FranckPetit/SébastienTixeuil Self-stabilizationandSensorNetworks

Self-stabilization and Sensor Networks

Franck Petit / Sébastien Tixeuil

UPMC

[email protected]

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

Sensor Networks

I processor + sensors + radio

I 2 AA batteries, on/off switch

I 3 LEDs for debugging

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Sensor Networks

While (batteries supply power)

I Collect, aggregate and reduce data

I log into memory

In spite of numerous fault modes

I Permanent sensor failures, node 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.

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

“Correct”

Time Configurations

Stabilization Time

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Complexity Criteria

Maximize useful lifetime of system

I “maximise useful”: correct quickly from illegitimate state

I Self-stabilization, scalability

I “maximise lifetime”: use minimal energy to preserve

System Specifics

I only one radio frequency

I no collision detect

I access technique: CSMA/CA

I use CRC to detect collision

I no directional send/receive

I msg. are small (30 bytes)

I radio range about 1 meter

I number of neighbors < 10

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

I unique node IDs (probably)

I cost a fewU(someday)

I slow processor (4 MHz)

I limited memory (4 KB RAM)

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

I several power modes available

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

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Sensor Networks and Self-stabilization TDMA Clustering Conclusion

The Model(s)

Self-stabilizing model

I Read neighborhood state,

I compute and update local state

Sensor Network model

I Read local state,

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

I Pros: reuse existing SS algorithms

I Cons: potentially inefficient, overhead

Design self-stabilizing algorithms for the sensor networks model

I Pros: potentially efficient

I Cons: ignore previous SS work

I [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

I Pros: reuse existing SS algorithms

I Cons: potentially inefficient, overhead

I [Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

I [Herman 03]Unison with collisions

Self-stabilization in Sensor Networks

Transform (i.e. Simulate) the self-stabilizing model into the sensor networks model

I Pros: reuse existing SS algorithms

I Cons: potentially inefficient, overhead

I [Herman 03]Cached Sensornet Transform

Design self-stabilizing algorithms for the sensor networks model

I Pros: potentially efficient

I Cons: ignore previous SS work

I [Herman 03]Unison with collisions

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cached Sensornet Transform

Basic Algorithm

I Each nodephas a variablevp

I Each neighborqofphas a variablecqvp

I cqvpis the cached value ofvpatq

I Wheneverpassignsvp,palso broadcasts the new value to the neighborhood

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 cqvp by 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 cqvp by vp

Example

Lemma (Closure)

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

Cached Sensornet Transform

Periodic retransmit

I Each nodepperiodically broadcastsvp to 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

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

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

Cached Sensornet Transform

Message Corruption

I Each neighborqofphas a Boolean variablebqvp I Ifqreceivesvpcorrectly,bqvpbecomes true

I G→Abecomes

for all neighborsqofp,bpvqandG→

A; for all neighborsqofp,bpvqbecomes false

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Example

I Ifqreceivesvpcorrectly,bqvpbecomes true

I G→Abecomes

for all neighborsqofp,bpvqandG→

A; for all neighborsqofp,bpvqbecomes false

Example

I Ifqreceivesvpcorrectly,bqvpbecomes true

I 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

I Ifqreceivesvpcorrectly,bqvpbecomes true

I G→Abecomes

for all neighborsqofp,bpvqandG→

A; for all neighborsqofp,bpvqbecomes false

Example

I Ifqreceivesvpcorrectly,bqvpbecomes true

I 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

I Ifqreceivesvpcorrectly,bqvpbecomes true

I G→Abecomes

for all neighborsqofp,bpvqandG→

A; for all neighborsqofp,bpvqbecomes false

Example

I Ifqreceivesvpcorrectly,bqvpbecomes true

I 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

I Ifqreceivesvpcorrectly,bqvpbecomes true

I G→Abecomes

for all neighborsqofp,bpvqandG→

A; for all neighborsqofp,bpvqbecomes false

Example

I Ifqreceivesvpcorrectly,bqvpbecomes true

I 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)

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

I Each nodephas a clock variablevp I For every neighborspandq,|vp−vq|61

I for every neighborq,vq>vp →vp:= vp+1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Unison

Specification

I Each nodephas a clock variablevp I For every neighborspandq,|vp−vq|61

Self-stabilizing Unison

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Example

Self-stabilizing Unison

I 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

I for every neighborq,vq>vp →vp:= vp+1

non activatable activatable legitimate

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Specification

I Each nodephas a clock variablevp I For every neighborspandq,|vp−vq|61

Self-stabilizing Unison

I for every neighborq,vq>vp →vp:= vp+1

I for every neighborq,cpvq>vp →vp:= vp+1

I Only correctly received messages update cached variables

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Unison with Collisions

Specification

I Each nodephas a clock variablevp I For every neighborspandq,|vp−vq|61

Self-stabilizing Unison with Collisions

I Only correctly received messages update cached variables

Unison with Collisions

Specification

I Each nodephas a clock variablevp I For every neighborspandq,|vp−vq|61

Self-stabilizing Unison with Collisions

I for every neighborq,cpvq>vp →vp:= vp+1

I 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

4 2

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

4 2

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 2

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 2

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

Unison with Collisions

Cache coherence weakening

I For every neighborspandq,cpvq6vq

Self-stabilizing Unison with collisions

I Unison and Weak cache coherence are preserved by program executions

I Unison and Weak cache coherence eventually hold

I 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

I [Herman 03]Cached Sensornet Transform

I Overhead is not upper bounded

Design self-stabilizing algorithms for the sensor

networks model

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMA

Motivation Algorithm stack Clustering

Density

Self-stabilizing Clustering Conclusion

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Towards an Intermediate Model

An atomic step at a node

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

Hypothesis

I Global clock, unique IDs

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

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

Towards an Intermediate Model

Solution

I TDMA to avoid collisions

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

I 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 . . .

I Algorithm messages are transmitted during the

Self-stabilizing TDMA for Sensors

I [Kulkarni, Arumugam 03]2-D Grids

I nodes are aware of their positions

I Not suitable for dynamic/faulty networks

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

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

I 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

Self-stabilizing TDMA for Sensors

2 0

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

I both are minimal,

I but second solution is better for time-slot assignment

Self-stabilizing TDMA for Sensors

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

I both are minimal,

I 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

N³ coloring→MIS

Variables propagation→N³coloring

CSMA/CA → Variables propagation

I Wait fixed delay

I to process received messages, and update local variables

I Wait random delay

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

I “Age” information to remove invalid data

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Shared variables → N

coloring

∃j∈N_i³, colorj =colori →colori := random(∆\ {colorj|j∈ N_i³})

I Stabilizes in expectedO(1)

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b

d e

g h

i a

b

d e

g h

i a

b

d e

g h

i a

b

d e

g h

i a

b

d e

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

c

d j

e

f

h i a

c

d j

e

f

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

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b

d e

g h

i

a

b c

d j

e

f

g h

i

a

b

d e

g h

i a

b

d e

g h

i a

b

d e

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

c

d j

e

f

h i

a

b c

d j

e

f

g h

i a

b c

d j

e

f

g h

i

a

c

d j

e

f

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

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

b c

d j

e

f

g h

i

a

b

d e

g h

i a

b

d e

g h

i a

b

d e

g h

i

a

b c

d j

e

f

g h

i

a

b

d e

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring → MIS

[Ikeda, Kamei, Kakugawa 02]

No parent in MIS→join MIS

a

c

d j

e

f

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

c

d j

e

f

h i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring + MIS → Minimal N

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

2,i 0

4,g

3,g

0 2,i

0

4,g

6,j 3,g

0 2,i

0 1

4,g

6,j 3,g

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring + MIS → Minimal N

coloring

MIS→send colors to dominated nodes

a

c

d j

e

f

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

coloring + MIS → Minimal N

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

2,i 0

4,g

3,g

1,i

0 2,i

0

4,g

6,j 3,g

0 2,i

0 1

4,g

6,j 3,g

0 2,i

0

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

N

coloring + MIS → Minimal N

coloring

MIS→send colors to dominated nodes

a

c

d j

e

f

h i

1,i

2,i 0

4,g

3,g

1,i

0 2,i

0

0,j

6,j 5

3,g

1,i

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

coloring + MIS → Minimal N

coloring

MIS→send colors to dominated nodes

a

b c

d j

e

f

g h

i

2,i 0

4,g

3,g

0 2,i

0

4,g

6,j 3,g

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

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

5 3 3 3

1 5

1 5

5

5 3 3 3

1 5

1 5

1 5

5 3 3 3

1 5

1 5

1 5

1

5 3 3 3

1 5

1 5

1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5

1 5

1 3

1

3 3

1 5

1 5

1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

5 3 3 3

1 5

1 5

1 5

5 3 3 3

1 5

1 5

1 5

1

5 3 3 3

1 5

1 5

1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5

1 5

1 3

1

3 3

1 5

1 5

1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

5 3 3 3

1 5

1 5

5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5

1

5 3 3 3

1 5

1 5

1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5

1 5

1 3

1

3 3

1 5

1 5

1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

5 3 3 3

1 5

1 5

5

5 3 3 3

1 5

1 5

1 5

5 3 3 3

1 5

1 5

1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5

1 5

1 3

1

3 3

1 5

1 5

1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3 1

5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

3 1

4 2 0 4

5 5

5

5 3 3 3

5 ¹₅

5

5 3 3 3

5 ¹₅

5

5 3 3 3

1 5

1 5

5

5 3 3 3

1 5

1 5

1 5

5 3 3 3

1 5

1 5

1 5

1

5 3 3 3

1 5

1 5

1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

1 5

1 5

1 5

1 5

1 3

1 3

1 3

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Minimal N

coloring → TDMA Schedule

2 0

1

4 2 0 4

5 5

5 5

5 3 3 3

5 ¹₅

5 5

5 3 3 3

5 ¹₅

5

1 5

5 3 3 3

1 5

1 5

5

1 5

5 3 3 3

1 5

1 5

1 5 1 5

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 3 3

1 5

1 5

1 5 1 5

1

5 3 ¹₃ 3

1 5

1 5

1 5 1 5

1 5

1 3

1

3 3

1 5

1 5

1 5 1 5

1 5

1 3

1 3

1 3

1 5

1 5 1 5

1 5

1 3

1 3

1 3

Outline

Sensor Networks and Self-stabilization Model(s)

Cached Sensornet Self-stabilizing Unison TDMA

Motivation Algorithm stack Clustering

Density

Self-stabilizing Clustering 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

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Density

ρ(u) = |{e= (v,w)∈E|w∈{u}∪Nu andv∈Nu}|

|Nu|

a

b c

d j

e

f

g h

i

a

b

d e

g h

i

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Density

ρ(u) = |{e= (v,w)∈E|w∈{u}∪Nu andv∈Nu}|

|Nu|

a

c

d j

e

f

h i a

c

d j

e

f

h i

Density

ρ(u) = |{e= (v,w)∈E|w∈{u}∪Nu andv∈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

a 1.33

c 1

d 1.75

j 1.5

e 1.8

f 1.33

h 1

i 1

a 1.33

b 1.75 c

1

d 1.75

j 1.5

e 1.8

f 1.33

g 1.5

h 1

i 1 a

1.33

b 1.75 c

1

d 1.75

j 1.5

e 1.8

f 1.33

g 1.5

h 1

i 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cluster Head Heuristics

a 1.33

b 1.75 c

1

d 1.75

j 1.5

e 1.8

f 1.33

g 1.5

h 1

i 1 a

1.33

b 1.75 c

1

d 1.75

j 1.5

e 1.8

f 1.33

g 1.5

h 1

i 1

a 1.33

b 1.75 1

d 1.75

e 1.8

g 1.5

h 1

i 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Cluster Head Heuristics

a 1.33

c 1

d 1.75

j 1.5

e 1.8

f 1.33

h 1

i 1 a

1.33

c 1

d 1.75

j 1.5

e 1.8

f 1.33

h 1

i 1 a

1.33

c 1

d 1.75

j 1.5

e 1.8

f 1.33

h 1

i 1

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Clustering

Basic Idea

I IdentifyNandN²

I Compute and broadcast density

I Attach to neighbor with higher density

I use identifiers to break ties

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Self-stabilizing Clustering

Basic Idea

I IdentifyNandN²

I Compute and broadcast density

I Attach to neighbor with higher density

Sensor Networks and Self-stabilization TDMA Clustering Conclusion

Faster Self-stabilizing Clustering

Basic Idea

I IdentifyNandN²

I Compute and broadcast density

I RandomL(1, 1)coloring withδ²colors

I This can be done in expectedO(1)time

I Attach to neighbor with higher density

I usecolorsto break ties