Distributed Approximation of Minimum Routing Cost Trees

We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph G over n nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a different (but short) message to each of its neighbors in each synchronous round. We provide a randomized (2 + e)-approximation with runtime \(\mathcal{O}(D+\frac{\log n}{\varepsilon})\) for unweighted graphs. Here, D is the diameter of G. This improves over both, the (expected) approximation factor \(\mathcal{O}(\log n)\) and the runtime \(\mathcal{O}(D\log^2 n)\) stated in [13].

[1]  A. Scott.  The optimal network problem: Some computational procedures , 1969 .

[2]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[3]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[4]  Chuan Yi Tang,et al.  A polynomial time approximation scheme for minimum routing cost spanning trees , 1998, SODA '98.

[5]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[6]  Chuan Yi Tang,et al.  Approximation algorithms for the shortest total path length spanning tree problem , 2000, Discret. Appl. Math..

[7]  Dahlia Malkhi,et al.  Efficient distributed approximation algorithms via probabilistic tree embeddings , 2008, PODC '08.

[8]  Ittai Abraham,et al.  Nearly Tight Low Stretch Spanning Trees , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[9]  Michael Elkin.  Low Stretch Spanning Trees , 2016, Encyclopedia of Algorithms.

[10]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[11]  Eilon Reshef.  Approximating Minimum Communication Cost Spanning Trees and Related Problems , 1999 .

[12]  Manuel Ricardo,et al.  A fast algorithm for computing minimum routing cost spanning trees , 2008, Comput. Networks.

[13]  Bang Ye Wu,et al.  A polynomial time approximation scheme for the two-source minimum routing cost spanning trees , 2002, J. Algorithms.

[14]  Jan Karel Lenstra,et al.  The complexity of the network design problem , 1978, Networks.

[15]  Roger Wattenhofer,et al.  Optimal distributed all pairs shortest paths and applications , 2012, PODC '12.

[16]  Dahlia Malkhi,et al.  Efficient distributed approximation algorithms via probabilistic tree embeddings , 2008, PODC '08.

[17]  Matteo Fischetti,et al.  Exact algorithms for minimum routing cost trees , 2002, Networks.

[18]  Shang-Hua Teng,et al.  Lower-stretch spanning trees , 2004, STOC '05.

[19]  Ching-Lueh Chang,et al.  Parallelized approximation algorithms for minimum routing cost spanning trees , 2007, ArXiv.

[20]  Michael Florian,et al.  Exact and approximate algorithms for optimal network design , 1979, Networks.

[21]  David Peleg.  Low Stretch Spanning Trees , 2002, MFCS.

[22]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[23]  Danupon Nanongkai,et al.  Distributed approximation algorithms for weighted shortest paths , 2014, STOC.

[24]  T. C. Hu.  Optimum Communication Spanning Trees , 1974, SIAM J. Comput..

[25]  Richard T. Wong,et al.  Worst-Case Analysis of Network Design Problem Heuristics , 1980, SIAM J. Algebraic Discret. Methods.