An O(log n) Dominating Set Protocol for Wireless Ad-Hoc Networks under the Physical Interference Model

1
An O(log n) Dominating Set Protocol for Wireless
Ad-Hoc Networks under the Physical Interference
Model

• Andrea W. Richa
• Arizona State University
• Joint work with Christian Scheideler and Paolo
  Santi

2
Wireless Ad-hoc Networks

• Mobile stations communicating over wireless
  medium
• Challenges
• design appropriate models
• design and analyze algorithms under these models

3
Wireless Ad-hoc Networks

• Wireless communication very difficult to model
  accurately
• Signal propagation
• Interference
• Mobility
• Physical Carrier Sensing
• Algorithms are very difficult to analyze under a
  very accurate model
• Find balance between accuracy and provability.

4
UDG What is the problem?

• Unit-Disk Graph (UDG)
• Given a transmission radius R, nodes u, v are
  connected iff d(u,v) R
• Problems
• Transmission range could be of highly nonuniform
  shape
• Does not consider interference

u
R
v
5
Packet Radio Network What is the problem?

• Can handle arbitrary transmission shapes
• Nodes u, v can communicate directly iff they are
  connected.
• Interference Model
• (interference range) (transmission range)
• Problem linear slowdown if interference range is
  larger than transmission range

6
PRN What is the problem?
rt
v
s
rt
rt
t
ri
n-2 nodes

• While in the PRN model, s can send a message to t
  in 2 steps, no uniform protocol can successfully
  send a message in expected o(n) number of steps
  linear slowdown

7
Bounded Interference Models

• Transmission and Interference Ranges
• Separate values.
• Interference range constant times bigger than
  transmission range.
• Problem?

u
ri
rt
does not cause interference at u (even if all
nodes outside transmit at the same time)
may cause interference at u
8
Physical Interference
Reality looks more like this
transmission range
u
interference
9
Bounded x Physical Interference Bad News

• Bad news
• Blough, Canali, Resta, and Santi08combined
  interference from far-away nodes cannot be
  neglected
• bounded interference model neglected
  interference can be two orders of magnitude
  greater than noise floor
• simulations 210 loss in throughput when
  interference from far away nodes taken into
  account
• (We will see some good news later)

10
Dominating Set Problem

• Classical dominating set problem
• Given a graph G(V,E) , find a subset U V of
  minimum size so that for every node v in V,
  either v is in U or v has a neighbor in U.

11
Dominating Set Problem
Wireless setting First formally analyzed for
unit disk graph model.
12
Is dominating set problem still relevant in
general setting?

• Studies fundamental problem of selecting local
  leaders of constant density that cover entire
  network area.
• Building block for many other problems in
  wireless networks.
• constant density at most a constant number of
  nodes in any constant size area.
• Our goalConstruct node set U of constant
  density via simple, local-control algorithm under
  the physical interference model so that all nodes
  v in V\U can receive messages from a node in U
  (i.e., U is coordinator set).

13
Bounded x Physical Interference Good news

• Blough, Canali, Resta and Santi 08If nodes
  have constant density, then physical (SINR)
  interference model reduces to bounded
  interference model.

14
Overview of Talk

• Our model
• Signal propagation
• Interference model
• Physical carrier sensing
• The dominating set problem
• Our contribution
• TWIN protocol
• Algorithm
• Analysis
• Future Work

15
Signal Propagation

• Log-normal shadowing model
• d0 reference distance
• ?gt2 path loss coefficient
• Signal loss at distance d in dB
• -10 log(d/d0)? X?
• for some Gaussian RV X?

16
Signal Propagation

• Log-normal shadowing model without X?
• P signal strength at d01
• signal strength at distance dgt1 P/d?

17
Signal Propagation

• Our model
• Non-symmetric function c(v,w) (1?)-1
  d(v,w), (1?) d(v,w)
• accounts for nonuniform variations of
  communication environment
• Received power (or signal strength) from v at w
  Pw(v)P/c(v,w)?

18
Signal Propagation

• random function c approximates well (a truncated
  form of) the log-normal shadowing model

19
Transmission Range

• forward error correction transition between
  being able to correctly receive a message
  (w.h.p.) and not being able to correctly receive
  a message (w.h.p.) is less than 1dB

sharp boundary
w
v
u
20
Physical Interference (SINR)

• u receives msg from vif and only if
  Pu(v) N?w Pu(w)N background noise
• Received power from v at w Pw(v)P/c(v,w)?

gt ?
v
u
21
Physical Carrier Sensing

• Provided by Clear Channel Assessment (CCA)
  Circuit
• Monitors the medium as a function of Received
  Signal Strength Indicator (RSSI)
• Energy Detection (ED) bit set to 1 if RSSI
  exceeds a certain threshold
• Has a register to set the threshold TSo v can
  check if N?w Pv(w) gt T

22
Overview of Talk

• Our model
• Signal propagation
• Interference model
• Physical carrier sensing
• Prior work and our Contribution
• TWIN protocol
• Algorithm
• Analysis
• Future Work

23
Prior Work

• Modelling
• Log-normal shadowing model and physical
  interference model common in physical layer
  community
• Gupta and Kumar 00, Grossglauser and Tse 01
  capacity of wireless networks
• Brar, Blough, Santi 06 and Moscibroda,
  Wattenhofer, Zollinger 06 transmission
  scheduling
• Goussevskaia, Moscibroda, Wattenhofer 08
  broadcasting
• Dominating sets
• Luby 85, Alzoubi et al 02, Dubhashi et al 03,
  Kuhn et al 03, Huang et al 04, UDG
• Kuhn et al 04, Partasarathy and Gandhi 04
  protocols for bounded interference model (runtime
  O(log2 n) )
• Kothapalli et al 05 protocol for more general
  bounded interference model with physical carrier
  sensing (runtime O(log4 n) )

24
Dominating Set Problem

• V set of n nodes of arbitrary distr. in IR2
• c non-symmetric cost function
• Find subset U of V of constant density so
• that for every v in V
• either v in U
• or there is a w in U with Pv(w) gt ?N.

v can receive msg from w
25
Our Contribution

• More general model for theoretical analysis
  (hopefully closer to reality)
• Theorem. TWIN protocol establishes a constant
  density dominating set in O(log n) time w.h.p.
• Main ideas
• Extensive use of physical carrier sensing
• Leaders emerge in twins (if possible)

26
Why Physical Carrier Sensing?

• Using physical carrier sensing, we can extract
  information from the network without relying on
  successful message transmissions
• quite often it is enough just to know if at least
  one node is sending a message, rather than
  receiving the message
• linear speedup
• It comes for free

27
Overview of Talk

• Our model
• Signal propagation
• Interference model
• Physical carrier sensing
• The dominating set problem
• Our contribution
• TWIN protocol
• Algorithm
• Analysis
• Future Work

28
TWIN Protocol

• Nodes do not need any prior knowledge
• All messages of constant size (signals)
• All nodes transmit with same power P
• Nodes may be
• inactive not in dominating set
• twin in dominating set twins come up in pairs
• active single isolated nodes which cannot form
  a twin pair but are still needed for coverage
• acc(v) counter (acc(v)gt0 iff v active)

29
TWIN Protocol

• Nodes operate in synchronized rounds that are
  continuously executed
• Stage 1 announcing active twins
• Stage 2 guessing the right density
• Stage 3 forming new twins

Diff frequency for each time slot no sync
round
stage 1
stage 2
stage 3
30
TWIN Protocol

• For every node v
• Initially, v is inactive and acc(v)0.
• Access probability pv may have any value in (0,
  pmax, where pmaxltlt1.
• D maximum density of twin nodes
• Stage 1 announcing active twins
• Active twin send ACTIVE signal with prob 1/D
• Inactive or active single if v receives ACTIVE
  signal, it terminates and becomes inactive

31
TWIN Protocol

• 0lt?lt1 constant inc/dec step for access
  probability
• Stage 2 guessing the right density
• Inactive or active single v chooses one of two
  time slots at random, say s (other slot s).
• Slot s v sends PING signal with prob pv.If not,
  v senses channel with threshold T
• Slot s v senses channel with threshold T
• v does not sense anything pvmin(1?)pv, pmax
• v senses busy channel pv(1?)-1pv
• If pvpmax then acc(v)acc(v)4, else
  acc(v)maxacc(v)-1,0 (0 inactive)

v is an active single
32
TWIN Protocol

• Stage 3 forming new twins
• Inactive or active single If v sent PING in slot
  s and received PING at slot sin stage 2, then it
  sends ACK in slot s of this stage. If it receives
  an ACK signal in slot s of this stage, v becomes
  an active twin.

PING
ACK
v
w
active twin
active twin, since w must have received
PINGfrom v only (otherwise no ACK from w)
PING
ACK
33
TWIN Protocol

• (Stage 3.) If v just became active twin, v sends
  NEW signal in last slot. If v is inactive or
  active single and senses a busy channel with
  threshold T, then v becomes inactive and
  terminates the protocol

NEW
v
z
inactive or active single
inactive
active twin
sensing range of v
34
Overview of Talk

• Our model
• Signal propagation
• Interference model
• Physical carrier sensing
• The dominating set problem
• Our contribution
• TWIN protocol
• Algorithm
• Analysis
• Towards self-stabilization
• Future Work

35
Analysis Overview

• probabilities pv quickly converge to constant in
  every transmission area
• low runtime constant chance of twins emerging
• constant twin density twins must receive ACKs,
  and NEW signals deactivate local neighborhood
• active singles nodes not covered and not having
  node to pair up with eventually become active
  single if density of active singles beyond
  certain constant, active twin will emerge

36
Getting Down to Constant Density

• Sensing area Rs(v)
• whenever a node in Rs(v) transmits, v will be
  able to sense transmission with threshold T
• Rs(v) R(v), where R(v) is the transmission
  area of node v
• Lemma After logarithmic many steps,
  ?w in RS(v) pw O(1)for a constant fraction
  of the rounds.

constant density
37
Bounding Far-away Interference
bounded interference

• A round is called good iff ?w in R(v) pw O(1)
  and the interference caused by nodes not in R(v)
  is less than T-N
• far-away noise will not trigger busy channel
• Lemma. For any constant egt0, at least (1- e)
  fraction of time steps are good for v w.h.p., if
  T sufficiently large.

38
Quick constant density coverage

• Lemma. After a logarithmic number of steps, for a
  pmax ltlt1, every node v will
• (coverage) either be an active single or have an
  active twin within its transmission range, whp.
  Moreover,
• (constant density) have at most a constant
  number of active singles and twins within its
  transmission range whp

39
Conclusion

• O(log n)-time protocol for dominating set under
  more realistic model
• should be implementable in most simple devices
• possible building block for many other
  applications on top of it
• Open questions
• self-stabilization
• How does protocol perform in practice???
• More robust form (jamming-resistant)

40
Is the model sufficiently realistic?

• Our interference model conservative
• signal cancellation
• different signal strengths
• bit recovery
• fading and other nondeterministic characteristics
  of the wireless signal

41
Towards Self-Stabilization

• Initial pvs can be arbitrary
• Initial acc(v)s can be arbitrary
• Problems
• Termination of protocol not allowed.Instead,
  node should just fall asleep for O(log n) many
  rounds.
• Initial density of active twins might be too
  high.
• Possible solution for 2. add another time slot
  in which active twins check their cumulative
  signal strength (random decision to send or
  sense)
• Problem time of stabilization cannot be bounded
  well, just eventual recovery

42
Questions?
43
Self-Stabilization

• wireless communication too complex no model will
  be able to accurately take into account all that
  can happen
• Problem What happens if things deviate from
  proposed model?
• Solution Protocols need to be self-stabilizing,
  i.e., they need to go back to a valid
  configuration for the model

44
(No Transcript)
45
(No Transcript)