Subscribe

Issue No.05 - May (2012 vol.61)

pp: 700-712

S. S. Iyengar , Comput. Sci. Dept., Louisiana State Univ., Baton Rouge, LA, USA

C. Busch , Comput. Sci. Dept., Louisiana State Univ., Baton Rouge, LA, USA

S. Srinivasagopalan , Comput. Sci. Dept., Louisiana State Univ., Baton Rouge, LA, USA

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2011.64

ABSTRACT

We consider the problem of constructing a single spanning tree for the single-sink buy-at-bulk network design problem for doubling-dimension graphs. We compute a spanning tree to route a set of demands along a graph G to or from a designated sink node. The demands could be aggregated at (or symmetrically distributed to) intermediate edges where the fusion cost is specified by a nonnegative concave function f. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number and location of data sources (or demands) and cost function at the edges. We present a deterministic, polynomial-time algorithm for constructing a spanning tree in low doubling-dimension graphs that guarantees a log

^{3}D-approximation over the optimal cost, where D is the diameter of the graph G. With a constant fusion-cost function, our spanning tree gives an O(log^{3}D)-approximation for every Steiner tree that includes the sink. We also provide a Ω(log n) lower bound for any oblivious tree in low doubling-dimension graphs. To our knowledge, this is the first paper to propose a single spanning tree solution to the single-sink buy-at-bulk network design problem (as opposed to multiple overlay trees).INDEX TERMS

trees (mathematics), computational complexity, deterministic algorithms, Steiner tree, oblivious spanning tree, doubling-dimension graph, single-sink buy-at-bulk network design problem, nonnegative concave function, deterministic algorithm, polynomial-time algorithm, optimal cost, constant fusion-cost function, Approximation methods, Approximation algorithms, Algorithm design and analysis, Steiner trees, Peer to peer computing, Extraterrestrial measurements, data structure., Spanning tree, buy-at-bulk, network design, approximation algorithm, doubling-dimension graph, data fusion

CITATION

S. S. Iyengar, C. Busch, S. Srinivasagopalan, "An Oblivious Spanning Tree for Single-Sink Buy-at-Bulk in Low Doubling-Dimension Graphs",

*IEEE Transactions on Computers*, vol.61, no. 5, pp. 700-712, May 2012, doi:10.1109/TC.2011.64REFERENCES

- [1] S. Srinivasagopalan, C. Busch, and S.S. Iyengar, “Brief Announcement: Universal Data Aggregation Trees for Sensor Networks in Low Doubling Metrics,”
Proc. Fifth Int'l Workshop Algorithmic Aspects of Wireless Sensor Networks (ALGOSENSORS '09), pp. 151-152, July 2009.- [2] C. Chekuri, M.T. Hajiaghayi, G. Kortsarz, and M.R. Salavatipour, “Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design,”
Proc. Ann. IEEE Symp. Foundations of Computer Science (FOCS '06), pp. 677-686, 2006.- [3] A. Goel and D. Estrin, “Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source Buy-at-Bulk,”
Proc. Ann. ACM-SIAM Symp. Discrete Algorithms (SODA '03), pp. 499-505, 2003.- [4] I. Abraham, C. Gavoille, A.V. Goldberg, and D. Malkhi, “Routing in Networks with Low Doubling Dimension,”
Proc. IEEE Int'l Conf. Distributed Computing Systems (ICDCS '06), p. 75, 2006.- [5] G. Konjevod, A.W. Richa, D. Xia, and H. Yu, “Compact Routing with Slack in Low Doubling Dimension,”
Proc. 26th Ann. ACM Symp. Principles of Distributed Computing (PODC '07), pp. 71-80, http://doi.acm.org/10.11451281100.1281113 , 2007.- [6] G. Konjevod, A.W. Richa, and D. Xia, “Dynamic Routing and Location Services in Metrics of Low Doubling Dimension,”
Proc. Int'l Symp. Distributed Computing (DISC '08), pp. 379-393, 2008.- [7] H.T.-H. Chan, A. Gupta, B.M. Maggs, and S. Zhou, “On Hierarchical Routing in Doubling Metrics,”
Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA '05), pp. 762-771, http://portal.acm.orgcitation.cfm?id=1070432.1070540 , 2005.- [8] T.-H.H. Chan and A. Gupta, “Small Hop-Diameter Sparse Spanners for Doubling Metrics,”
Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithm (SODA '06), pp. 70-78, http://doi. acm.org/10.11451109557.1109566 , 2006.- [9] J. Kleinberg, A. Slivkins, and T. Wexler, “Triangulation and Embedding Using Small Sets of Beacons,”
J. ACM, vol. 56, no. 6, pp. 1-37, 2009.- [10] P. Fraigniaud, “The Inframetric Model for the Internet,” technical report, 2007.
- [11] P. Fraigniaud, E. Lebhar, and Z. Lotker, “A Doubling Dimension Threshold $\theta (\log \log n)$ for Augmented Graph Navigability,”
Proc. Ann. European Symp. (ESA), pp. 376-386, 2006.- [12] F. Kuhn, T. Moscibroda, and R. Wattenhofer, “On the Locality of Bounded Growth,”
Proc. Ann. ACM Symp. Principles of Distributed Computing (PODC '05), pp. 60-68, 2005.- [13] J. Gao, L.J. Guibas, N. Milosavljevic, and D. Zhou, “Distributed Resource Management and Matching in Sensor Networks,”
Proc. Int'l Conf. Information Processing in Sensor Networks (IPSN '09), pp. 97-108, Apr. 2009.- [14] S. Funke, L. Guibas, A. Nguyen, and Y. Wang, “Distance-Sensitive Information Brokerage in Sensor Networks,”
Proc. Int'l Conf. Distributed Computing in Sensor Networks (DCOSS '06), pp. 234-251, 2006.- [15] S.V. Pemmaraju and I.A. Pirwani, “Energy Conservation via Domatic Partitions,”
Proc. Int'l Symp. Mobile Ad Hoc Networking and Computing (MobiHoc '06), pp. 143-154, 2006.- [16] F.S. Salman, J. Cheriyan, R. Ravi, and S. Subramanian, “Approximating the Single-Sink Link-Installation Problem in Network Design,”
SIAM J. Optimization, vol. 11, no. 3, pp. 595-610, 2000.- [17] A.A. Hagberg, D.A. Schult, and P.J. Swart, “Exploring Network Structure, Dynamics, and Function Using NetworkX,”
Proc. Seventh Python in Science Conf. (SciPy '08), pp. 11-15, Aug. 2008.- [18] Y. Mansour and D. Peleg, “An Approximation Algorithm for Minimum-Cost Network Design,” technical report, Inst. of Science, Rehovot 1998.
- [19] Y. Bartal, “Competitive Analysis of Distributed Online Problems - Distributed Paging,” PhD dissertation, 1994.
- [20] B. Awerbuch and Y. Azar, “Buy-at-Bulk Network Design,”
Proc. Ann. Symp. Foundations of Computer Science (FOCS '97), pp. 542-547, 1997.- [21] Y. Bartal, “On Approximating Arbitrary Metrices by Tree Metrics,”
Proc. Ann. ACM Symp. Theory of Computing (STOC '98), pp. 161-168, 1998.- [22] S. Guha, A. Meyerson, and K. Munagala, “A Constant Factor Approximation for the Single Sink Edge Installation Problems,”
Proc. ACM Symp. Theory of Computing (STOC '01), pp. 383-388, 2001.- [23] K. Talwar, “The Single-Sink Buy-at-Bulk lp Has Constant Integrality Gap,”
Proc. Ninth Int'l IPCO Conf. Integer Programming and Combinatorial Optimization, pp. 475-486, 2002.- [24] A. Kumar, A. Gupta, and T. Roughgarden, “A Constant-Factor Approximation Algorithm for the Multicommodity Rent-or-Buy Problem,”
Proc. 43rd Symp. Foundations of Computer Science (FOCS '02), p. 333, 2002.- [25] A. Gupta, A. Kumar, M. Pál, and T. Roughgarden, “Approximation via Cost Sharing: Simpler and Better Approximation Algorithms for Network Design,”
J. ACM, vol. 54, no. 3, p. 11, 2007.- [26] R. Jothi and B. Raghavachari, “Improved Approximation Algorithms for the Single-Sink Buy-at-Bulk Network Design Problems,”
J. Discrete Algorithms, vol. 7, no. 2, pp. 249-255, 2009.- [27] A. Frangioni and B. Gendron, “0-1 Reformulations of the Multicommodity Capacitated Network Design Problem,”
Discrete Applications Math., vol. 157, no. 6, pp. 1229-1241, 2009.- [28] B. Gendron, T.G. Crainic, and A. Frangioni, “Multicommodity Capacitated Network Design,”
Telecomm. Network Planning.- [29] T. Öncan, “Design of Capacitated Minimum Spanning Tree with Uncertain Cost and Demand Parameters,”
Information Sciences: An Int'l J., vol. 177, no. 20, pp. 4354-4367, 2007.- [30] L. Jia, G. Noubir, R. Rajaraman, and R. Sundaram, “Gist: Group-Independent Spanning Tree for Data Aggregation in Dense Sensor Networks,”
Proc. IEEE Int'l Conf. Distributed Computing in Sensor Systems (DCOSS), pp. 282-304, 2006.- [31] L. Jia, G. Lin, G. Noubir, R. Rajaraman, and R. Sundaram, “Universal Approximations for Tsp, Steiner Tree, and Set Cover,”
Proc. 37th Ann. ACM Symp. Theory of Computing (STOC '05), pp. 386-395, 2005.- [32] A. Goel and I. Post, “An Oblivious o(1)-Approximation for Single Source Buy-at-Bulk,”
Proc. Ann. IEEE Symp. Foundations of Computer Science, pp. 442-450, 2009.- [33] A. Gupta, M.T. Hajiaghayi, and H. Räcke, “Oblivious Network Design,”
Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithm, pp. 970-979, 2006.- [34] J. Chuzhoy, A. Gupta, J.S. Naor, and A. Sinha, “On the Approximability of Some Network Design Problems,”
ACM Trans. Algorithms, vol. 4, no. 2, pp. 1-17, 2008.- [35] M. Imase and B.M. Waxman, “Dynamic Steiner Tree Problem,”
SIAM J. Discrete Math., vol. 4, no. 3, pp. 369-384, http://link.aip.org/link/?SJD/4/3691, 1991.- [36] T. Nieberg, “Independent and Dominating Sets in Wireless Communication Graphs,” PhD dissertation, Univ. of Twente, Apr. 2006.
- [37] A. Gupta, R. Krauthgamer, and J.R. Lee, “Bounded Geometries, Fractals, and Low-Distortion Embeddings,”
Proc. Ann. IEEE Symp. Foundations of Computer Science (FOCS '03), p. 534, 2003.- [38] M.R. Garey and D.S. Johnson, “The Rectilinear Steiner Tree Problem is Np-Complete,”
SIAM J. Applied Math., vol. 32, no. 4, pp. 826-834, http://www.jstor.org/stable2100192, 1977. |