Sparse Power Efficient Topology for Wireless Networks

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Sparse Power Efficient Topology for Wireless
Networks

• X.Y. Li, P.J. Wan, Y. Wang, O. Frieder
• Proceedings of the 35th Annual Hawaii
  International Conference, 2002 Pages3839 - 3848

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Outline

• Introduction
• Several well-know proximity graphs
• The Algorithms
• Experimental Results
• Summary and Future Work

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Introduction

• Limited resource in wireless ad hoc networks is
  crucial for network operations.
• One efficient only a linear number of links using
  a localized construction methods.
• The sparseness of a constructed networks topology
  should not compromise too much on the power
  consumptions on both unicast and
  broadcast/multicast communications.
• A network topology is said to be power efficient
  if there is a power efficient route to connect
  any two nodes.
• There is a tradeoff between the sparseness of the
  topology and its power efficiency.

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• Sparseness
• The number of edges
• The number of edges is a O(n)
• The nodes degree
• The maximum node degree is bounded by a constant
• Power efficiency
• Length stretch factor
• A subgraph H is called a length spanner of a
  graph G if there is a positive real constant ?
  such that for any two nodes u and v, the shortest
  path lH(u,v) in H is at most ? times of the
  shortest path lG(u,v) in G.
• Power stretch factor
• A subgraph H is called a power spanner of a graph
  G if there is a positive real constant ? such
  that for any two nodes u and v, the minimum power
  path pH(u,v) in H is at most ? times of the
  minimum power path pG(u,v) in G.

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G
H
v
v
u
u
pG(u,v) 13
pG(u,v) 10
Maximal node degree (G) 6
Maximal node degree (G) 3
Power stretch factor of H with respect to G
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• This paper consider the following network
• Consist a set V of wireless nodes distributed in
  a two-dimensional plane
• Each node has an omni-directional antenna
• All node have the maximum transmission rnage
  equal to one unit
• Power-attenuation mode is uvß, 2?ß?4
• This network can be modeled as a unit disk graph
  UDG(V)
• there is a edge uv if and only if uv is at
  most one

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• Lemma 2
• For any H ? G with length stretch factor ?, the
  power stretch factor is at most ?ß
• It implies that a graph with a constant bounded
  length stretch factor must also have a constants
  bounded power stretch factor.
• But the reverse is not necessary true.

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Several well-know proximity graphs

• RNG (Constrained Relative neighborhood graph)
• It has a edge uv if and only if uv ? 1 and
  there is no point w ?V such that uw ? uv
  and vw ? uv

u
v
GG can be computed locally
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• GG (Constrained Gabriel graph)
• It has a edge uv if and only if uv ? 1 and
  there is no point w ?V such that w is in the open
  disk using uv as a diameter

u
v
RNG can be computed locally
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• YGk (Constrained Yao graph) with an parameter k,
  k ? 6

YG can be computed locally
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• EMST ? RNG ? GG
• EMST ? RNG ? YG

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• RNG, GG, YGk can be computed locally using 1-hop
  information (information within the maximum
  transmission range of a node)
• Localized algorithm
• Every node u can exactly decide all edges
  incident on u based only the information of all
  nodes within a constants hops of u
• This paper discusses several combinations of the
  GG and YG using information from constant hops.

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The Algorithms

• First Yao then Gabriel graph GYGk(V)
• First Gabriel then Yao YGGk(V)
• The second phase in two algorithm can further
  improve the sparseness
• But, the theoretical results are remaining the
  same as YG(V)
• Lemma 4 if UDG(V) is connected and k gt 6,
  GYGk(V) and YGGk(V) are connected

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?
?
?
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• YGk, YGGk and GYGk have a bounded power stretch
  factor and out-degree k for each.
• But some nodes may have a very large in-degree.

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• YGk (Yao and Sink)
• Replace the directed star consisting of all links
  towards a node u by a directed tree T(u) as the
  sink.

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• YYk (Yao plus Reverse Yao Graph)

• Theorem YYk is strongly connected

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Experimental Results

• Node degree

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• Power stretch factor

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Future Work

• It is still open problem whether YY has a bounded
  power stretch factor theoretically
• Consider incorporating the receiver cost c into
  Power-attenuation model to reflect the actual
  transmission cost
• Design localized algorithms with lower power
  stretch factor and maximum node degree.

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The END