Embedding Paths into Trees: VM Placement to Minimize Congestion
- 3 Citations
- 1.8k Downloads
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.
Keywords
Virtual Machine Full Version Quadratic Assignment Problem Gateway Node Reachability GraphPreview
Unable to display preview. Download preview PDF.
References
- 1.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)Google Scholar
- 2.Awerbuch, B., Singh, T.: Online algorithms for selective multicast and maximal dense trees. In: STOC (1997)Google Scholar
- 3.Bansal, N., Lee, K.W., Nagarajan, V., Zafer, M.: Minimum congestion mapping in a cloud. In: PODC, pp. 267–276 (2011)Google Scholar
- 4.Chowdhury, N.M.M.K., Rahman, M.R., Boutaba, R.: Virtual network embedding with coordinated node and link mapping. In: INFOCOM (2009)Google Scholar
- 5.Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group steiner tree problem. In: SODA (1998Google Scholar
- 6.Goel, A., Hezinger, M., Plotkin, S.: Online throughput-comptetitive algorithm for multicast routing and admission control. In: SODA (1998)Google Scholar
- 7.Hassin, R., Levin, A., Sviridenko, M.: Approximating the minimum quadratic assignment problems. ACM Transactions on Algorithms (2009)Google Scholar
- 8.Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Structures & Algorithms 20(3), 465–482 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
- 9.Meng, X., Pappas, V., Zhang, L.: Improving the scalability of data center networks with traffic-aware virtual machine placement. In: INFOCOM (2010)Google Scholar
- 10.Racke, H.: Minimizing congestion in general networks. In: FOCS (2002)Google Scholar
- 11.Rastogi, R., Silberschatz, A., Yener, B.: Secondnet: a data center network virtualization architecture with bandwidth guarantees. In: Co-NEXT Workshop (2010)Google Scholar
- 12.Yu, M., Yi, Y., Rexford, J., Chiang, M.: Rethinking virtual network embedding: Substrate support for path splitting and migration. In: SIGCOMM (2008)Google Scholar