Symmetric Connectivity With Minimum Power Consumption in Radio Networks

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Symmetric Connectivity With Minimum Power
Consumptionin Radio Networks

• G. Calinescu (IL-IT)
• I.I. Mandoiu (UCSD)
• 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
Nodes transmit messages within a range depending
on their battery power, e.g., agb cgb,d
ggf,e,d,a
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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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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

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Our results

• General graph formulation
• Similarity to Steiner tree problem
• t-restricted decompositions
• Improved approximation results
• 1ln2 ? ? 1.69 ?
• 15/8 for a practical greedy algorithm
• Efficient exact algorithm for Min-Power Symmetric
  Unicast
• 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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Size-restricted Tree Decompositions

• A t-restricted decomposition Q of tree T is a
  partition into edge-disjoint sub-trees with at
  most t vertices
• Power-cost of Q sum of power costs of sub-trees
  
• ?t supT min p(Q)Q t-restricted decomposition
  of T / p(T)
• E.g., ?2 2

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Size-restricted Tree Decompositions

• Theorem For every T and t, there exists a
  2t-restricted decomposition Q of T such that p(Q)
  ? (11/t) p(T)
• ?t ? 1 1 / ?log k?
• ?t ? 1 when t ? ?

• Theorem For every T, there exists a 3-restricted
  decomposition Q of T such that p(Q) ? 7/4 p(T)
• ?3 ? 7/4

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Gain of a Sub-tree

• t-restricted decompositions are the analogue of
  t-restricted Steiner trees
• Fork sub-tree of size 2 pair of edges sharing
  an endpoint
• The gain of fork F w.r.t. a given tree T
  decrease in power cost obtained by
• adding edges in fork F to T
• deleting two longest edges in two cycles of TF

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Approximation Algorithms

• For a sub-tree H of G(V,E) the gain w.r.t.
  spanning tree T is defined by
• gain(H) 2 c(T) 2 c(T/H)
  p(H)
• where G/H G with H contracted to a single
  vertex
• Camerini, Galbiati Maffioli 92 / Promel
  Steger 00
• ?3 ? ? 7/4 ? approximation
• t-restricted relative greedy algorithm
  Zelikovsky 96
• 1ln2 ? ? 1.69 ? approximation
• Greedy triple (fork) contraction algorithm
  Zelikovsky 93
• (?2 ?3) / 2 ? 15/8 approximation

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Greedy Fork Contraction Algorithm

• 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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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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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
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Percent Improvement Over MST
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Runtime (CPU seconds)
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Percent Improvement Over MST
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Summary and Ongoing Research

• Graph-based algorithms handle practical
  constraints
• Obstacles, power level upper-bounds
• Improved approximation algorithms based on
  similarity to Steiner tree problem in graphs
• Ideas extend to Min-Power Symmetric Multicast
• Ongoing research
• -- Every tree has 3-decomposition with at most
  5/3 times larger power-cost
• 5/3? approximation using Camerini et al. 92 /
  Promel Steger 00
• 11/6 approximation factor for greedy
  fork-contraction algorithm

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Symmetric Connectivity With Minimum Power
Consumptionin Radio Networks

• G. Calinescu (IL-IT)
• I.I. Mandoiu (UCSD)
• A. Zelikovsky (GSU)