1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 23 Nov 2005 06th Lecture Christian Schindelhauer schindel@upb.de

2. Search Algorithms, WS 2004/05 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Theory of Wireless Routing

3. Search Algorithms, WS 2004/05 3 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

4. Search Algorithms, WS 2004/05 4 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 on a line with coordinates 0,d/n, 2d/n,...,d –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. u v

5. Search Algorithms, WS 2004/05 5 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

6. Search Algorithms, WS 2004/05 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Dilation  Is it possible to optimize congestion and dilation at the same time?  Scenario: – A grid of n nodes (for a square number n) –Demand: W/n 2 packets between each pair of nodes  Optimal path system w.r.t. dilation –send all packets directly from source to target –Dilation: 1 –Congestion:  (W) if the distance from source to target is at least (3/4) n, then the communication disks cover the grid So, a constant fraction of all W messages interfere with each other  Good path system w.r.t. congestion –send all packets on the shortest path with unit steps first horizontal and then vertical –Congestion: On all horizontal lines at most packets can interfere each other Influence of horizontal on vertical lines increases the congestion by at most a factor of 2. –Dilation:

7. Search Algorithms, WS 2004/05 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Dilation  Is it possible to optimize congestion and dilation at the same time?  Scenario: – A grid of n nodes (for a square number n) –Demand: W/n 2 packets between each pair of nodes  Good path system w.r.t. dilation –Build a spanning tree in H-Layout with diameter O(log n) –Dilation: O(log n) –Congestion:  (W (log n))  Theorem –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

8. Search Algorithms, WS 2004/05 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Communication Load

9. Search Algorithms, WS 2004/05 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load (equivalent description)

10. Search Algorithms, WS 2004/05 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load versus Load

11. Search Algorithms, WS 2004/05 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Communication Load versus Congestion and Area

12. Search Algorithms, WS 2004/05 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Trade-Off between Dilation and Congestion

13. Search Algorithms, WS 2004/05 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Tradeoff between Dilation and Congestion Dilation Congestion  n sites on a grid  Between each pair of sites demand of W/n 2 packets any basic network Grid Tree

14. Search Algorithms, WS 2004/05 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Congestion versus Energy  Is it possible to optimize congestion and energy at the same time?  Scenario: –The vertex set U ,n for a  [0,0.5] consists of two horizontal parallel line graphs line graphs 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   Then, there are n other nodes equdistantly placed between the blue nodes with distance  /n vertices are equidistantly placed between the blue nodes  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

15. Search Algorithms, WS 2004/05 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy and Congestion are incompatible

16. Search Algorithms, WS 2004/05 16 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*) either n 1/3 or

17. Search Algorithms, WS 2004/05 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Yao-Graph YG 6 Choose nearest neighbor in each sector c-spanner, i.e. constant stretch-factor distributed construction c-spanner: for every pair of nodes u,v there exists a path P s.t. ||P|| ≤ c · ||u,v|| c-Spanner [Chew86]

18. Search Algorithms, WS 2004/05 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Spanner Graphs and Yao-Graphs  Definition –A c-Spanner is a graph where for every pair of nodes u,v there exists a path P s.t. ||P|| ≤ c · ||u,v||.  Motivation: –Short paths –Energy optimal paths  Example of a Spanner-Graph: –Yao-graph  Defintion Yao-Graph (Theta-Graph) –Given a node set V –Define for each node k sectors S 1 (u), S 2 (u),..., S k (u) of angle  = 2  /k with same orientation –The Yao-Graph consists of all edges E = (u,v | exists i  {1,..,k}: v  S i (u) and for all v’  S i (u): ||u,v’|| ≥ ||u,v|| } YG 6

19. Search Algorithms, WS 2004/05 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Yao-Graph is a c-Spanner  Theorem: –The Yao-Graph is a c-Spanner for more than 6 sectors.  Proof –By induction over the distance of nodes –...

20. Search Algorithms, WS 2004/05 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Weaker Spanning Weak-Spanner [FMS97]...sufficient for allowing routing which approximates minimal congestions by a factor of O(Int(G) g(V)) [Meyer auf der Heide, S, Volbert, Grünewald 02] Power-Spanner [LWW01, GLSV02]...approximates energy-optimal path-system for every pair of nodes u,v exists a path inside the disk C( u, c · ||u,v|| ) for every pair of nodes u,v exists path P s.t. |P| ≤ c · |P opt | |P| = Σ |v i, v i+1 | d v u P opt P

21. Search Algorithms, WS 2004/05 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Spanners, Weak Spanners, Power Spanners  Theorem –Every c-Spanner is a c-weak spanner.  Proof: –exercise  Theorem –Every c-weak-Spanner is a c’-power Spanner when d  2.  Proof: –straightforward for d>2 –involved construction for d=2

22. Search Algorithms, WS 2004/05 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Koch Curve is not a Spanner  Koch-Curves: Koch 0, Koch 1, Koch 2,...  Theorem –The Koch Curve is not a c-Spanner  Theorem –The Koch Curve is a weak 1-Spanner.

23. Search Algorithms, WS 2004/05 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Yao-Family Yao-Graph Spanner ⊇ SparsY Sparsified Yao-Graph use only the shortest ingoing edges weak- & power-Spanner, constant in-degree ⊇ SymmY Symmetric Yao-Graph only symmetric edges not a spanner, nor weak spanner, yet power-spanner Disadvantage: Unbounded in-degree Interferences !

24. 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention End of 6th lecture Next lecture:Mi 30 Nov 2005, 4pm, F1.110 Next exercise class: Tu 29 Oct 2005, 1.15 pm, F2.211 or Th 01 Dec 2005, 1.15 pm, F1.110