algorithmic models for sensor networks

Algorithmic Models for Sensor Networks

Stefan Schmid and Roger Wattenhofer

WPDRTS, Island of Rhodes, Greece, 2006

Stefan Schmid, ETH Zurich @ WPDRTS 2006 2

Algorithmic Models

• Why are models needed?- Formal proofs of correctness, efficiency, real-time guarantees, …

- Common basis to compare results?

• A typical problem in sensor networks:

Find the destination!source

destination


Finding a Destination


Finding a Destination Efficiently: Backbone


Backbone

• Idea: Some nodes become backbone nodes (gateways). Each node can access and be accessed by at least one backbone node.

• Routing:

1. If source is not agateway, transmitmessage to gateway

2. Gateway acts asproxy source androutes message onbackbone to gatewayof destination.

3. Transmission gatewayto destination.


(Connected) Dominating Set

• A Dominating Set DS is a subset of nodes such that each node is either in DS or has a neighbor in DS.

• A Connected Dominating Set CDS is a connected DS, that is, there is a path between any two nodes in CDS that does not use nodes that are not in CDS.

• A CDS is a good choicefor a backbone.

• It might be favorable tohave few nodes in the CDS. This is known as theMinimum CDS problem.


A Famous Dominating Set…


Algorithm 1

0.20.5

0.2

0.80

0.2

0.3

0.1

0.3

0

Input:

Local Graph

Fractional

Dominating Set

Dominating

Set

Connected

Dominating Set

0.5

Phase C:

Connect DS

by “tree” of

“bridges”

Phase B:

Probabilistic

algorithm

Phase A:

Distributed

linear program


Algorithm 1: Phase A


Algorithm 1: Phase B

Each node applies the following algorithm:

1. Calculate (= maximum degree of neighbors in distance 2)

2. Become a dominator (i.e. go to the dominating set) with probability

3. Send status (dominator or not) to all neighbors

4. If no neighbor is a dominator, become a dominator yourself

From phase A Highest degree in distance 2


Algorithm 2: Idea

trans

miss

ion ra

dius


Algorithm 2

1. Beacon your position

2. If, in your virtual grid cell, you are the node closest to the center of the cell, then join the DS, else do not join.

3. That’s it.


The model determines the distributed

complexity of clustering

Comparison

Algorithm 1

• Algorithm computes DS

• k2+O(1) transmissions/node• O(O(1)/k log ) approximation

• Quite complex!• Performance OK

Algorithm 2

• Algorithm computes DS

• 1 transmission/node• O(1) approximation

• Easy!• Performance great!

Gen

eral

Gra

ph!

No

Posi

tion

Info

rmat

ion!

Uni

t Dis

k G

raph

Onl

y!

Req

uire

s G

PS D

evic

e!


Relation Between Algorithms and Models

too pessimistic too optimistic

General

GraphUDG

GPS

UDG

Distances

Bounded

IndependenceUBG

Distances

too realistic too simplistic

Message

Passing

Models

Physical Signal

Propagation

Radio Network

Model

Unstructured Radio

Network Model

UDG, no

Distances

Time:

Approximation:


Let‘s Talk about Models!

• Why models for sensor networks?- Allows precise evaluation of algorithms- Analysis of correctness and efficiency (proofs)

• Goal of model designer?- Simplifications and abstractions- But close to reality!


Let’s Talk about Models!

• Model for what?- Connectivity- Interference- Algorithm type- Node distribution - Energy consumption- etc.!


Let’s Talk about Models!

• Algorithmic models often inspired by- “Connections” => Graph Theory- Transmission ranges, interference, … => Geometry

• Goal of our paper:- Survey of simple algorithmic models - “higher level abstractions“

We ask:

- How are models related to each other?

- When should which model be preferred?


Connectivity: Unit Disk Graph

• Which nodes are adjacent to a given node v?

• Example: Unit Disk Graph (UDG)- Classic Model from computational geometry- {u,v} 2 E , |u,v| · 1

• Pro- Very simple- Analytically tractable- Realistic for unobstructed environments

• Contra- Too simple- Not realistic for inner-city networks with many buildings etc.


Connectivity: Unit Disk Graph

R

R

Unit Disk GraphUnit Disk Graph


Connectivity: Quasi Unit Disk Graph

• More realistic: Quasi UDG (QUDG)- two radii- {u,v} 2 E , |u,v| · - {u,v} 2 E , |u,v| > 1- otherwise: It depends!

• It depends…- … on an adversary,- … on probabilistic model,- etc.!

• Advantage: More flexible and realistic than UDG!


Connectivity: Drawbacks of QUDG

• How realistic is QUDG?- if there is a wall…

- … u and v can be close but not adjacent- => QUDG model requires very small

• However, although if there are walls, connectivity typically still adheres to certain geometric constraints!- Resort to general connectivity graphs too pessimistic!

• Observation: Even in complex environments, the neighbors of a node are often also neighboring (cf wall example)

- Motivation for Bounded Independence Graph!


Connectivity: Bounded Independence Graph

• Bounded Independence Graph (BIG)

• Size of any independent set grows polynomially with the hop distance r

- typically: in O(rc) for constant c¸ 2


Connectivity : Unit Ball Graph

• Finally, there are many interesting UDG generalizations

• Example: Unit Ball Graph (UBG)- Nodes are assumed to form a doubling metric- The set of nodes at distance r of a node u can be covered by a constant number of balls of radius r/2 around other nodes, for all r

- i.e., Bu(r) µi=1…c Bui(r/2), 8 r


Connectivity: Unit Ball Graph

S

1 11


Connectivity Put into Perspective (1)

• Fact: UDG is a QUDG- = 1

• However, in the QUDG with constant , the set of nodes in radius r can always be covered by a constant number of balls of radius r/2 and hence:

• Fact: QUDG is a UBG

UDG

QUDG

UBG


Connectivity Put into Perspective (2)

• Fact: The UBG is a BIG.

- The size of the independent sets of any UBG is polynomially bounded.

• Fact: A BIG is of course a special kind of a general graph (GG).

QUDG

UBG

BIG

GG

UDG


More Models!

• Interference- Which senders can disturb the reception of which other peers?

• Node distribution

• Location information- GPS / Galileo device, etc.?

• Etc.!

vs vs


Choice of Model (1)

• Which model to choose?


Choice of Model (2)

• Which model to choose?

too pessimistic too optimistic

General

GraphUDG

Quasi

UDG

d

1

Bounded

Independence

Unit Ball

Graph


Choice of Model (3)

• Note: An algorithm which is correct in a “higher” model is also correct in a “lower” model in our figure.

Robustness and correctness properties of an algorithm should be proven in a model as high as possible!

• For efficiency considerations, however, a less conservative and more idealistic model might be fine!

• And: Study of simpler models might give insights into how algorithms for general model could look like!


Conclusion

• Our paper…- … surveys models for connectivity, interference, etc.

- … mostly simplistic models (“high-level”) only.

- … a first step to put things into perspective!

• Models… - … influence design, performance, correctness of algorithms! - … are more sophisticated than some years ago.- … still require lot of research.- … are not even completely known by experts!- … raise interesting questions of how they are related!


Thank you for your attention!


Questions? / Feedback?

algorithmic models for sensor networks

Documents

set ds

connected ds

connected dominating

subset of nodes

backbone nodes gateways

phase beach node

node closest

following algorithm