Fault Tolerant Deployment and Topology Control in Wireless Networks

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Fault Tolerant Deployment and Topology Control
in Wireless Networks
MOBIHOC 2003
X.-Y. Li, P.-J. Wan, Y. Wang, and C.-W. Yi Dept.
of Computer Science Illinois Institute of
Technology

• Presented by Ning Li
• November 22, 2009

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Outline

• Introduction
• Related Work
• Fault Tolerant Deployment
• Topology Control for Fault Tolerance
• Simulations

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Introduction

• k-connected (k-vertex connect) if for each pair
  of vertices, there are k mutually vertex disjoint
  (except end-vertices) paths connecting them.
  Equivalently, a graph is k-connected if there is
  no a set of k-1 nodes whose removal will
  partition the network into at least two
  components.
• k-edge connected if for each pair of vertices,
  there are k mutually edge disjoint paths
  connecting them.
• Fault Tolerance a k-connected wireless network
  can sustain the failure of k-1 nodes.

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Point Process

• Uniform Random Point Process Xn consists of n
  independent points each of which is uniformly and
  independently distributed over a region ?.
• Homogeneous Poisson Process the number of points
  in a region is a random variable depending only
  on the volume of the region

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Related Work (1)

• P. Gupta and P. R. Kumar. Critical power for
  asymptotic connectivity in wireless networks.
  Stochastic Analysis, Control, Optimization and
  Applications, 1998.
• Assumptions
• n nodes are independently and uniformly
  distributed in a disk of unit area in ?2
• Assume a common range rn
• Two nodes can communicate with each other if the
  distance is ? rn.
• Objective Find the critical range rn.
• Results Let , the network is
  asymptotically connected w.p.1 if and is
  asymptotically disconnected w.p.1 if

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Related Work (2)

• M. Penrose. The longest edge of the random
  minimal spanning tree. Annals of Applied
  Probability, 1997.
• Results The longest edge Mn of the minimum
  spanning tree of n points randomly and uniformly
  distributed in a unit area square C satisfies
  that

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Related Work (3)

• F. Xue and P. R. Kumar, The number of neighbors
  needed for connectivity of wireless networks. To
  appear in Wireless Networks.
• Assumptions
• n nodes are independently and uniformly
  distributed in a square of unit area in ?2
• Each node is connected to its ?n nearest
  neighbors.
• Objective Find ?n such that the resulting
  network topology is asymptotically connected.
• Results ?(log n) neighbors are necessary and
  sufficient, i.e., there are two constants such
  that (c10.074 and
  c2gt2/log(4/e)5.1774)

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Related Work (4)

• M. Penrose. On k-connectivity for a geometric
  random graph. Random Structures and Algorithms,
  1999.
• Assumptions
• n nodes are independently and uniformly
  distributed in a unit cube in ?d
• Two nodes can communicate with each other if the
  distance is ? r (common range).
• Results Let ?n denote the minimum r at which the
  graph is k-connected and let ?n denote the
  minimum r at which the graph has minimum degree
  k, then P?n?n?1 as n??.

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Fault Tolerant Deployment

• How is the transmission range related to the
  number of nodes in a fixed area such that the
  resulted network can sustain k fault nodes with
  high probability?
• Consider a homogeneous Poisson point process of
  rate n, denoted by Pn, on a unit-area square C.

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Preliminaries

• Let denote the expected number of points of
  Pn with degree k in a graph of G(Pn, r).
• Let D(x,r) be the disk centered at x with radius
  r.
• Given a point x, let vr(x) be the area of the
  intersection of D(x,r) with the unit-area square
  C.
• The probability that point x has degree k is
• We have

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Preliminaries
Where is the minimum radius r at which
the graph G(Pn, r) has the minimum degree at
least k1.
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Lower Bound
is a monotone increasing function of ,
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Upper Bound

• Region I
• Region II

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Upper Bound

• Let w(x) be any function s.t. w(x) vr(x) and is
  monotone increasing of r.
• Let
• Thus,
• Let r be the solution of
• Let r be the solution of
• r r.

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Upper Bound
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Topology Control for Fault Tolerance

• Given a k-fault-tolerant deployment of wireless
  nodes, find a localized method to control the
  network topology such that the resulting topology
  is still fault tolerant but with much fewer
  communication links maintained.
• We show that the constructed topology has only
  linear number of links and is a length spanner.
• Let ?G(u,v) be the shortest path connecting u and
  v in a weighted graph G, and ?G(u,v) be the
  length of ?G(u,v). Then a graph G is a t-spanner
  of a graph H if V(G)V(H) and, for any two nodes
  u and v of V(H), ?G(u,v)t?H(u,v). We also
  call t the length stretch factor of the spanner G.

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Yao Graph

• Yao Graph at each node u, any p equal-separated
  rays originated at u define p equal cones. In
  each cone, choose the shortest (directed) edge uv
  ? G, if there is any, and add a directed link uv.
  Ties are broken arbitrarily.
• For fault tolerance in each cone, node u chooses
  the k1 closest nodes in that cone, if there is
  any, and add directed links from u to these nodes.

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Yp,k1 is a spanner
Proof Need to show that Let We have Stretch
ratio
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Simulations Probability of k-connected
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Probability of a graph with minimal degree k is
k-connected
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