E. AlthausMax-Plank-Institut fur Informatik G. CalinescuIllinois Institute of Technology I.I. MandoiuUC San Diego S. Prasad Georgia State University N. TchervenskiIllinois Institute of Technology A. ZelikovskyGeorgia State University Power Efficient Range Assignment in Ad-hoc Wireless Networks

2 Cumberland 2003 Outline Motivation Previous work Approximation results Experimental Study

3 Cumberland 2003 Ad Hoc Wireless Networks Applications in battlefield, disaster relief, etc No wired infrastructure Battery operated  power conservation critical

4 Cumberland 2003 Power Attenuation Model Signal power falls inversely proportional to d k, k  [2,4]  Transmission range radius ~ k-th root of power Omni-directional antennas Uniform power attenuation coefficient k Uniform transmission efficiency coefficients Uniform receiving sensitivity thresholds  Transmission range = disk centered at the node  Symmetric power requirements Power(u,v) = Power(v,u)

5 Cumberland 2003 Asymmetric Connectivity Power ranges b a c d g f e Connectivity graph a b d g f e c Multi-hop ACK! a b d g f e c

6 Cumberland 2003 Symmetric Connectivity  Per link acknowledgements a b d g f e c Asymmetric Connectivity Increase range of “b” by 1 Decrease range of “g” by 2 a b d g f e c Symmetric Connectivity

7 Cumberland 2003 Given: set of nodes, coefficient k Find: power levels for each node s.t. –Symmetrically connected path between any two nodes –Total power is minimized Problem Formulation

8 Cumberland 2003 a b d g f e c h Power-cost of a Tree Node power = power required by longest edge Tree power-cost = sum of node powers

9 Cumberland 2003 Given: set of nodes, coefficient k Find: spanning tree with minimum power-cost Reformulation of Min-power Problem

10 Cumberland 2003 Previous Work d Max power objective –MST is optimal [Lloyd et al. 02] Total power objective –NP-hardness [Clementi,Penna,Silvestri 00] –MST gives factor 2 approximation [Kirousis et al. 00] –1+ln2  1.69 approximation [Calinescu,M,Zelikovsky 02]

11 Cumberland 2003 Our results 5/3 approximation factor –NP-hard to approximate within log(#nodes) for asymmetric power requirements Optimum branch-and-cut algorithm –practical up to nodes New heuristics + experimental study

12 Cumberland 2003 MST Algorithm Power cost of the MST is at most 2 OPT (1) power cost of any tree is at most twice its cost p(T) =  u max v~u c(uv)   u  v~u c(uv) = 2 c(T) (2) power cost of any tree is at least its cost (1) (2) p(MST)  2 c(MST)  2 c(OPT)  2 p(OPT)

13 Cumberland 2003 Tight Example 1+  1   11  Power cost of MST is n Power cost of OPT is n/2 (1+  ) + n/2   n/2 n points

14 Cumberland 2003 Gain of a Fork Fork = pair of edges sharing an endpoint Gain of fork F = decrease in power cost obtained by –adding F’s edges to T –deleting longest edges from the two cycles of T+F Gain = =3 a b d g f e c 12 2 h a b d g f e c 12 2 h (+3) (+1) 13 (+3) 2(-10)

15 Cumberland 2003 Approximation Algorithms Every tree can be decomposed into a union of forks s.t. sum of power-costs = at most 5/3 x tree power-cost  Min-Power Symmetric connectivity can be approximated within a factor of 5/3 +  for every  >0

16 Cumberland 2003 Greedy Fork Contraction Algorithm Start with MST Find fork with max gain Contract fork Repeat Greedy Fork Contraction has approximation ratio at most 11/6 = (2+5/3)/2

17 Cumberland 2003 Experimental Setting Random instances with up to 100 points Compared algorithms –Edge switching

18 Cumberland 2003 Edge Switching Heuristic a b d g f e c 12 2 h

19 Cumberland 2003 Edge Switching Heuristic Delete edge a b d g f e c 12 2 h

20 Cumberland 2003 Edge Switching Heuristic Delete edge Reconnect with min increase in power-cost 2 a b d g f e c 12 2 h

21 Cumberland 2003 Experimental Setting Random instances with up to 100 points Compared algorithms –Edge switching –Distributed edge switching –Edge + fork switching –Incremental power-cost Kruskal –Branch and cut –Greedy fork-contraction

22 Cumberland 2003 Percent Improvement Over MST

23 Cumberland 2003 Percent Improvement Over MST

24 Cumberland 2003 Runtime (CPU seconds)

25 Cumberland 2003 Summary Efficient algorithms that reduce power consumption compared to MST algorithm Can be modified to handle obstacles, power level upper- bounds, etc. Ongoing research -Improved approximations / hardness results -Multicast -Dynamic version of the problem (still constant factor)