1. 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

2. Outline Motivation Previous work Approximation results
Experimental Study
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3. Ad Hoc Wireless Networks
Applications in battlefield, disaster relief, etc
No wired infrastructure
Battery operated  power conservation critical
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4. 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)
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5. 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
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6. Symmetric Connectivitya 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
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7. 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
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8. 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
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9. Reformulation of Min-power Problem
Given: set of nodes, coefficient k
Find: spanning tree with minimum power-cost
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10. 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
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11. 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 nodes
New heuristics + experimental study
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12. 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)
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13. 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
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14. 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 8 10 13
13(+3)
13 (+1)
13 (+3)
2(-10)
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15. 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
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16. Experimental Setting Random instances with up to 100 points
Compared algorithms
Edge switching
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17. Edge Switching Heuristica b d g f e c 12 2 h 4 15 10 13 2
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18. Edge Switching Heuristic
Delete edge a b d g f e c 12 2 h 4 13 2
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19. 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
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20. 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
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21. Greedy Fork Contraction Algorithm
Start with MST
Find fork with max gain
Contract fork
Repeat
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22. Percent Improvement Over MST
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23. Percent Improvement Over MST
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24. Runtime (CPU seconds)
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25. 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)
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