mesh
1
Introduction
The following study was motivated by a question concerning ATM networks. In such networks, the traffic requests (or demands) are routed via virtual paths (VPs) (see the book on ATM [KG96]). This set of virtual paths has to be chosen to satisfy two opposite goals : minimizing the hop count (number of VPs which are used to route a request) and minimizing the load or congestion (the number of VPs sharing the same physical link). These two minimization objectives are contradictory. The problem can be formulated for general networks, where the traffic is routed in a virtual (or logical) network that has to be embedded into a physical one. The design of a virtual topology with given hop count and load is a difficult problem which has been considered by many authors and is called the VPL (Virtual Path Layout) design problem. We follow the model introduced in [CGZ94, GZ94] and refer the reader to the survey of Zaks [Zak97] for more details. This VPL problem consists in finding a virtual graph of given diameter D and its embedding in the physical graph such that the congestion (maximum load) is minimized. Here we consider the case where the physical network is a mesh (or a torus). Results concerning other physical graphs and set of requests can be found in [CGZ94, GZ94]. This problem 1
has already been studied by Becchetti et al. (see [BBGG98]) where authors give an asymptoticaly optimal result for D 3. We give a solution for D 2 and show that the load is optimal using the properties of the projective planes. The key point of our method is to find a partition of the vertices in some sets corresponding to the points of a projective plane and to link this sets in the virtual graph according to the lines of this projective plane. The nice properties of the projective plane strucure allows us to have a balanced congestion (by finding a good embedding) and to guarantee that the diameter of the virtual graph is 2. We denote by π G D the minimum of the maximum load of the embeddings over all virtual graphs of diameter D in G. We establish the following result for square meshes or tori T n n , if n is 4 such that n q2 q 1 with q a power of a prime, π T n n 2 Θ n3 . As we think that the method of solving the problem is more interesting than the result itself, we can ask the following question for general graphs, with D 2, does this kind of structure (which induces some projective plane) is always optimal or close to the optimal ? One should make the most of this technique to design efficient algorithms to solve this problem. As the proposed solution for D 3 are usually hierarchic, one could want to find a distributed structure similar to the projective plane one (maybe found in code theory or packing theory) which guarantees the diameter constraint and allows to have an optimal maximum load.
References [BBGG98] L. Becchetti, P. Bertolazzi, C. Gaibisso, and G. Gambosi. On the design of efficient ATM routing schemes. Journal of Theoretical Computer Science, Volume 270, Issues 1-2:341–359, 1998. [CGZ94]
I. Cidon, O. Gerstel, and S. Zaks. A scalable approach to routing in ATM networks. In Proc. of the 8th International Workshop on Distributed Algorithms, WDAG ’94, volume 857 of Lecture Notes in Computer Science, pages 209–222. Springer Verlag, 1994.
[GZ94]
O. Gerstel and S. Zaks. The Virtual Path Layout problem in fast networks (extended abstract). In Proceedings of the Thirteenth Annual ACM Symposium on Principles of Distributed Computing, pages 235– 243, Los Angeles, California, 14–17 August 1994.
[KG96]
D. Kofman and M. Gagnaire. R´eseaux haut d´ebit : r´eseaux ATM, r´eseaux locaux, r´eseaux tout-optiques. InterEditions, 1996.
[Zak97]
S. Zaks. Path Layout in ATM networks. Lecture Notes in Computer Science, 1338:144–177, 1997. 2
