Abstract
In the store-and-forward routing problem, packets have to be routed along given paths such that the arrival time of the latest packet is minimized. A groundbreaking result of Leighton, Maggs and Rao says that this can always be done in time \(O(\textrm{congestion} + \textrm{dilation})\), where the congestion is the maximum number of paths using an edge and the dilation is the maximum length of a path. However, the analysis is quite arcane and complicated and works by iteratively improving an infeasible schedule. Here, we provide a more accessible analysis which is based on conditional expectations. Like [LMR94], our easier analysis also guarantees that constant size edge buffers suffice.
Moreover, it was an open problem stated e.g. by Wiese [Wie11], whether there is any instance where all schedules need at least \((1+\varepsilon)\cdot(\textrm{congestion}+\textrm{dilation})\) steps, for a constant ε > 0. We answer this question affirmatively by making use of a probabilistic construction.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J.H.: The probabilistic method, 3rd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., Hoboken (2008); With an appendix on the life and work of Paul Erdős
Clementi, A.E.F., Di Ianni, M.: On the hardness of approximating optimum schedule problems in store and forward networks. IEEE/ACM Trans. Netw. 4(2), 272–280 (1996)
Dubhashi, D.P., Panconesi, A.: Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, Cambridge (2009)
Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Infinite and Finite Sets (Colloq., Keszthely, 1973; Dedicated to P. Erdős on his 60th Birthday). Colloq. Math. Soc. János Bolyai, vol. II, 10, pp. 609–627. North-Holland, Amsterdam (1975)
Feige, U., Scheideler, C.: Improved bounds for acyclic job shop scheduling. Combinatorica 22(3), 361–399 (2002)
Koch, R., Peis, B., Skutella, M., Wiese, A.: Real-Time Message Routing and Scheduling. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 217–230. Springer, Heidelberg (2009)
Leighton, F.T., Maggs, B.M., Rao, S.B.: Packet routing and job-shop scheduling in O(congestion + dilation) steps. Combinatorica 14(2), 167–186 (1994)
Leighton, T., Maggs, B., Richa, A.W.: Fast algorithms for finding O (congestion + dilation) packet routing schedules. Combinatorica 19(3), 375–401 (1999)
Mastrolilli, M., Svensson, O.: Hardness of approximating flow and job shop scheduling problems. J. ACM 58(5), 20 (2011)
Moser, R.A., Tardos, G.: A constructive proof of the general lovász local lemma. J. ACM 57(2) (2010)
Mitzenmacher, M., Upfal, E.: Probability and computing. Randomized algorithms and probabilistic analysis. Cambridge University Press, Cambridge (2005)
Meyer auf der Heide, F., Vöcking, B.: Shortest-path routing in arbitrary networks. J. Algorithms 31(1), 105–131 (1999)
Ostrovsky, R., Rabani, Y.: Universal o(congestion + dilation + log1 + epsilon n) local control packet switching algorithms. In: STOC, pp. 644–653 (1997)
Peis, B., Skutella, M., Wiese, A.: Packet Routing: Complexity and Algorithms. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 217–228. Springer, Heidelberg (2010)
Peis, B., Wiese, A.: Universal Packet Routing with Arbitrary Bandwidths and Transit Times. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 362–375. Springer, Heidelberg (2011)
Rabani, Y., Tardos, É.: Distributed packet switching in arbitrary networks. In: STOC, pp. 366–375 (1996)
Scheideler, C.: Universal Routing Strategies for Interconnection Networks. LNCS, vol. 1390. Springer, Heidelberg (1998)
Srinivasan, A., Teo, C.: A constant-factor approximation algorithm for packet routing and balancing local vs. global criteria. SIAM J. Comput. 30(6), 2051–2068 (2000)
Wiese, A.: Packet Routing and Scheduling. Dissertation, TU Berlin (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rothvoß, T. (2013). A Simpler Proof for \(O(\textrm{Congestion} + \textrm{Dilation})\) Packet Routing. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-36694-9_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36693-2
Online ISBN: 978-3-642-36694-9
eBook Packages: Computer ScienceComputer Science (R0)