Algorithms for Radio Networks Winter Term 2005/2006 21 Dec 2005 10th Lecture

1. Algorithms for Radio NetworksWinter Term 2005/200621 Dec 200510th Lecture Christian Schindelhauer schindel@upb.de

2. 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. Radio Broadcasting without ID • Theorem There is no deterministic broadcasting algorithm for the radio broadcasting problem (without id) • Proof: Consider the following graph: • Blue node sends (at any time)a message to the neighbors • 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 • Red node does not receive any message.

4. A simple random algorithm (I) • Every station uses the following algorithm • Simple-Random(t) begin ifmessage m is available then for i ← 1 tot do r ← result of a fair coin toss (0/1 with prob. 1/2) ifr = 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 2Δ/Δ (D+ log n).

5. 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. 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. 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. 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. Changing the Game: New Models • Probabilistic mode: • Number n of players is known • The maximal degree Δis known • But no ID • Restriction: What if the maximal degree is not known? • Corollary • BGI-Broadcast informs all nodes with probability1- 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. 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. 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. 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)= 1 stays informed in the next roundd • P(3,1)= 1 etc. • ...

13. 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. Survey of Randomized Broadcasting Algorithms • Lower bounds for random algorithms concerning expected round time: • Alon, Bar-Noy, Linial, Peleg, 1991(log2n) for diameter D=1 • Kushiletz, Mansour, 1998(D log (n/D)) • Expected round time of random algorithms • Gaber, Mansour, 2003 O(D+ log5 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) + log2 n) • Bar-Yehuda, Goldreich, Itai, 1992 O(n log n) if D is unknown

15. 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(n11/6) • Chlebus, Gasieniec, Östlin, Robson, 2000 O(n3/2) • Chrobak, Gasieniec, Rytter, 2001, O(n log2 n) • Kowalski, Pelc, 2002 O(n log n log D)

16. Thanks for your attention!End of 11th lectureNext lecture: We 18 Jan 2006, 4pm, F1.110Next exercise class: Th 19 Jan 2006, 1.15 pm, F2.211 or Tu 24 Jan 2006, 1.15 pm, F1.110Next mini exam Mo 13 Feb 2006, 2pm, FU.511