Constant Density Spanners for Wireless AdHoc Networks

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Constant Density Spanners for Wireless Ad-Hoc
Networks

• Discrete Mathematics and Algorithms Seminar
• Melih Onus
• April 5 2005

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Ad-Hoc Networks

• Mobile Devices communicating via radio
• Network without centralized control
• The wireless units, or nodes, are represented by
  a graph, and two nodes are connected by an edge
  if and only if they are within transmission range
  of each other

• Transmissions of messages interfere at a node if
  at least two of its neighbors transmit a message
  at the same time.
• A node can only receive a message if it does not
  interfere with any other message.

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Unit Disk Graph Model

• In theory, its assumed that nodes form a unit
  disk graph

• Two nodes can communicate if they are within
  Euclidean distance 1 (equal transmission ranges)

• Problems In reality
• Signal propagation of real antennas not clear-cut
  disk

• The transmission range of a message is not the
  same as its interference range

• Thus, algorithms designed for unit disk graph
  model may not work well in practice

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Our communication model

• The transmission range of a message is not the
  same as its interference range
• The transmission and interference areas of a node
  are not necessarily disk-shaped
• Provides a realistic model for physical carrier
  sensing

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Our communication model

• A set V of nodes are distributed in an arbitrary
  way in a 2-dimensional Euclidean space

• For a given cost function c and given
  transmission range rt, transmission area of u is
  v?V c(u,v) ? rt

• For given interference range ri, interference
  area of v is u?V c(u,v) ? ri

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Transmission Interference Area

• Node u is guaranteed receive a message from a
  node v in its transmission area as long as there
  is no other node w ? V in its interference area
  that transmits a message at the same time

.
.
u
.
ok!
v
w
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Transmission Range

• Nodes can communicate if distance ? rt/(1 ?)

• Nodes cannot communicate if distance gt rt/(1- ?)

.
u
rt/(1?)

• In range (rt/(1- ?), rt/(1 ?)), it is
  unspecified whether massage arrives

rt/(1-?)
Cost Function c(v,w) ? (1- ?)d(v,w), (1
?)d(v,w)
d(v,w) the Euclidean distance between v and w
? ? 0,1), fixed constant
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Physical carrier sensing

• Nodes cannot only send and receive messages but
  they can also perform physical carrier sensing

.

• Nodes can set their sensing threshold T

.

• Sensing range grows monotonically with T

u
ok!
.
v
w
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Carrier Sense Transmission Interference Areas

• For a given carrier sensing threshold T, carrier
  sensing transmission area of u is v?V c(u,v)
  ? rst(T)

• For a given carrier sensing threshold T, carrier
  sensing interference area of u is v?V c(u,v)
  ? rsi(T)

rst(T) carrier sensing transmission(CST) range
rsi(T) carrier sensing interference(CSI) range
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Carrier Sensing

• If node v transmits a message and v is in the
  CST range of node u, then u senses the message
  transmission

.

• If node u senses a message transmission, then
  there is at least one node w in the CSI area of
  u that transmitted a message

w
.
u
.
ok!
v
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Dominating set

• A dominating set (DS) is a subset of nodes such
  that either a node is in DS or has a neighbor in
  DS.
• A minimum dominating set (MDS) is a DS with
  smallest possible number of nodes

A
B
D
G
G
B
D
A
E
E
C
C
F
F
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Our Results

• The nodes do not know the total number of nodes
• The dominating set protocol generates a constant
  approximation of a MDS in O(log4 n) communication
  rounds, with high probability
• If physical carrier sensing is not available and
  the nodes have no estimate of the size of the
  network, then ?(n) are necessary for obtaining a
  constant approximation of MDS. (Jurdzinski,
  Stachowiak 2002)

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Preliminary Scenario

• rstrt, so CST area is equal to transmission area
• rsiri, so CSI area is equal to interference area

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Preliminary DS Algorithm

• Nodes can either be active or inactive
• The active nodes are the candidates for the
  dominating set
• Algorithm
• If v is active, then v sends out an ACTIVE
  signal. If v is inactive and v did not sense any
  ACTIVE signal, it becomes active again.
• If v is active, then v sends out a LEADER signal
  with probability ½. If v decides not to send out
  a LEADER signal, but senses a LEADER signal from
  at least one other node, then v becomes inactive.
  

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Example I
Active
A
C
A
C
Inactive
Active signal
E
E
Leader signal
Transmission range
B
D
B
D
Interference range
Dominating Set B, C
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Example II
C will sense leader signal of B
Active
A
C
A
C
Inactive
Active signal
E
E
Leader signal
Transmission range
B
D
B
D
Interference range
B is not a dominating set
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Ideas

• There may be active nodes within range rt at the
  end of the algorithm, but at most constant number
  of them
• Distributed Coloring Each node divides the time
  into time frames of k slots for a given constant
  k
• There is no active node with same
  time slot within range ri of an active
  node
• Two different sensing threshold

k is number of active nodes in CSI area of a node
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Sensing Thresholds

• The nodes use two different sensing thresholds,
  Ta and Ti, depending on their state
• The sensing threshold Ta has a CSI range of rt
• The sensing threshold Ti has a CST range of ri

rs
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DS Algorithm

• Time Step I
• If v is active and in its active slot, then v
  sends out an ACTIVE signal
• If v is inactive and v did not sense any ACTIVE
  signal for the last k slots using a sensing
  threshold of Ta, v senses with threshold Ti, and
  if it does not sense anything, it becomes active
  and declares the current slot number as its
  active slot
• If v did sense some ACTIVE signal in one of the
  last k slots, it just performs sensing with
  threshold Ta and records the outcome

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DS Algorithm

• Time Step II
• If v is active and is in its active slot, then v
  sends out a LEADER message containing its ID with
  some fixed probability p
• If v decides not to send out a LEADER message but
  it either senses a LEADER message with threshold
  Ta or receives a LEADER message, v becomes
  inactive.

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Why k slots?

• If an inactive node v sensed an active signal,
  there is at least one active node u in its
  carrier sense interference area
• There is at most constant number of active nodes
  in carrier sense interference area of a node, say
  k
• Choose k as k gt k
• Then, if there is an active node in carrier sense
  interference area of u, but there is no active
  node in its transmission area, then v will be
  active at this slot

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Analysis

• If there is no active node in transmission area
  of an active node u, then u will stay as active
  forever, since inactive nodes cannot be active in
  its slot.
• If u become active after v, then c(u, v) gt rs,
  since u will sense all k slots before becoming
  active.

rs is the CST range when CSI range is equal to rt
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Analysis

• A node u is called leader if it is active and
  there is no other active node v of same color
  with c(u,v) ? rt
• Lemma Every connected component of active nodes
  needs a most O( log n) steps, w.h.p., until every
  node in it either becomes inactive or becomes a
  leader

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Analysis

• Lemma At any time, if active, nonleading nodes
  cover an area A?(log3n), the number of leaders
  emerging from these nodes is ?(A/log2n), w.h.p.
• Theorem If all nodes are initially inactive,
  then after O(log4 n) rounds of the algorithm, the
  leaders form a static dominating set of constant
  density, with high probability.

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Assumptions

• Fixed identification numbers of any form are not
  required
• The nodes do not know the total number of nodes
• We only require that the mobile hosts can
  synchronize up to some reasonably small time
  difference, which can be done, for example, with
  the help of GPS signals

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Constant density spanner

• Constant density spanner Given a graph G find
  subgraph G of G such that distance of two nodes
  in G is less than a constant factor of original
  distance
• Dominating Set
• Distributed Coloring
• Gateway Selection

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Conclusion

• More realistic transmission and interference
  model
• New communication model that considers physical
  sensing
• Polylogarithmic constant approximation DS
  algorithm under the realistic wireless model

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References

• K. Kothapalli, C Scheideler, M. Onus, A. Richa.
  Constant Density Spanners For Wireless Ad-hoc
  Networks, submitted to SPAA 05
• T. Jurdzinski, G. G. Stachowiak. Probabilistic
  algorithms for the wakeup problem in single hop
  radio networks, ISAAC 535-549, 2002
• Fabian Kuhn, Thomas Moscibroda, and Roger
  Wattenhofer, Initializing Newly Deployed Ad Hoc
  and Sensor Networks, MOBICOM, Philadelphia, USA,
  September 2004.