1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 16 Nov 2005 5th Lecture Christian Schindelhauer schindel@upb.de

2. Alg. for Radio Networks, WS05/06 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A Simple Physical Network Model Homogenous Network of –n radio stations s 1,..,s n on the plane Radio transmission –One frequency –Adjustable transmission range Maximum range > maximum distance of radio stations Inside the transmission area of sender: clear signal or radio interference Outside: no signal –Packets of unit length

3. Alg. for Radio Networks, WS05/06 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Routing Problem Given: –n points in the plane, V=(v 1,..,v n ) representing mobile nodes of a mobile ad hoc network –the complete undirected graph G = (V,E) as possible communication network representing a MANET where every connection can be established Routing problem: –f : V  V  N, where f(u,v) packets have to be sent from u to v, for al u,v  V –Find a path for each packet of this routing problem in the complete graph The union of all path systems is called the Link Network or Communication Network

4. Alg. for Radio Networks, WS05/06 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Formal Definition of Interference Let D r (u) the disk of radius u with center u in the plane Define for an edge e={u,v} D(e) = D r (u)  D r (v) The set of edges interfering with an edge e = {u,v} of a communication network N is defined as: The Interference Number of an edge is given by |Int(e)| The Interference Number of the Network is max{|Int(e} | e  E}

5. Alg. for Radio Networks, WS05/06 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Formal Definition of Congestion The Congestion of an edge e is defined as: The Congestion of the path system P is defined as The Dilation D(P) of a path system is the length of the longest path.

6. Alg. for Radio Networks, WS05/06 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy The energy for transmission of a message can be modeled by a power over the distance d between sender and transceiver Two energy models: –Unit energy accounts only the energy for upholding an edge Idea: messages can be aggregated and sent as one packet –Flow Energy Model: every message is counted separately

7. Alg. for Radio Networks, WS05/06 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion, Energy and Dilation Congestion Energy Dilation Maximum number of hops (diameter of the network) Sum of energy consumed in all routes Maximum number of packets interfering at an edge

8. Alg. for Radio Networks, WS05/06 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Trade-offs For a given a set of nodes –the complete network optimizes dilation –the MST optimizes unit energy –the Gabriel Graph optimizes flow energy (with an energy-optimal path system) Is it possible to construct a network (with a path system) that optimizes –Energy and Dilation? –Congestion and Dilation? –Congestion and Energy?

9. Alg. for Radio Networks, WS05/06 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy versus Dilation Is it possible to optimize energy and dilation at the same time? Scenario: – n+1 equidistant nodes u 0,..., u n, distance d/n – Demand: W packets from u 0 to u n Energy-optimal path system (D=n): Dilation-optimal path system (D=1): u0u0 unun d u0u0 unun u0u0 unun

10. Alg. for Radio Networks, WS05/06 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy versus Dilation n+1 nodes, demand: W packets from u 0 to u n Theorem: In this scenario we observe for all path systems: Proof: –Consider only one packet and its path with hop distances d 1 d 2,.., d m with –The term is minimized if d 1 =d 2 =,.., =d m =d/m. –Then the energy is d 2 /m and the claim follows. –For the flow energy sum over all W packets.

11. Alg. for Radio Networks, WS05/06 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Tradeoff between Energy and Dilation Energy E Dilation D Demand of W packets between u and v any basic network u v

12. Alg. for Radio Networks, WS05/06 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Dilation Theorem 11 * Given a grid G n of nodes For any path system in this scenario we observe Proof strategy: –Vertically split the square into three equal rectangles –Consider only 1/9 of the traffic from the leftmost to the rightmost rectangle –Define the communication load of an area –Proof that the communication load is a lower bound for congestion –Minimize the communication load for a given dilation between the rectangles GnGn *) Meyer auf der Heide et al. “Congestion, Energy and Dilation in Radio Networks, TOCS 2004

13. Alg. for Radio Networks, WS05/06 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Communication Load Interference function for a point x and an edge e: Communication load for a point x and a bounded Region R: x

14. Alg. for Radio Networks, WS05/06 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load (equivalent description) Devide region R into elementary regions Communication load for a an elementary region R’: elementary Regions

15. Alg. for Radio Networks, WS05/06 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load versus Load Lemma 6 x How many non-interfering regions can overlap at x? x AiAi

16. Alg. for Radio Networks, WS05/06 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load versus Congestion and Area Lemma 7

17. Alg. for Radio Networks, WS05/06 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Trade-Off between Dilation and Congestion Theorem 11 S

18. Alg. for Radio Networks, WS05/06 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Dilation Congestion  n sites on a grid  Between each pair of sites demand of W/n 2 packets any basic network Grid Tree Tradeoff between Dilation and Congestion

19. Alg. for Radio Networks, WS05/06 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Energy Is it possible to optimize congestion and dilation at the same time? Scenario: –node set U ,n for a  [0,1/2] with n  blue nodes on each line –neighbored (and opposing) blue vertices have distance  /n  –Vertical pairs of opposing vertices of the line graphs have demand W/n  –n - n  other nodes equdistantly placed between the blue nodes with distance  /n  /n  demand: W/n 

20. Alg. for Radio Networks, WS05/06 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Energy Best path system w.r.t. Congestion –One hop communication between blue nodes: Congestion: O( W/n  ) –Unit-Energy: :  (  2 n -  ) –Flow-Energy:  (W  2 n -  ) Best path w.r.t Energy: –U-shaped paths –Unit-Energy: O (  2 n -1 ) –Flow-Energy: O (  2 n - 1 W) –Congestion:  (W) Choose  =1/3  /n  demand: W/n 

21. Alg. for Radio Networks, WS05/06 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy and Congestion are incompatible Theorem 13:

22. Alg. for Radio Networks, WS05/06 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Incompatibility of Congestion and Energy Congestion Energy  n 1/3 blue sites  One packet demand between all vertical pairs of blue sites C* = O(1) E*=O(1/n) C  (n 1/3 C*) O(1/n 2/3 ) any link network E  (n 1/3 E*) eithern 1/3 or

23. 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention End of 5th lecture Mini-Exam:Mon 21 Nov 2005, 2pm, FU.511 Next lecture:Wed 16 Nov 2005, 4pm, F1.110