Inspiteofnumerousfaultmodes Definition(Self-stabilizingSystem) Definition(Self-stabilizingSystem) Maximizeusefullifetimeofsystem While(batteriessupplypower) Self-stabilizationandSensorNetworks Definition(ClassicalSystem, Definition(ClassicalSystem, a.k.a. a.k

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization and Sensor Networks

Sébastien Tixeuil

LRI - CNRS 8623 & INRIA Grand Large Université Paris Sud, France

[email protected]

August 9, 2007

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMAMotivation

Algorithm stack Clustering

Density

Self-stabilizing Clustering Simulation Results Conclusion

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Sensor Networks

�processor + sensors + radio

�2 AA batteries, on/off switch

�3 LEDs for debugging

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Sensor Networks While (batteries supply power)

�Collect, aggregate and reduce data

�log into memory

In spite of numerous fault modes

�Permanent sensor failures, node failures

�restarts, radio failures

�transient faults, reconfigurations

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Distributed Systems Definition (Classical System,

a.k.a.

Non stabilizing)

Starting from aparticularinitial configuration, the system immediatelyexhibits correct behavior.

Definition (Self-stabilizing System)

Starting fromanyinitial configuration, the system eventuallyreaches a configuration from with its behavior is correct.

�Self-stabilization permits to recover from transient failures

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Distributed Systems Definition (Classical System,

a.k.a.

Non stabilizing)

Starting from aparticularinitial configuration, the system immediatelyexhibits correct behavior.

Definition (Self-stabilizing System)

Starting fromanyinitial configuration, the system eventuallyreaches a configuration from with its behavior is correct.

�Self-stabilization permits to recover from transient failures

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization

“Correct”

Time Configurations

Stabilization Time

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Complexity Criteria

Maximize useful lifetime of system

�“maximise useful”: correct quickly from illegitimate state

�Self-stabilization, scalability

�“maximise lifetime”: use minimal energy to preserve batteries

�local vs. global preserving

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

System Specifics

�only one radio frequency

�no collision detect

�access technique: CSMA/CA

�use CRC to detect collision

�no directional send/receive

�msg. are small (30 bytes)

�radio range about 1 meter

�number of neighbors < 10

�could be large number of nodes (perhaps > 100000)

�unique node IDs (probably)

�cost a few�(someday)

�slow processor (4 MHz)

�limited memory (4 KB RAM)

�item nodes have real-time clocks≡drift between 1 msec and 100 msec per second

�several power modes available

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

c d

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f h g

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a b

c d

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a b

c d

e

f h g

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a b

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

c d

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f h g

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a b

c d

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c d

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

c d

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f h g

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a b

c d

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f h g

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a b

c d

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f h g

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

a b

c d

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f h g

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a b

c d

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f h g

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a b

c d

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s) Self-stabilizing model

�Read neighborhood state,

�compute and update local state

Sensor Network model

�Read local state,

�compute and broadcast to neighborhood

�Collisions may appear

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

�Pros: reuse existing SS algorithms

�Cons: potentially inefficient, overhead

�[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

�Pros: potentially efficient

�Cons: ignore previous SS work

�[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

�Pros: reuse existing SS algorithms

�Cons: potentially inefficient, overhead

�[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

�Pros: potentially efficient

�Cons: ignore previous SS work

�[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

�Pros: reuse existing SS algorithms

�Cons: potentially inefficient, overhead

�[Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

�Pros: potentially efficient

�Cons: ignore previous SS work

�[Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Basic Algorithm

�Each nodephas a variablevp

�Each neighborqofphas a variablecqvp

�cqvpis the cached value ofvpatq

�Wheneverpassignsvp,palso broadcasts the new value to the neighborhood

�Whenever a neighborqofpreceivesvp,qupdatescqvp

accordingly

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Definition (Cache coherence)

For all neighborspandq,cqvp=vp

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Closure)

If started from a cache coherent state, and without collisions, the self-stabilizing model is simulated by replacing all occurrences of cqvpby vp

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Periodic retransmit

�Each nodepperiodically broadcastsvpto its neighborhood

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

Lemma (Convergence)

If started from an arbitrary state, and without collisions, a cache coherent state is eventually reached

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Message Corruption

�Each neighborqofphas a Boolean variablebqvp

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

�Ifqreceivesvpcorrectly,bqvpbecomes true

�G→Abecomes

for all neighborsqofp,bpvqandG→ A; for all neighborsqofp,bpvqbecomes false

a b

c cbva

cavb

ccva

cavc

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Periodic Retransmit Message Corruption Lemma (Self-stabilization)

If started from an arbitrary state, the self-stabilizing model is eventually simulated

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

�Each nodephas a clock variablevp

�For every neighborspandq,|vp−vq|�1

Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

�Each nodephas a clock variablevp

�For every neighborspandq,|vp−vq|�1

Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 0 8 1 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 0 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 1 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 2

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 2 2 8 2 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 3 3 8 3 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 4 4 8 4 4

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 5 5 8 5 5

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 6 6 8 6 6

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

7 7 6 8 7 7

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

8 7 6 8 8 8

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

8 7 7 8 9 8

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example Self-stabilizing Unison

�for every neighborq,vq�vp→vp:=vp+1

non activatable activatable legitimate

8 8 7 8 9 9

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions Specification

�Each nodephas a clock variablevp

�For every neighborspandq,|vp−vq|�1

Self-stabilizing Unison

with Collisions

�for every neighborq,vq�vp→vp:=vp+1

�for every neighborq,cpvq�vp→vp:=vp+1

�Only correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions Specification

�Each nodephas a clock variablevp

�For every neighborspandq,|vp−vq|�1

Self-stabilizing Unison

with Collisions

�for every neighborq,cpvq�vp→vp:=vp+1

�Only correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions Specification

�Each nodephas a clock variablevp

�For every neighborspandq,|vp−vq|�1

Self-stabilizing Unison

with Collisions

�for every neighborq,cpvq�vp→vp:=vp+1

�Only correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 2

1 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 3

2 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 4

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

1 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

2 4

3 2

4 2

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

3 4

3 3

4 3

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

non activatable activatable legitimate lower than value strictly greater

4 4

3 4

4 4

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions Cache coherence weakening

�For every neighborspandq,cpvq�vq

Self-stabilizing Unison with collisions

�Unison and Weak cache coherence are preserved by program executions

�Unison and Weak cache coherence eventually hold

�Some extra work is expected to get bounded clock values

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilization in Sensor Networks Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

�[Herman 03]Cached Sensornet Transform

�Overhead is not upper bounded

Design self-stabilizing algorithms for the sensor networks model

�[Herman 03]Unison with collisions

�Proof in the model is specific to the problem

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMA

Motivation Algorithm stack Clustering

Density

Self-stabilizing Clustering Simulation Results Conclusion

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Towards an Intermediate Model An atomic step at a node

�Compute new state, write new state at all neighbors (no collision)

Hypothesis

�Global clock, unique IDs

Solution

�TDMA to avoid collisions

�assume synchronised, real-time clocks (to enable TDMA slotted time)

�but TDMA implemented using CSMA/CA as basic, underlying model

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Towards an Intermediate Model

Solution

�TDMA to avoid collisions

�assume synchronised, real-time clocks (to enable TDMA slotted time)

�but TDMA implemented using CSMA/CA as basic, underlying model

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

TDMA Scheduling

1 2 3 4 5 1 2 3 4 5 1 2 . . .

�Algorithm messages are transmitted during the

“overhead” periods

�TDMA slot assignment is the output of our algorithm

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing TDMA for Sensors

�[Kulkarni, Arumugam 03]2-D Grids

�nodes are aware of their positions

�Not suitable for dynamic/faulty networks

�[Herman,Tixeuil 04]General graphs of bounded degree

�Randomized algorithm, self-stabilizing in expected O(1)time, to assign TDMA slots

�Solution is a protocol stack based on variable propagation, minimal coloring ofN², MIS construction, and mapping colors↔TDMA slots

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

2 0

3 1

4 2 0 4

2 0

3 1

4 2 1 3

5 5

5 5

5 3 3 3

5 5

5 5

5 5 4 3

�both are minimal,

�but second solution is better for time-slot assignment

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

2 0

3 1

4 2 0 4

2 0

3 1

4 2 1 3

5 5

5 5

5 3 3 3

5 5

5 5

5 5 4 3

�both are minimal,

�but second solution is better for time-slot assignment

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Overview

N²minimal coloring→TDMA schedule

N³coloring + MIS→N²minimal coloring

N³coloring→MIS

Variables propagation→N³coloring

CSMA/CA→Variables propagation

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

CSMA/CA → Variables propagation

�Wait fixed delay

�to process received messages, and update local variables

�Wait random delay

�to allow Aloha-style analysis for probability of collisions among neighbors

�“Age” information to remove invalid data

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Shared variables → N

³

coloring Previous L(1,0)& L(1,1) Coloring

�[Ghosh, Karaata 93]Planar graphs L(1,0)

�[Sur, Srimani 93]Bipartite graphs L(1,0)

�[Gradinariu,Tixeuil 00]General graphs L(1,0)

�[Gradinariu, Johnen 01]Colors of sizen²L(1,1)

Our algorithm

∃j∈N³_i, colorj=colori→colori:=random(∆\ {colorj|j∈ Ni³})

�Stabilizes in expectedO(1)

�Output an ID-based DAG of constant height

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring + MIS → Minimal N

²

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0 1

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring + MIS → Minimal N

²

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i 1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0 1

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring + MIS → Minimal N

²

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0 1

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring + MIS → Minimal N

²

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

1

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

³

coloring + MIS → Minimal N

²

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

1

4,g 0,j

6,j 5

3,g

1,i

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹₅

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

15

5 3 3 3

15 1

5

5

1 5

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹₃ 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

1 5 1 5

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹₃ 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

1 5 1 5

15 3 3 3

15 1

5

15 15

15 3 ¹₃ 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

1

5 1

5

15 15

1

5 3 ¹₃ 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

1

5 1

5

15 15

1

5 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

1 3

5 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

1

5 1

5

15 15

1

5 1

3 1

3 1

3

15 1

5

15 15

15 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

²

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹5

5 5

5 3 3 3

5 ¹5

5

15

5 3 3 3

15 1

5

5

15

5 3 3 3

15 1

5

15 15

5 3 3 3

15 1

5

15 15

15 3 3 3

15 1

5

15 15

15 3 ¹3 3

15 1

5

15 15

15 1

3 1

3 3

15 1

5

15 15

15 1

3 1

3 1

3

1

5 1

5

15 15

1

5 1

3 1

3 1

3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMAMotivation

Algorithm stack Clustering

Density

Self-stabilizing Clustering Simulation Results Conclusion

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Motivation

Clusters for routing

MANET routing protocols are flat, thus not scalable Cluster-heads have extra responsibility for the routing of message

Cluster-heads should be stable

Handle departures and removals Handle node mobility

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Density

ρ(u) =|{e= (v,w)∈E|w∈{u}∪Nuandv∈Nu}|

|Nu|

a

b c

d j

e

f

g h

i

a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Density

ρ(u) =|{e= (v,w)∈E|w∈{u}∪Nuandv∈Nu}|

|Nu|

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Density

ρ(u) =|{e= (v,w)∈E|w∈{u}∪Nuandv∈Nu}|

|Nu|

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cluster Head Heuristics

1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i

1.33a

b 1.75 c1

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i 1.33a

b 1.75 c1

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cluster Head Heuristics

1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i 1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i

1.33a

b 1.75 c1

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cluster Head Heuristics

1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i 1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i 1.33a

1.75b 1c

1.75d j 1.5

1.8e f 1.33

g 1.5

h1 1i