Power Efficient
Range Assignment in
Ad-hoc Wireless Networks
E. Althaus
G. Calinescu
I.I. Mandoiu
S. Prasad
N. Tchervenski
A. Zelikovsky
Max-Plank-Institut fur Informatik
Illinois Institute of Technology
UC San Diego
Georgia State University
Illinois Institute of Technology
Georgia State University
Outline
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Motivation
Previous work
Approximation results
Experimental Study
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Ad Hoc Wireless Networks
• Applications in battlefield, disaster relief, etc
• No wired infrastructure
• Battery operated  power conservation critical
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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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Asymmetric Connectivity
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Connectivity graph
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Multi-hop ACK!
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Power ranges
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Symmetric Connectivity
Asymmetric Connectivity
Symmetric Connectivity
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Increase range of “b” by 1
Decrease range of “g” by 2
Per link acknowledgements
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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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Power-cost of a Tree
Node power = power required by longest edge
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Tree power-cost = sum of node powers
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Reformulation of Min-power Problem
• Given: set of nodes, coefficient k
• Find: spanning tree with minimum power-cost
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Previous Work
• 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]
d
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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 35-40 nodes
• New heuristics + experimental study
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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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Tight Example
n points
1
1+ 

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
1+ 
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
1+ 
Power cost of MST is n
Power cost of OPT is n/2 (1+ ) + n/2   n/2
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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
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Gain = 10-3-1-3=3
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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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Experimental Setting
• Random instances with up to 100 points
• Compared algorithms
– Edge switching
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Edge Switching Heuristic
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Edge Switching Heuristic
• Delete edge
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Edge Switching Heuristic
• Delete edge
• Reconnect with min increase in power-cost
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Experimental Setting
• Random instances with up to 100 points
• Compared algorithms
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Edge switching
Distributed edge switching
Edge + fork switching
Incremental power-cost Kruskal
Branch and cut
Greedy fork-contraction
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Greedy Fork Contraction Algorithm
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Start with MST
Find fork with max gain
Contract fork
Repeat
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Percent Improvement Over MST
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Percent Improvement Over MST
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Runtime (CPU seconds)
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Summary
• Efficient algorithms that reduce power consumption compared
to MST algorithm
• Can be modified to handle obstacles, power level upperbounds, etc.
• Ongoing research
- Improved approximations / hardness results
- Multicast
- Dynamic version of the problem (still constant factor)
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