Abstract
Modern cloud infrastructure providers allow customers to rent computing capability in the form of a network of virtual machines (VMs) with bandwidth guarantees between pairs of VMs. Typical requests are in the form of a chain of VMs with an uplink bandwidth to the gateway node of the network (rooted path requests), and most data center architectures route network packets along a spanning tree of the physical network. VMs are instantiated inside servers which reside at the leaves of this network, leading to the following optimization problem: given a rooted tree network T and a set of rooted path requests, find an embedding of the requests that minimizes link congestion.
Our main result is an algorithm that, given a rooted tree network T with n leaves and set of weighted rooted path requests, embeds a 1 − ε fraction of the requests with congestion at most poly(logn, logθ,ε − 1)·OPT (approximation is necessary since the problem is NP-hard). Here OPT is the congestion of the optimal embedding and θ is the ratio of the maximum to minimum weights of the path requests. We also obtain an O(Hlogn/ε 2) approximation if node capacities can be augmented by a (1 + ε) factor (here H is the height of the tree). Our algorithm applies a randomized rounding scheme based on Group Steiner Tree rounding to a novel LP relaxation of the set of subtrees of T with a given number of leaves that may be of independent interest.
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
Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: Online load balancing with applications to machine scheduling and virtual circuit routing. In: STOC (1993)
Awerbuch, B., Singh, T.: Online algorithms for selective multicast and maximal dense trees. In: STOC (1997)
Bansal, N., Lee, K.W., Nagarajan, V., Zafer, M.: Minimum congestion mapping in a cloud. In: PODC, pp. 267–276 (2011)
Chowdhury, N.M.M.K., Rahman, M.R., Boutaba, R.: Virtual network embedding with coordinated node and link mapping. In: INFOCOM (2009)
Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group steiner tree problem. In: SODA (1998
Goel, A., Hezinger, M., Plotkin, S.: Online throughput-comptetitive algorithm for multicast routing and admission control. In: SODA (1998)
Hassin, R., Levin, A., Sviridenko, M.: Approximating the minimum quadratic assignment problems. ACM Transactions on Algorithms (2009)
Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Structures & Algorithms 20(3), 465–482 (2002)
Meng, X., Pappas, V., Zhang, L.: Improving the scalability of data center networks with traffic-aware virtual machine placement. In: INFOCOM (2010)
Racke, H.: Minimizing congestion in general networks. In: FOCS (2002)
Rastogi, R., Silberschatz, A., Yener, B.: Secondnet: a data center network virtualization architecture with bandwidth guarantees. In: Co-NEXT Workshop (2010)
Yu, M., Yi, Y., Rexford, J., Chiang, M.: Rethinking virtual network embedding: Substrate support for path splitting and migration. In: SIGCOMM (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dutta, D., Kapralov, M., Post, I., Shinde, R. (2012). Embedding Paths into Trees: VM Placement to Minimize Congestion. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-33090-2_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33089-6
Online ISBN: 978-3-642-33090-2
eBook Packages: Computer ScienceComputer Science (R0)