Power Efficient Range Assignment in Ad-hoc Wireless Networks E. Althaus Max-Plank-Institut fur Informatik G. Calinescu Illinois Institute of Technology I.I. Mandoiu UC San Diego S. Prasad Georgia State University N. Tchervenski Illinois Institute of Technology A. Zelikovsky Georgia State University

Outline Motivation Previous work Approximation results Experimental Study WCNC 2003

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

Power Attenuation Model Signal power falls inversely proportional to dk, 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) WCNC 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 WCNC 2003

Symmetric Connectivity a 2 3 1 b d g f e c Asymmetric Connectivity Increase range of “b” by 1 Decrease range of “g” by 2 a 2 1 b d g f e c Symmetric Connectivity Per link acknowledgements WCNC 2003

Problem Formulation 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 WCNC 2003

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

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

Previous Work Max power objective Total 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] d WCNC 2003

Our results 5/3 approximation factor Optimum branch-and-cut algorithm NP-hard to approximate within log(#nodes) for asymmetric power requirements Optimum branch-and-cut algorithm practical up to 35-40 nodes New heuristics + experimental study WCNC 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 maxv~uc(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) WCNC 2003

Tight Example  Power cost of MST is n 1+  1  Power cost of MST is n Power cost of OPT is n/2 (1+ ) + n/2   n/2 n points WCNC 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 = 10-3-1-3=3 a b d g f e c 12 2 h 8 10 13 13(+3) 13 (+1) 13 (+3) 2(-10) WCNC 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 WCNC 2003

Experimental Setting Random instances with up to 100 points Compared algorithms Edge switching WCNC 2003

Edge Switching Heuristic a b d g f e c 12 2 h 4 15 10 13 2 WCNC 2003

Edge Switching Heuristic Delete edge a b d g f e c 12 2 h 4 13 2 WCNC 2003

Edge Switching Heuristic Delete edge Reconnect with min increase in power-cost a b d g f e c 12 2 h 4 13 15 2 WCNC 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 WCNC 2003

Greedy Fork Contraction Algorithm Start with MST Find fork with max gain Contract fork Repeat WCNC 2003

Percent Improvement Over MST WCNC 2003

Percent Improvement Over MST WCNC 2003

Runtime (CPU seconds) WCNC 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) WCNC 2003