Efficient distributed approximation algorithms via probabilistic tree embeddings

  • Authors:

  • Maleq Khan;Fabian Kuhn;Dahlia Malkhi;Gopal Pandurangan;Kunal Talwar

  • Affiliations:

  • VBI, Virginia Tech, Blacksburg, VA, USA;ETH Zurich, Zurich, Switzerland;Microsoft Research, Mountain view, CA, USA;Purdue University, West Lafayette, IN, USA;Microsoft Research, Mountain View, CA, USA

  • Venue:
  • Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
  • Year:
  • 2008

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Abstract

We present a uniform approach to design efficient distributed approximation algorithms for various network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol, Rao, and Talwar (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain expected O(log n)-approximate distributed algorithms for the generalized Steiner forest problem, the minimum routing cost spanning tree problem, and the $k$-source shortest paths problem in arbitrary networks. The time complexities of our algorithms are within a polylogarithmic factor of the optimum. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that might be of independent interest. Assuming a global order on the nodes of a network, the LE list of a node stores the smallest node (w.r.t. the given order) within every distance $d$. Assuming a random order on the nodes, we give an almost-optimal distributed algorithm for computing LE lists on weighted graphs. For unweighted graphs, our LE lists computation has asymptotically optimal time complexity O(D), where D is the diameter of the network. As a byproduct, we get an improved synchronous leader election algorithm for general networks.