1. 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/2006 21 Dec 2005 10th Lecture Christian Schindelhauer schindel@upb.de

2. Algorithms for Radio Networks 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Radio Broadcasting  Broadcasting –A sender distributes a message to n radio stations  Radio Broadcasting –Undirected Graph G=(V,E) describes possible connections If edge {u,v} exists, u can transmit to v and vice versa If no edge exists, then there is no reception and no interference –One frequency, stations communicate in a round model –If more than one neighbored station send at the same time, no signal is received (not even an interference signal)  Main problem: –Graph G=(V,E) is unknown to the participants –Distributed algorithm avoiding conflicts

3. Algorithms for Radio Networks 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Radio Broadcasting without ID  Theorem There is no deterministic broadcasting algorithm for the radio broadcasting problem (without id)  Proof: Consider the following graph: 1.Blue node sends (at any time) a message to the neighbors 2.As soon they are informed, they behave completely synchronously –because they use the same algorithm –so, they send (or do not send) always at the same time 3.Red node does not receive any message.

4. Algorithms for Radio Networks 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A simple random algorithm (I)  Every station uses the following algorithm  Simple-Random(t) begin if message m is available then for i ← 1 to t do r ← result of a fair coin toss (0/1 with prob. 1/2) if r = 1 then send m to all neighbors fi od fi end  Theorem For appropriate c>1 we have: Simple-Random informs the complete network with probability of at least 1-O(n  k ) within time c 2 Δ / Δ ( D+ log n).

5. Algorithms for Radio Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Extending the Deterministic Model  Model too restrictiv  New deterministic model: –Every of the n players knows his unique id number from the set {1,..,n}  Probabilistic model: –Die number n of players is known –The maximal degree Δ is known –But no ID is available

6. Algorithms for Radio Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Decay (I)  Idee: randomized thinning out of the players Decay(k,m) begin j ← 1 repeat j ← j + 1 Send message to all neighbors r ← result of fair coin toss (0/1 with prob. 1/2) until r=0 oder j > k end

7. Algorithms for Radio Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Decay (II)  d neighbors are informed  All d neighbors start simultanously (k,m)  P(k,d):Prob. that message is received by d neighbors within at most k rounds: Lemma For d≥2 :

8. Algorithms for Radio Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer BGI-Broadcast [Bar-Yehuda, Goldreich, Itai 1987]  All informed players have synchronized round counters, i.e. –Time is attached to each message –and incremented in each round BGI-Broadcast( Δ,  ) begin k ← 2 log Δ t ← 2 log (N/  ) wait until message arrives for i ← 1 to t do wait until (Time mod k) = 0 Decay(k,m) od end Theorem BGI-Broadcast informs all nodes with probability 1-  in time O((D+log(n/  )) log Δ)

9. Algorithms for Radio Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Changing the Game: New Models  Probabilistic mode: –Number n of players is known –The maximal degree Δ is known –But no ID  Restriction: What if the maximal degree is not known? –Corollary BGI-Broadcast informs all nodes with probability 1-  in time O((D+log(n/  )) log n)  Determinististic model: –Each of the n players knows a unique identifier (id) of the set {1,..,n} and knows n

10. Algorithms for Radio Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Determinism versus Probabilism  Theorem For every distributed deterministic Radio-Broadcasting algorithm using IDs there is a graph with D=2 that cannot be completely informed within time n-2.  Theorem BGI-Broadcast informs all nodes with probability 1-  in time O((D+log(n/  )) log Δ) for any e>0.  Theorem For any constant  >0 BGI-Broadcast informs all nodes of a graph with D=2 with probability 1-  in time O((log n) 2 ).

11. Algorithms for Radio Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Decay  d neighbors are informed  All d neighbors start simultanously (k,m)  P(k,d):Prob. that message is received from d neighbors within at most k rounds: Lemma For d≥2 :

12. Algorithms for Radio Networks 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof of Lemma (Part I)  P(k,d):Prob. that the message is received from d neighbors within at most k rounds  0 neighbored players are informed: –P(1,0)= 0 Chance of being informed in the first round by nobody –P(2,0)= 0 –P(3,0)= 0 –...  1 neighbored player is informed: –P(1,1)= 1 One player cannon cause any conflict –P(2,1)= 1stays informed in the next roundd –P(3,1)= 1etc. –...

13. Algorithms for Radio Networks 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof of Lemma (Part I)  P(k,d): –Prob. that the message is received from d neighbors within at most k rounds  2 neighbored players are informed: –P(2,1)= 0 Two nodes send in the first round. No chance –P(2,2)= P(no player continues) P(1,0) + P(one player continues) P(1,1) + P(two players continue) P(1,1) = 1/4 P(1,0) + 1/2 P(1,1) + 1/4 P(2,1) = 0 + 1/2 + 0 = 1/2

14. Algorithms for Radio Networks 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Survey of Randomized Broadcasting Algorithms  Lower bounds for random algorithms concerning expected round time: –Alon, Bar-Noy, Linial, Peleg, 1991  (log 2 n) for diameter D=1 –Kushiletz, Mansour, 1998  (D log (n/D))  Expected round time of random algorithms –Gaber, Mansour, 2003O(D+ log 5 n) if the network is known –Bar-Yehuda, Goldreich, Itai, 1992 O((D+log n) log n) (presented here) –Czumaj, Rytter, 2003:O(D log (n/D) + log 2 n) –Bar-Yehuda, Goldreich, Itai, 1992 O(n log n) if D is unknown

15. Algorithms for Radio Networks 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Survey of Deterministic Algorithms  Lower bounds for deterministic algorithms concerning expected round time: –Bar-Yehuda, Goldreich, Itai, 1992  (n) (presented here)  Worst case time of deterministic algorithms –Chlebus, Gasieniec, Gibbons, Pelc, Rytter, 1999 O(n 11/6 ) –Chlebus, Gasieniec, Östlin, Robson, 2000 O(n 3/2 ) –Chrobak, Gasieniec, Rytter, 2001, O(n log 2 n) –Kowalski, Pelc, 2002O(n log n log D)

16. 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention! End of 11th lecture Next lecture:We 18 Jan 2006, 4pm, F1.110 Next exercise class: Th 19 Jan 2006, 1.15 pm, F2.211 or Tu 24 Jan 2006, 1.15 pm, F1.110 Next mini examMo 13 Feb 2006, 2pm, FU.511