Embedding Paths into Trees: VM Placement to Minimize Congestion

  • Debojyoti Dutta
  • Michael Kapralov
  • Ian Post
  • Rajendra Shinde
Conference paper

DOI: 10.1007/978-3-642-33090-2_38

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)
Cite this paper as:
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

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/ε2) 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Debojyoti Dutta
    • 1
  • Michael Kapralov
    • 2
  • Ian Post
    • 2
  • Rajendra Shinde
    • 2
  1. 1.Cisco SystemsUSA
  2. 2.Stanford UniversityStanfordUSA

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