Power Efficient Range Assignment in Ad-hoc Wireless Networks

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Power Efficient Range Assignment in Ad-hoc
Wireless Networks
ES0036

• E. Althous (MPI)
• G. Calinescu (IL-IT)
• I.I. Mandoiu (UCSD)
• S. Prasad (GSU)
• N. Tchervinsky (IL-IT)
• A. Zelikovsky (GSU)

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Ad Hoc Wireless Networks

• Applications in battlefield, disaster relief,
  etc.
• No wired infrastructure
• Battery operated ? power conservation critical
• Omni-directional antennas Uniform power
  detection thresholds
• ?Transmission range disk centered at the node
• Signal power falls inversely proportional to dk
• ?Transmission range radius kth root of node
  power

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Asymmetric Connectivity
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Range radii
Strongly connected
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Nodes transmit messages within a range depending
on their battery power, e.g., agb cgb,d
ggf,e,d,a
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Message from a to b has multi-hop
acknowledgement route
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Symmetric Connectivity

• Per link acknowledgements ? symmetric
  connectivity
• Two nodes are symmetrically connected iff they
  are within transmission range of each other

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Min-power Symmetric Connectivity Problem

• Given set S of nodes (points in Euclidean
  plane), and coefficient k
• Find power levels for each node s.t.
• There exist symmetrically connected paths between
  any two nodes of S
• Total power is minimized

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Results

• Previous results
• Max power objective
• MST is optimal Lloyd et al. 02
• Total power objective
• NP-hardness Clementi,PennaSilvestri 00
• MST gives factor 2 approximation Kirousis et al.
  00
• Our results
• General graph formulation
• Improved approximation results
• 5/3 ?
• 11/6 for a practical greedy algorithm
• New ILP formulation
• Several swapping heuristics
• Experimental study

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Graph Formulation

• Power cost of a node maximum cost of the
  incident edge
• Power cost of a tree sum of power costs of its
  nodes
• Min-Power Symmetric Connectivity Problem in
  Graphs
• Given edge-weighted graph G(V,E,c), where c(e)
  is the power required to establish link e
• Find spanning tree with a minimum power cost

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MST Algorithm

• Theorem The power cost of the MST is at most 2
  OPT
• Proof
• 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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Greedy Fork Contraction Algorithm

• Fork F is the set of two adjacent edges
• Gain of fork F, gain(F), is by how much
  inserting of F and removing other two edges
  improves the power cost
• Input Graph G(V,E,cost) with edge costs
• Output Low power-cost tree spanning V
• TfMST(G)
• Hf?Repeat forever
• Find fork F with maximum gain
• If gain(F) is non-positive, exit loop
• HfH U F
• TfT/F
• Output T ? H

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Edge Swapping Heuristic

• For each edge do
• Delete an edge
• Connect with min increase in power-cost
• Undo previous steps if no gain

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Remove edge 10 power cost decrease -6
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Reconnect components with min increase in
power-cost 5
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Integer Linear Program Formulation

• yuv range variable, 1 if for uv is maximum
  weight edge from u in tree T
• xuv tree variable, 1 if uv is in tree T

- choose a single power range - power range
connects endpoints - connectivity requirement
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Experimental Study

• Random instances up to 100 points
• Compared algorithms
• branch and cut based on novel ILP formulation
  Althaus et al. 02
• Greedy fork-contraction
• Incremental power-cost Kruskal
• Edge swapping
• Delaunay graph versions of the above

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Percent Improvement Over MST
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Runtime (CPU seconds)
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Percent Improvement Over MST