# On the Power Assignment Problem in Radio Networks

- 73 Downloads
- 69 Citations

## Abstract

Given a finite set *S* of points (i.e. the stations of a radio network) on a *d*-dimensional Euclidean space and a positive integer 1≤*h*≤|*S*|−1, the MIN *D*D *H*-RANGE ASSIGNMENT problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided that the transmission ranges of the stations ensure the communication between any pair of stations in at most *h* hops.

Two main issues related to this problem are considered in this paper: the trade-off between the power consumption and the number of hops; the computational complexity of the MIN *D*D *H*-RANGE ASSIGNMENT problem.

As for the first question, we provide a lower bound on the minimum power consumption of stations on the plane for constant *h*. The lower bound is a function of |*S*|, *h* and the minimum distance over all the pairs of stations in *S*. Then, we derive a *constructive* upper bound as a function of |*S*|, *h* and the maximum distance over all pairs of stations in *S* (i.e. *the diameter* of *S*). It turns out that when the minimum distance between any two stations is “not too small” (i.e. well spread instances) the upper bound matches the lower bound. Previous results for this problem were known only for very special 1-dimensional configurations (i.e., when points are arranged on a line at unitary distance) [Kirousis, Kranakis, Krizanc and Pelc, 1997].

As for the second question, we observe that the tightness of our upper bound implies that MIN 2D *H*-RANGE ASSIGNMENT restricted to well spread instances admits a polynomial time approximation algorithm. Then, we also show that the same approximation result can be obtained for random instances. On the other hand, we prove that for *h*=|*S*|−1 (i.e. the *unbounded* case) MIN 2D *H*-RANGE ASSIGNMENT is NP-hard and MIN 3D *H*-RANGE ASSIGNMENT is APX-complete.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]P. Alimonti and V. Kann, Hardness of approximating problems on cubic graphs, in:
*Proc. of 3rd Italian Conf. on Algorithms and Complexity*, Lecture Notes in Computer Science, Vol. 1203 (Springer, 1997) pp. 288–298.Google Scholar - [2]E. Arikan, Some complexity results about packet radio networks, IEEE Transactions on Information Theory 30 (1984) 456–461.Google Scholar
- [3]G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela and M. Protasi,
*Complexity and Approximation — Combinatorial Optimization Problems and their Approximability Properties*(Springer, 1999).Google Scholar - [4]B.S. Baker, Approximation algorithms for NP-complete problems on planar graphs, Journal of ACM 41 (1994) 153–180.Google Scholar
- [5]P. Berman and M. Karpinski, On some tighter inapproximability results, Electronic Colloquium on Computational Complexity, TR-29 (1998) http://www.eccc.uni-trier.de/eccc/.Google Scholar
- [6]I. Chlamtac and A. Farago, Making transmission schedules immune to topology changes in multihop packet radio networks, IEEE/ACM Transactions on Networking 2 (1994) 23–29.Google Scholar
- [7]B.S. Chlebus, L. Gasienec, A.M. Gibbons, A. Pelc and W. Ritter, Deterministic broadcasting in unknown radio networks, in:
*Proc. of 11th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA)*(2000) pp. 861–870.Google Scholar - [8]A.E.F. Clementi, P. Penna and R. Silvestri, Hardness results for the power range assignment problem in packet radio networks, in:
*Proc. of II International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (RANDOM/APPROX'99)*, Lecture Notes in Computer Science, Vol. 1671 (1999) pp. 197–208.Google Scholar - [9]A.E.F. Clementi, P. Penna and R. Silvestri, The power range assignment problem in radio networks on the plane, in:
*Proc. of XVII Symposium on Theoretical Aspects of Computer Science (STACS'00)*, Lecture Notes in Computer Science, Vol. 1770 (2000) pp. 651–660.Google Scholar - [10]P. Eades, A. Symvonis and S. Whitesides, Two algorithms for three dimensional orthogonal graph drawing, in:
*Proc. of Graph Drawing '96*, Lecture Notes in Computer Science, Vol. 1190 (1996) pp. 139–154.Google Scholar - [11]A. Ephemides and T. Truong, Scheduling broadcast in multihop radio networks, IEEE Transactions on Communications 30 (1990) 456–461.Google Scholar
- [12]M.R. Garey and D.S. Johnson,
*Computers and Intractability — A Guide to the Theory of NP-Completness*(Freeman, New York, 1979).Google Scholar - [13]J.D. Gibson (ed.),
*The Mobile Communications Handbook*(CRC Press, 1996).Google Scholar - [14]P. Gupta and P.R. Kumar, Critical power for asympotic connectivity in wireless networks, in:
*Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming*, eds. W.M. McEneany, G. Yin and Q. Zhang (Birkhäuser, Boston, 1998) pp. 547–566.Google Scholar - [15]G. Kant, Drawing planar graphs using the canonical ordering, Algorithmica — Special Issue on Graph Drawing 16 (1996) 4–32. (Extended abstract appeared in Proc. of 33th IEEE FOCS (1992).)Google Scholar
- [16]L. M. Kirousis, E. Kranakis, D. Krizanc and A. Pelc, Power consumption in packet radio networks, in:
*Proc. of 14th Annual Symposium on Theoretical Aspects of Computer Science (STACS)*, Lecture Notes in Computer Science, Vol. 1200 (1997) pp. 363–374.Google Scholar - [17]R. Mathar and J. Mattfeldt, Optimal transmission ranges for mobile communication in linear multihop packet radio netwoks, Wireless Networks 2 (1996) 329–342.Google Scholar
- [18]K. Pahlavan and A. Levesque,
*Wireless Information Networks*(Wiley-Interscience, New York, 1995).Google Scholar - [19]C.H. Papadimitriou,
*Computational Complexity*(Addison-Wesley, 1994).Google Scholar - [20]C.H. Papadimitriou and M. Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Science 43 (1991) 425–440.Google Scholar
- [21]P. Piret, On the connectivity of radio networks, IEEE Transactions on Information Theory 37 (1991) 1490–1492.Google Scholar
- [22]P. Raghavan and R. Motwani,
*Randomized Algorithms*(Cambridge Univ. Press, 1995).Google Scholar - [23]S. Ramanathan and E. Lloyd, Scheduling boradcasts in multi-hop radio networks, IEEE/ACM Transactions on Networking 1 (1993) 166–172.Google Scholar
- [24]R. Ramaswami and K. Parhi, Distributed scheduling of broadcasts in radio network, in:
*Proc. of INFOCOM*(1989) pp. 497–504.Google Scholar - [25]S. Ulukus and R.D. Yates, Stochastic power control for cellular radio systems, IEEE Transactions on Communications 46(6) (1996) 784–798.Google Scholar
- [26]L. Valiant, Universality considerations in VLSI circuits, IEEE Transactions on Computers 30 (1981) 135–140.Google Scholar