1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 25 Jan 2006 13th Lecture Stefan Rührup sr@upb.de

2. Algorithms for Radio Networks 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Modeling Worst Case Mobility Problem: –Mobile users –positions are known at the moment (t=0), but not in the future (t=∆). –How to adjust the transmission range? Reasonable restriction for the worst case –velocity bound  pedestrian model –acceleration bound  vehicular model transmission range

3. Algorithms for Radio Networks 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Modeling Worst Case Mobility V:Pedestrian Model ↔ Maximum velocity ≤ v max A:Vehicular Model ↔ Maximum accelaration ≤ a max

4. Algorithms for Radio Networks 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Modeling Worst Case Mobility V:Pedestrian Model ↔ Maximum velocity ≤ v max A:Vehicular Model ↔ Maximum acceleration ≤ a max

5. Algorithms for Radio Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Mobile Ad Hoc Network Basic idea: Maintain the network for a time interval Δ As a start: synchronous round model In every round of duration Δ –Determine positions (speed vectors) of possible communication partners –Establish (stable) communication links –Update routing information –Do the job, i.e. packet delivery, video streams, telephone,…

6. Algorithms for Radio Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Mobility Model Velocity bounded mobility model Pedestrians move with a (known) bounded velocity For all i  {1,…,n}: Acceleration bounded mobility model Cars move with a (known) bounded acceleration For all i  {1,…,n}: Technical assumptions: –polynomial distances and speeds –complete knowledge of position

7. Algorithms for Radio Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Taming Mobility in the Link Layer General Concept –Increase transmission distance to guarantee a certain life span Δ for each link –Apply the mobility model (i.e. a max or v max is known) transmission range start position (t 0 ) end position (t 0 +Δ)

8. Algorithms for Radio Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer vxvx x Example Trains moving in opposite directions Transmission range only sufficient in the static case If we take the velocity into account: We need a modified distance measure

9. Algorithms for Radio Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Given the positions u,w and the velocity bound v max Maximum distance after the time interval ∆: Velocity bounded (pedestrian) model uncertainty uw

10. Algorithms for Radio Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Transmission range for the velocity bounded (pedestrian) model: end Walking range start end Sa- tisfies triangle inequality Pedestrian Model transmission range for node u and t=0

11. Algorithms for Radio Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Acceleration bound a max Positions u,v and speed vectors u’,v’ known Maximum distance after time interval ∆: With given distances and we can approximate by a constant factor. Vehicular Model uncertainty due to acceleration uw velocity

12. Algorithms for Radio Networks 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer end start end Sa- tisfies triangle inequality Vehicular Model Transmission range for the acceleration bounded (vehicular) model:

13. Algorithms for Radio Networks 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication links (edges) can interfere –edges interfere, if one node is located within the transmission radius of a node of the other edge Velocity bounded model: interference, if Acceleration bounded model: interference, if and interference p q e e’ Interferences

14. Algorithms for Radio Networks 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer An edge g interferes with edge e in the 1. pedestrian (v) model 2. vehicular (a) model No interference Interference g g e e e e g e g p q q p Interferences more formally

15. Algorithms for Radio Networks 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Interference number of a network What is the maximum number of interfering edges in a network? Interference number is influenced by the transmission range and the positions of the nodes How many nodes can meet in one place and form a crowd?  Definition of crowdedness Crowdedness gives a lower bound for interferences In both mobility models we observe for all connected graphs G(V,E): Int(G)  crowd(S) - 1

16. Algorithms for Radio Networks 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Crowdedness of node set –natural lower bound on network parameters (like diversity) Pedestrian (v) model: –Maximum number of nodes that can collide with a given node in time span [0,Δ] Vehicular (a) model: –Maximum number of nodes that may move to node u meeting it with zero relative speed in time span [0,Δ] crowd(S) := max u  S crowd(u) Crowdedness more formally

17. Algorithms for Radio Networks 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Transmission ranges defined for two mobility models Basis for –the definition of interferences –construction of a network topology –analyisis: interference number, congestion Mobility Models

18. Algorithms for Radio Networks 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hierarchical Grid Graph (pedestrian model) Start with grid of box size Δ v max For O(log n) rounds do –Determine a cluster head per box –Build up star-connections from all nodes to their cluster heads –Erase all non cluster heads –Connect neighbored cluster heads –Increase box size by factor 2 od

19. Algorithms for Radio Networks 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Interference number of the network G: Int(G) := max e  E(G) |Int(e)| Load and path system –Set of all message paths defines path system P –l(e) := # number of packets sent over edge e according to P Congestion of an edge (with respect to a path system P ) –Congestion of a network G: Mobile spanner –A network is a mobile spanner, if for all u,w exists a path connecting u and v with bounded length Load, Congestion and Spanners

20. Algorithms for Radio Networks 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A graph G is called a mobile spanner, if for all nodes there is a path in G with Congestion of a path system: A mobile spanner G approximates an optimal path system Mobile Spanner

21. Algorithms for Radio Networks 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Results Lemma In both mobility models α  {v,a} every mobile spanner is also a mobile power spanner, i.e. for some ß≥1 for all u,w  S there exists a path (u=p 0,p 1,…,p k =w) in G such that:

22. Algorithms for Radio Networks 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Results Theorem Given a mobile spanner G for any of our mobility models then –for every path system P in a complete network C –there exists a path system P‘ in G such that Theorem The Hierarchical Grid Graph constitutes a mobile spanner with at most O(crowd(V) + log n) interferences (for both mobility models). The Hierarchical Grid Graph can be built up in O(crowd(V) + log n) parallel steps using radio communication

23. Algorithms for Radio Networks 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hierarchical Grid Graph (vehicular model) Algorithm: –Consider coordinates (x(s i ),y(s i ),x(s‘ i ),y(s‘ i )) –Start with four-dimensional grid with rectangular boxes of size (6Δ²a max, 6Δ²a max,2Δv max,2Δv max ) –Use the same algorithm as before x vxvx t=0 x vxvx t= Δ x vxvx t=2 Δ

24. Algorithms for Radio Networks 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Stable Basic Networks under Worst Case Mobility Corollary There exist distributed algorithms that construct a mobile network G for velocity bounded and acceleration bounded model with the following properties: 1.G allows routing approximating the optimal congestion by O(log² n) 2.Energy-optimal routing can be approximated by a factor of O(1) 3.G approximates the minimal interference number by O(log n) 4.The degree is O(crowd(S)+ log n) 5.The diameter is O(log n) Still no routing can satisfy small congestion and energy at the same time!

25. Algorithms for Radio Networks 25 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Applications of mobility models Various protocols for mobile networks more or less influenced by mobility (esp. routing) How to evaluate and compare protocols? –simulation of mobile networks –mobility models as benchmarks traces or synthetic models?

26. Algorithms for Radio Networks 26 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Requirements imitation of realistic motion mobility model should not be too complex ⇒ simplifications (depends on the network model) e.g., cellular network: –random walk on network cells –speed and direction vectors can be neglected many application-specific mobility models

27. Algorithms for Radio Networks 27 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Parameters Bounds on velocity, acceleration or the change of direction Is the motion independent of other mobile hosts? –entity mobility models –group mobility models Degree of randomness –sharp turns possible or smoothed motion? Granularity –macroscopic view (→ cellular networks) –microscopic view (→ ad hoc networks)

28. Algorithms for Radio Networks 28 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Brownian Motion (microscopic view) –speed an direction are chosen randomly in each time step (uniformly from [v min,v max ] and [0,  ]) Random Walk –macroscopic view –memoryless –e.g., for cellular networks –movement from cell to cell –choose the next cell randomly –residual probability [Camp et al. 2002] Brownian Motion, Random Walk

29. Algorithms for Radio Networks 29 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer [Camp et al. 2002] [Johnson, Maltz 1996] Random Waypoint Mobility Model move directly to a randomly chosen destination choose speed uniformly from [v min,v max ] stay at the destination for a predefined pause time

30. Algorithms for Radio Networks 30 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer adjustable degree of randomness velocity: direction: Gauss-Markov Mobility Model [Liang, Haas 1999] mean random variable gaussian distribution tuning factor [Camp et al. 2002] α=0.75

31. Algorithms for Radio Networks 31 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Concept map of mobility models [Bettstetter 2001]

32. 32 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention! End of 13th lecture Next lecture:We 1 Feb 2006, 4pm, F1.110 Next exercise class: Th 26 Jan 2006, 1.15 pm, F2.211 or Tu 31 Jan 2006, 1.15 pm, F1.110 Next mini examMo 13 Feb 2006, 2pm, FU.511