Critical Density Thresholds and Complexity in Wireless Networks Bhaskar Krishnamachari School of...

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)

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

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

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

Complex SystemsComplex Systems Ordered global behavior and structure “emerging” from

multiple local interactions.

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

Connectivity in a Multi-hop NetworkConnectivity in a Multi-hop Network

All nodes increasing transmission range R simultaneously

Phase Transition for ConnectivityPhase Transition for Connectivity

Communication radius RProb

abilit

undesirableregime

desirableregime

Energy-efficient operating point

n = 20

Communication radius R

abilit

xx11 xx22 xx33 xx44

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.

Phase Transition for Bi-ConnectivityPhase Transition for Bi-Connectivity

2 node-disjoint paths between all pairs of nodes.

n = 100

abilit

Bernoulli Random Graphs G(n,p)Bernoulli Random Graphs G(n,p)

Studied by mathematicians since 50’s (Erdos, Renyi, Bollobas, Spencer, others)

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

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

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?

abilit

n = 100 n = 100

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

Communication between sensors within range RCommunication between sensors within range R

sensorsensor

targettarget

Communication graphCommunication graph

sensorsensor

targettarget

sensorsensor

targettarget

Targets visible within range STargets visible within range S

Visibility graphVisibility graph

sensorsensor

targettarget

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

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

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.

Communication/Interference range R

n = 1001D model

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:

n = 1001D model

99% connectivity threshold

Operating point w/maximum throughput

Communication/Interference range R

n = 302D model

n = 1002D model

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)

Min-k-NeighborMin-k-Neighbor

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

Critical Threshold for Min-k-NeighborCritical Threshold for Min-k-Neighbor

Critical Threshold for Min-k-neighborCritical Threshold for Min-k-neighbor

n = 100 n = 100

Probability that all nodes have at least 2 neighbors

Probabilistic FloodingProbabilistic Flooding

abilit

Query forwarding probability q

Each node forwards packet with probability q

n = 100 n = 100

Resource-efficientoperating point

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))

Phase Transition in 3-SATPhase Transition in 3-SAT

0 2 3 4 5Ratio of Constraints to Variables6 7 8

400050 var 40 var 20 var

0.02 3 4 5

Ratio of Constraints to Variables6 7 8

50% sat0.8

Selman et al. (1994):

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

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.

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.

Hamiltonian Cycle FormationHamiltonian Cycle Formation

Does there exist a cycle in the network graph that visits each node exactly once?

n = 100

Conflict Free Resource AllocationConflict Free Resource Allocation

Allocating channels with 1-hop constraint.

3 channels

Interference Range

More bandwidth

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

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:

Mean computational cost

Probability of Tracking

Sensing range Communication rangeSensing range Communication range

Mean communication cost

Probability of Tracking

Sensing range Communication range Communication rangeSensing range

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.

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)

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