Algorithmic Models for Sensor Networks
Stefan Schmid and Roger Wattenhofer
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?
• A typical problem in sensor networks:
Find the destination!
source
destination
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Finding a Destination
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Finding a Destination Efficiently: Backbone
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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 a
gateway, transmit
message to gateway
2. Gateway acts as
proxy source and
routes message on
backbone to gateway
of destination.
3. Transmission gateway
to destination.
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(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 choice
for a backbone.
• It might be favorable to
have few nodes in the
CDS. This is known as the
Minimum CDS problem.
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A Famous Dominating Set…
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Algorithm 1
Input:
Local Graph
Fractional
Dominating Set
Dominating
Set
Connected
Dominating Set
0.2
0.2
0
0.5
0.3
0.3
0
0.5 0.2
Phase A:
Distributed
linear program
0.8
0.1
Phase B:
Probabilistic
algorithm
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Phase C:
Connect DS
by “tree” of
“bridges”
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Algorithm 1: Phase A
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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
From phase A
Highest degree in distance 2
3. Send status (dominator or not) to all neighbors
4. If no neighbor is a dominator, become a dominator yourself
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Algorithm 2: Idea
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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.
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Comparison
Algorithm 1
Algorithm 2
• Algorithm computes DS
• Algorithm computes DS
• k2+O(1) transmissions/node
• O(O(1)/k log ) approximation
• 1 transmission/node
• O(1) approximation
• Quite complex!
• Performance OK
• Easy!
• Performance great!
The model determines the distributed
complexity of clustering
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Relation Between Algorithms and Models
Bounded
General
Graph Independence
UDG, no
Distances
UBG
Distances
UDG
Distances
UDG
GPS
too optimistic
too pessimistic
Time:
Approximation:
too simplistic
too realistic
Physical Signal
Propagation
Unstructured Radio
Network Model
Radio Network
Model
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Message
Passing
Models
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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!
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Let’s Talk about Models!
• Model for what?
- Connectivity
- Interference
- Algorithm type
- Node distribution
- Energy consumption
- etc.!
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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?
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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
Unit Disk Graph
R
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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!
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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!
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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
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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
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Connectivity: Unit Ball Graph
S
1
1
1
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Connectivity Put into Perspective (1)
• Fact: UDG is a QUDG
-=1
UBG
• 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:
QUDG
• Fact: QUDG is a UBG
UDG
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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).
GG
BIG
UBG
QUDG
UDG
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More Models!
• Interference
- Which senders can disturb the reception of
which other peers?
• Node distribution
vs
vs
• Location information
- GPS / Galileo device, etc.?
• Etc.!
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Choice of Model (1)
• Which model to choose?
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Choice of Model (2)
• Which model to choose?
General
Graph
UDG
too optimistic
too pessimistic
Bounded
Independence
Unit Ball
Graph
Quasi
UDG
1
d
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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!
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Conclusion
• 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!
• Our paper…
- … surveys models for connectivity, interference, etc.
- … mostly simplistic models (“high-level”) only.
- … a first step to put things into perspective!
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Thank you for your attention!
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Questions? / Feedback?
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