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Critical Density Thresholds and Complexity in Wireless Networks
Bhaskar Krishnamachari
School of Electrical and Computer Engineering Cornell University
Department of Electrical EngineeringUniversity of Southern California (Fall 2002)
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Large Scale Wireless Networks: The VisionLarge Scale Wireless Networks: The Vision The “many - tiny” principle: wireless networks of thousands of inexpensive
miniature devices capable of computation, communication and sensing For smart spaces, environmental monitoring ...
Berkeley MoteBerkeley MoteFrom Pister et al., Berkeley Smart Dust Project
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Tomorrow’s Embedded Wireless SystemsTomorrow’s Embedded Wireless Systems
2From Manges et al., Oak Ridge National Laboratory, Instrumentation and Controls Division
ORNL Telesensor ChipORNL Telesensor Chip22
Berkeley Dust MoteBerkeley Dust Mote11
1From Pister et al., Berkeley Smart Dust Project
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ChallengesChallenges Unattended Operation: adaptive, self-configuration mechanisms Severe energy constraints : no battery replacement Large Scale : possibly thousands of nodes Are there qualitatively new phenomena at this scale?
Berkeley MoteBerkeley MoteFrom Pister et al., Berkeley Smart Dust Project
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Complex SystemsComplex Systems Ordered global behavior and structure “emerging” from
multiple local interactions.
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From the Oxford Cryodetector Group
Phase TransitionsPhase Transitions
Emergent structure - abrupt change in a global system property at a critical level of local interactions.
Example: SuperconductivityExample: Superconductivity
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Connectivity in a Multi-hop NetworkConnectivity in a Multi-hop Network
© 2002 Bhaskar Krishnamachari
All nodes increasing transmission range R simultaneously
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© 2002 Bhaskar Krishnamachari
Phase Transition for ConnectivityPhase Transition for Connectivity
Communication radius RProb
abilit
y (N
etwo
rk is
Con
nect
ed)
Communication radius RProb
abilit
y (N
etwo
rk is
Con
nect
ed)
undesirableregime
desirableregime
Energy-efficient operating point
n = 20
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© 2002 Bhaskar Krishnamachari
Phase Transition for ConnectivityPhase Transition for Connectivity
Communication radius R
Prob
abilit
y (N
etwo
rk is
Con
nect
ed)
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© 2002 Bhaskar Krishnamachari
Phase Transition for ConnectivityPhase Transition for Connectivity
xx11 xx22 xx33 xx44
RR
2D model of Gupta & Kumar (1998) shows log n density threshold; involves continuum percolation theory. Simpler 1D Model. Consider Poisson arrivals with rate . Connectivity between nodes within range R.
Probability that first n nodes form a connected network:
This model also shows an analogous phase transition with an O(log n) density threshold function.
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© 2002 Bhaskar Krishnamachari
Phase Transition for Bi-ConnectivityPhase Transition for Bi-Connectivity
2 node-disjoint paths between all pairs of nodes.
n = 100
Communication radius R
Prob
abilit
y (N
etwo
rk is
Con
nect
ed)
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© 2002 Bhaskar Krishnamachari
Bernoulli Random Graphs G(n,p)Bernoulli Random Graphs G(n,p)
Studied by mathematicians since 50’s (Erdos, Renyi, Bollobas, Spencer, others)
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© 2002 Bhaskar Krishnamachari
Phase Transitions in Random GraphsPhase Transitions in Random Graphs
A number of zero-one laws developed over the years. E.g. (Fagin 1976): for all first order graph properties A,
(Friedgut 1996): ALL monotone graph properties undergo sharp phase transitions.
Examples: k-Connectivity k-Colorability Hamiltonian cycle
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© 2002 Bhaskar Krishnamachari
Fixed Radius Random Graphs G(n,R)Fixed Radius Random Graphs G(n,R)
A model for multi-hop wireless networks Density parameter: communication radius R
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© 2002 Bhaskar Krishnamachari
Phase Transition for Hamiltonian Cycle FormationPhase Transition for Hamiltonian Cycle Formation
Does there exist a cycle in the network graph that visits each node exactly once?
Communication radius RProb
abilit
y (H
amilt
onia
n cy
cle e
xist
s)
n = 100 n = 100
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© 2002 Bhaskar Krishnamachari
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
Given: Multiple sensors and targets Sensors can only communicate locally Sensors can only “see” targets locally Need 3 communicating sensors tracking each target
sensosensorrtargettarget
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© 2002 Bhaskar Krishnamachari
Communication between sensors within range RCommunication between sensors within range R
sensorsensor
targettarget
RR
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
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© 2002 Bhaskar Krishnamachari
Communication graphCommunication graph
sensorsensor
targettarget
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
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© 2002 Bhaskar Krishnamachari
sensorsensor
targettarget
SS
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
Targets visible within range STargets visible within range S
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Visibility graphVisibility graph
sensorsensor
targettarget
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
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© 2002 Bhaskar Krishnamachari
Decision problem: can all targets can be tracked Decision problem: can all targets can be tracked by three communicating sensors ?by three communicating sensors ?
sensorsensor
targettarget
A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem
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© 2002 Bhaskar Krishnamachari
Phase Transition for Sensor Tracking Problem Phase Transition for Sensor Tracking Problem
Probability of Tracking all Targets
Sensing range Communication range
s = 17s = 17t = 5t = 5
%%
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1
23
2
© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
No nodes within 2 hops of each other can be allocated the same channel.
Are k channels enough ? Less likely with more edges in network graph, i.e. with higher transmission range.
NP-Hard in general, polynomial-time for 1D model.
1
23
1
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© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
Communication/Interference range R
Pr
obab
ility
(k c
hann
els
suffi
ce)
n = 1001D model
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© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
If k channels are used, and W is the aggregate available bandwidth/data-rate, the maximum network-wide throughput is T = n(W/k)
Let 1 = max(# of 1-hop neighbors), 2 = max(# of 2-hop neighbors), then:
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Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
n = 1001D model
99% connectivity threshold
Operating point w/maximum throughput
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© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
Communication/Interference range R
Pr
obab
ility
(k c
hann
els
suffi
ce)
n = 302D model
99% connectivity threshold
29
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Communication/Interference radius R© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
n = 302D model
99% connectivity threshold
Operating point w/maximum throughput
Expe
cted
nor
mal
ized
netw
ork
thro
ughp
ut T
/W
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Communication/Interference radius R© 2002 Bhaskar Krishnamachari
Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation
n = 1002D model
99% connectivity threshold
Operating point w/maximum throughput
Expe
cted
nor
mal
ized
netw
ork
thro
ughp
ut T
/W
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Min-k-neighbor property: all nodes have at least k neighbors.
Let Ai = event that node i has at least k neighbors, then
where (for R 0.5, ignoring edge effects)
© 2002 Bhaskar Krishnamachari
Min-k-NeighborMin-k-Neighbor
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The min-k-neighbor property undergoes a zero-one phase transition asymptotically at transmission radius
For finite n, the probability that all neighbors have at least k neighbors is > 1 - , if
© 2002 Bhaskar Krishnamachari
Critical Threshold for Min-k-NeighborCritical Threshold for Min-k-Neighbor
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Critical Threshold for Min-k-neighborCritical Threshold for Min-k-neighbor
n = 100 n = 100
Probability that all nodes have at least 2 neighbors
Communication radius R
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© 2002 Bhaskar Krishnamachari
Probabilistic FloodingProbabilistic Flooding
Prob
abilit
y( a
ll no
des r
ecei
ve p
acke
t)
Query forwarding probability q
Each node forwards packet with probability q
n = 100 n = 100
Resource-efficientoperating point
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© 2002 Bhaskar Krishnamachari
Phase Transitions and Computational Complexity: Phase Transitions and Computational Complexity: Constraint SatisfactionConstraint Satisfaction
In the early 90’s, AI researchers found phase transitions in NP-complete constraint satisfaction problems like SAT (Cheeseman et al. 1991, Selman et al. 1992, Hogg et al. 1994)
SAT: given a binary logic formula in CNF (conjunction of OR clauses), is there a truth assignment which makes the formula true?
((a a b b c c) ) ( ( d d e e f f ) ) ( ( a a c c ee))
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Phase Transition in 3-SATPhase Transition in 3-SAT
0 2 3 4 5Ratio of Constraints to Variables6 7 8
1000
3000
Cost
of C
ompu
tati
on
2000
400050 var 40 var 20 var
0.02 3 4 5
Ratio of Constraints to Variables6 7 8
0.2
0.6
Prob
abili
ty o
f Sol
utio
n
0.4
50% sat0.8
1.0
Selman et al. (1994):
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Phase Transition in 3-SATPhase Transition in 3-SAT
(Friedgut 1999): constant ck s.t. , formulas with at most (1-)ckn clauses are satisfiable w.h.p. and formulas with at least (1+)ckn are unsatisfiable w.h.p.
Critical ratio of clauses to variables for 3-SAT: empirically ~ 4.24, theoretically between 3.125 and 4.601
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© 2002 Bhaskar Krishnamachari
Worst-case vs. Average ComplexityWorst-case vs. Average Complexity
3-SAT remains NP-complete even for random instances with ratios between 1/3 and 7(n2-3n+2).
(Frieze et al. 1996): Polynomial heuristic finds satisfying solutions w.h.p. if ratio less than 3.006.
(Bearne et al. 1998): Can prove unsatisfiability in polynomial time w.h.p. if ratio more than n/log(n).
(Williams and Hogg, 1994): First-order, algorithm-independent analysis of CSPs showing that average computational effort peaks at phase transition point.
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© 2002 Bhaskar Krishnamachari
Distributed Constraint Satisfaction Distributed Constraint Satisfaction in Wireless Networksin Wireless Networks
Many NP-complete problems in wireless networks can be formulated as distributed constraint satisfaction problems (DCSP).
A DCSP consists of agents, each with a set of variables they need to set values to. There are intra-agent constraints and inter-agent constraints on these values.
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Hamiltonian Cycle FormationHamiltonian Cycle Formation
Does there exist a cycle in the network graph that visits each node exactly once?
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© 2002 Bhaskar Krishnamachari
Hamiltonian Cycle FormationHamiltonian Cycle Formation
Communication radius R
n = 100
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© 2002 Bhaskar Krishnamachari
Hamiltonian Cycle FormationHamiltonian Cycle Formation
Communication radius R
n = 100
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Conflict Free Resource AllocationConflict Free Resource Allocation
Allocating channels with 1-hop constraint.
3 channels
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Conflict Free Resource AllocationConflict Free Resource Allocation
3 channels
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Conflict Free Resource AllocationConflict Free Resource Allocation
Interference Range
More bandwidth
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© 2002 Bhaskar Krishnamachari
Decision problem: can all targets can be tracked Decision problem: can all targets can be tracked by three distinct communicating sensors ?by three distinct communicating sensors ?
sensorsensor
targettarget
Sensor Tracking ProblemSensor Tracking Problem
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© 2002 Bhaskar Krishnamachari
Sensor Tracking ProblemSensor Tracking Problem Decision Problem: Can each target be tracked by a
set of three distinct communicating sensors?
NP-complete for arbitrary communication and visibility graphs, polynomial solvable when the communication graph is complete.
Can be formulated as a DCSP:
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Sensor Tracking ProblemSensor Tracking Problem
Mean computational cost
Probability of Tracking
Sensing range Communication rangeSensing range Communication range
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Sensor Tracking ProblemSensor Tracking Problem
Mean communication cost
Probability of Tracking
Sensing range Communication range Communication rangeSensing range
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ConclusionsConclusions
Phase transition analysis appears to be a promising approach for determining resource-efficient operating points for various global network properties.
Also helpful in reducing the average computational cost of distributed algorithms for constraint satisfaction in wireless networks.
It may be possible to incorporate this approach into online, self-configuring mechanisms.
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AcknowledgementsAcknowledgements
Professor Stephen Wicker (Cornell ECE) Professor Bart Selman (Cornell CS) Dr. Ramon Bejar (Cornell CS) Dr. Carla Gomes (Cornell CS) Marc Pearlman (Cornell ECE)
© 2002 Bhaskar Krishnamachari
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End