1. Algorithmic Models for Sensor Networks Stefan Schmid and Roger Wattenhofer WPDRTS, Island of Rhodes, Greece, 2006

2. Stefan Schmid, ETH Zurich @ WPDRTS 20062 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

3. Stefan Schmid, ETH Zurich @ WPDRTS 20063 Finding a Destination

4. Stefan Schmid, ETH Zurich @ WPDRTS 20064 Finding a Destination Efficiently: Backbone

5. Stefan Schmid, ETH Zurich @ WPDRTS 20065 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.

6. Stefan Schmid, ETH Zurich @ WPDRTS 20066 (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.

7. Stefan Schmid, ETH Zurich @ WPDRTS 20067 A Famous Dominating Set…

8. Stefan Schmid, ETH Zurich @ WPDRTS 20068 Algorithm 1 0.2 0.5 0.2 0.8 0 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

9. Stefan Schmid, ETH Zurich @ WPDRTS 20069 Algorithm 1: Phase A

10. Stefan Schmid, ETH Zurich @ WPDRTS 200610 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

11. Stefan Schmid, ETH Zurich @ WPDRTS 200611 Algorithm 2: Idea transmission radius

12. Stefan Schmid, ETH Zurich @ WPDRTS 200612 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.

13. Stefan Schmid, ETH Zurich @ WPDRTS 200613 The model determines the distributed complexity of clustering Comparison Algorithm 1 Algorithm computes DS k 2 +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! General Graph! No Position Information! Unit Disk Graph Only! Requires GPS Device!

14. Stefan Schmid, ETH Zurich @ WPDRTS 200614 Relation Between Algorithms and Models too pessimistic too optimistic General Graph UDG GPS UDG Distances Bounded Independence UBG Distances too realistic too simplistic Message Passing Models Physical Signal Propagation Radio Network Model Unstructured Radio Network Model UDG, no Distances Time: Approximation:

15. Stefan Schmid, ETH Zurich @ WPDRTS 200615 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!

16. Stefan Schmid, ETH Zurich @ WPDRTS 200616 Let’s Talk about Models! Model for what? - Connectivity - Interference - Algorithm type - Node distribution - Energy consumption - etc.!

17. Stefan Schmid, ETH Zurich @ WPDRTS 200617 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?

18. Stefan Schmid, ETH Zurich @ WPDRTS 200618 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.

19. Stefan Schmid, ETH Zurich @ WPDRTS 200619 Connectivity: Unit Disk Graph R R Unit Disk Graph

20. Stefan Schmid, ETH Zurich @ WPDRTS 200620 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!

21. Stefan Schmid, ETH Zurich @ WPDRTS 200621 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!

22. Stefan Schmid, ETH Zurich @ WPDRTS 200622 Connectivity: Bounded Independence Graph Bounded Independence Graph (BIG) Size of any independent set grows polynomially with the hop distance r - typically: in O(r c ) for constant c ¸ 2

23. Stefan Schmid, ETH Zurich @ WPDRTS 200623 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., B u (r) µ i=1…c B u i (r/2), 8 r

24. Stefan Schmid, ETH Zurich @ WPDRTS 200624 Connectivity: Unit Ball Graph S 111

25. Stefan Schmid, ETH Zurich @ WPDRTS 200625 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

26. Stefan Schmid, ETH Zurich @ WPDRTS 200626 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

27. Stefan Schmid, ETH Zurich @ WPDRTS 200627 More Models! Interference - Which senders can disturb the reception of which other peers? Node distribution Location information - GPS / Galileo device, etc.? Etc.! vs

28. Stefan Schmid, ETH Zurich @ WPDRTS 200628 Choice of Model (1) Which model to choose?

29. Stefan Schmid, ETH Zurich @ WPDRTS 200629 Choice of Model (2) Which model to choose? too pessimistic too optimistic General Graph UDG Quasi UDG d 1 Bounded Independence Unit Ball Graph

30. Stefan Schmid, ETH Zurich @ WPDRTS 200630 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!

31. Stefan Schmid, ETH Zurich @ WPDRTS 200631 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!

32. Stefan Schmid, ETH Zurich @ WPDRTS 200632 Thank you for your attention!

33. Stefan Schmid, ETH Zurich @ WPDRTS 200633 Questions? / Feedback?