Power Assignment Problems in Wireless Communication: Covering Points by Disks, Reaching few Receivers Quickly, and Energy-Efficient Travelling Salesman Tours

A fundamental class of problems in wireless communication is concerned with the assignment of suitable transmission powers to wireless devices/stations such that the resulting communication graph satisfies certain desired properties and the overall energy consumed is minimized. Many concrete communication tasks in a wireless network like broadcast, multicast, point-to-point routing, creation of a communication backbone, etc. can be regarded as such a power assignment problem. This paper considers several problems of that kind; the first problem was studied before in [1,6] and aims to select and assign powers to kout of a total of nwireless network stations such that all stations are within reach of at least one of the selected stations. We show that the problem can be (1 + i¾?) approximated by only looking at a small subset of the input, which is of size , i.e. independent of nand polynomial in kand 1/i¾?. Here ddenotes the dimension of the space where the wireless devices are distributed, so typically d≤ 3 and describes the relation between the Euclidean distance between two stations and the power consumption for establishing a wireless connection between them. Using this coresetwe are able to improve considerably on the running time of $n^{((\alpha/\epsilon)^{O(d)})}$ for the algorithm by Bilo et al. at ESA'05 ([6]) actually obtaining a running time that is linearin n. Furthermore we sketch how outliers can be handled in our coreset construction. The second problem deals with the energy-efficient, bounded-hop multicast operation: Given a subset Cout of a set of nstations and a designated source node swe want to assign powers to the stations such that every node in Cis reached by a transmission from swithin khops. Again we show that a coreset of size independent of nand polynomial in k, |C|, 1/i¾?exists, and use this to provide an algorithm which runs in time linear in n. The last problem deals with a variant of non-metric TSP problem where the edge costs are the squared Euclidean distances; this problem is motivated by data aggregation schemes in wireless sensor networks. We show that a good TSP tour under Euclidean edge costs can be very bad in the squared distance measure and provide a simple constant approximation algorithm, partly improving upon previous results in [5], [4].

[1]  Sariel Har-Peled.  Clustering Motion , 2004, Discret. Comput. Geom..

[2]  Xiang-Yang Li,et al.  Minimum-energy broadcast routing in static ad hoc wireless networks , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[3]  Christoph Ambühl,et al.  An Optimal Bound for the MST Algorithm to Compute Energy Efficient Broadcast Trees in Wireless Networks , 2005, ICALP.

[4]  Andrea E. F. Clementi,et al.  Efficient Algorithms for Low-Energy Bounded-Hop Broadcast in Ad-Hoc Wireless Networks , 2004, STACS.

[5]  Christoph Amb.  An Optimal Bound for the MST Algorithm to Compute Energy Efficient Broadcast Trees in Wireless Networks , 2005 .

[6]  Esther M. Arkin,et al.  Minimum-cost coverage of point sets by disks , 2006, SCG '06.

[7]  Andrea E. F. Clementi,et al.  On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs , 2001, STACS.

[8]  Klaus Jansen,et al.  Polynomial-time approximation schemes for geometric graphs , 2001, SODA '01.

[9]  Vittorio Bilò,et al.  Geometric Clustering to Minimize the Sum of Cluster Sizes , 2005, ESA.

[10]  Anthony Ephremides,et al.  Energy-Efficient Broadcast and Multicast Trees in Wireless Networks , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[11]  Andrzej Pelc,et al.  Power consumption in packet radio networks , 2000, Theor. Comput. Sci..

[12]  Tomás Feder,et al.  Optimal algorithms for approximate clustering , 1988, STOC '88.

[13]  P. Orponen,et al.  On Approximation Preserving Reductions: Complete Problems and Robust Measures (Revised Version) , 1987 .

[14]  Michael A. Bender,et al.  Performance guarantees for the TSP with a parameterized triangle inequality , 2000, Inf. Process. Lett..

[15]  Stefan Funke,et al.  Bounded-Hop Energy-Efficient Broadcast in Low-Dimensional Metrics Via Coresets , 2007, STACS.

[16]  Samir Khuller,et al.  Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets , 1998, Inf. Comput..

[17]  Thomas Andreae,et al.  On the traveling salesman problem restricted to inputs satisfying a relaxed triangle inequality , 2001, Networks.