E' AlthausMaxPlankInstitut fur Informatik

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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

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Outline

• 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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Symmetric Connectivity

• ?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
d
Tree power-cost sum of node powers
f
c
g
b
a
h
e
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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
• 1ln2 ? 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 maxvuc(uv) ? ?u ?vu
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
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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 Fs edges to T
• deleting longest edges from the two cycles of TF

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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
?gt0
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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 (25/3)/2

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Experimental Setting

• Random instances with up to 100 points
• Compared algorithms
• Edge switching

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Edge Switching Heuristic
2
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Edge Switching Heuristic

• Delete edge

2
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Edge Switching Heuristic

• Delete edge
• Reconnect with min increase in power-cost

2
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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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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
  upper-bounds, etc.
• Ongoing research
• Improved approximations / hardness results
• Multicast
• Dynamic version of the problem (still constant
  factor)