algorithmic models for sensor networks
DESCRIPTION
Stefan Schmid and Roger Wattenhofer. Algorithmic Models for Sensor Networks. WPDRTS, Island of Rhodes, Greece , 2006. Algorithmic Models. Why are models needed? - Formal proofs of correctness, efficiency, real-time guarantees, … - Common basis to compare results?. - PowerPoint PPT PresentationTRANSCRIPT
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 5
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.
Stefan Schmid, ETH Zurich @ WPDRTS 2006 6
(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.
Stefan Schmid, ETH Zurich @ WPDRTS 2006 8
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 10
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 12
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.
Stefan Schmid, ETH Zurich @ WPDRTS 2006 13
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!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 14
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:
Stefan Schmid, ETH Zurich @ WPDRTS 2006 15
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!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 16
Let’s Talk about Models!
• Model for what?- Connectivity- Interference- Algorithm type- Node distribution - Energy consumption- etc.!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 17
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?
Stefan Schmid, ETH Zurich @ WPDRTS 2006 18
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.
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Connectivity: Unit Disk Graph
R
R
Unit Disk GraphUnit Disk Graph
Stefan Schmid, ETH Zurich @ WPDRTS 2006 20
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!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 21
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!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 22
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 23
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 25
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 26
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
Stefan Schmid, ETH Zurich @ WPDRTS 2006 27
More Models!
• Interference- Which senders can disturb the reception of which other peers?
• Node distribution
• Location information- GPS / Galileo device, etc.?
• Etc.!
vs vs
Stefan Schmid, ETH Zurich @ WPDRTS 2006 29
Choice of Model (2)
• Which model to choose?
too pessimistic too optimistic
General
GraphUDG
Quasi
UDG
d
1
Bounded
Independence
Unit Ball
Graph
Stefan Schmid, ETH Zurich @ WPDRTS 2006 30
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!
Stefan Schmid, ETH Zurich @ WPDRTS 2006 31
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!