CS547: Wireless Networking

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CS547 Wireless Networking

• Lecture 5 Max-Life Power Scheduling

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Network Model

• Distributed over a plane
• Equal maximum transmission range
• Topology G unit-disk graph

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Energy Conservation via Adjustable Transmission
Range
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Range Assignment Induced by A Subgraph H

• Set of neighbors of u
• Transmission range of u
• Transmission power of u
• Power cost of H

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Max-Life Range Assignment for Property P

• Input A unit-disk graph G ?P and the total
  energy b(u) of each node u.
• Output a collection F of subgraphs H ?P of G
  and their lives such that for
  each node u,
• Measure

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Example Network Life of Connectivity
H1
H2
G
b(u) 91625 for all u, ? 2
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Packing Linear Program Formulation

• V constraints a solution with at most V
  subgraphs
• P variables exponential

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Minimum Power Assignment for Property P with
Unit-costs

• Input a unit-disk graph G ?P, and the unit-cost
  c(u) of each node u.
• Output a subgraph H ?P of G
• Measure

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Modified Garg-Köneman Algorithm

• Initialize
• While Dlt1
• Find a topology H ?P using an (approximation)
  algorithm A. on the instance (G, c)
• Compute the bottleneck node v with the minimum
  b(v)/pH(v)
• If H ?F, then lH ? lH b(v)/pH(v) else F ? F
  ?H, lH ? b(v)/pH(v).
• For each node u, c(u) ? c(u) 1?(b(v)/pH(v)/
  b(u)/pH(u)
• D ??ub(u) c(u)
• Output

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Approximation Ratio

• If A is a ?-approximation algorithm for Minimum
  Power Assignment for Property P with Efficiency,
  then Garg-Köneman Algorithm with A is a ?(1
  ?)-approximation algorithm for Max-Life Power
  Assignment for Property P

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A General Approximation Scheme for Min-Power
Assignment

• For each edge uv of G, define a weight
• Find a subgraph H ?P of G with small weight

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Power vs. Weight

• For any graph H,

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Power vs. Weight

• Let be the golden ratio
• Lemma 1 Let H be a min-power spanning tree of G
  with minimal total Euclidean length. Then
• Lemma 2 Let H be a min-power biconnected
  spanning subgraph of G with minimal total
  Euclidean length. Then

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Approximation Algorithms for Connectivity And
Biconnectivity

• Connectivity Find the MST of
• Biconnectivity Using the 2-approximation
  algorithm to find a biconnected spanning subgraph
  H of

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A Property of Golden Ratio

• Let xyz be a triangle with
  . Then

1, g
1, g
gt1
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Proof of Lemma 1

• Only need to prove that
• Only need to consider ? 2

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Proof of Lemma 1

• Partition the neighborhood into outer-closed
  inner-open annuli of geometrically decreasing
  radii
• Each annulus contains at most 9 neighbors
  edge-switching argument

u
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Edge Classification w.r.t. Rooted Spanning Tree
tree edge
no cross edge ?DFS tree
back edge
cross edge
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Characterization of Biconnectivity

• Root of the DFS tree has exactly one child
• Each maximal subtree rooted at u ? root has at
  least one back edge from one of its nodes to an
  ancestor of u.
• Minimal chord-free

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Biconnectivity-Preserving Edge-Switching
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Proof of Lemma 2

• Only need to prove that

u
u


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Open Problems k-Connectivity for k ? 3

• O(k)-approx. without unit-costs
• O(k)-approx. with unit-costs
• classification of edges
• edge-switching preserving k-connectivity

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Open Problems Broadcast

• Spanning arborescence
• Constant-approx. without unit-costs
• log-approx. with unit-costs
• Constant-approx. with unit-costs

s
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Open Problems Multicast

• Spanning Steiner arborescence
• Constant-approx. without unit-costs
• Directed Steiner tree based approx. with
  unit-costs
• log. or constant-approx. with unit-costs