1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/ Dec th Lecture Christian Schindelhauer

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

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.

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

Algorithms for Radio Networks 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A Simple Random Algorithm (II)  D: diameter of the graph  Δ: degree Lemma A neighbor node will be informed in each round with a probability of at least Δ 2  Δ, if at least one neighbor is informed. Proof: 1.The probability that exactly one of the m ≤ Δ informed neighbors sends (without interference) is: Prob. that m-1 neighbors do not send Prob. that one neighbor sends m number of possible senders

Algorithms for Radio Networks 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Analyzing Random Algorithms A neighbor node is in formed in a round with probability p ≥ Δ 2  Δ, if at least one neighbor is informed. –Consider any node with distance D to the source –Let (u,u 1,u 2,..,u D ) be a path from the source u to this node –We underestimate the real information flow and consider only the information flow on the shortest path p 1p1p p p p p 1p1p1p1p 1p1p 1 1p1p

Algorithms for Radio Networks 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A Markov-Process for the Spread of Information Time p 1p1p p 1p1p p 1p1p p 1p1p p 1p1p 1 1 p 1p1p p 1p1p Path Lemma For all α>1 and β ≥ 0: If on a path of length D a message advances independently with probability p and stops with probability 1-p, then the probability that the information has not reached the end of the path after t steps, where is at most

Algorithms for Radio Networks 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof of Lemma (Where we stopped last time...)  Let X t denote the distance which the message advances in t rounds  Then, the probability that we advance exactly i steps is defined as  Since it is a probability we have  The probability that we advance at most D steps is given by  We have seen that the quotient of these probababilities is:

Algorithms for Radio Networks 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof of the Lemma (continued)  Consider an upper bound S for i: i≤S≤t and choose  Then,  and  This implies

Algorithms for Radio Networks 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof of Lemma (completed)  Let S=D+   Then for

Algorithms for Radio Networks 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Run Time of Simple-Random  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).  Beweis: –Consider node and a path to the source of length at most D –Consider information flow on the path p ≥ Δ 2  Δ –Use α=2 and β = (k+1) log n –The above lemma shows the probability to succeed on one path is given by at most –To upperbound the probability of failure for all n nodes we have to multiply this term by n yielding –This error bounds refers to the time bound

Algorithms for Radio Networks 12 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 knwos 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

Algorithms for Radio Networks 13 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

Algorithms for Radio Networks 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Decay (II)  d neighbors are informed  Alle 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 :

Algorithms for Radio Networks 15 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

Algorithms for Radio Networks 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer BGI-Broadcast Analyse (I) Theorem BGI-Broadcast informs all nodes with probability 1-  in time O((D+log(n/  )) log Δ) Beweis: –Assume t is arbitrarily large –Consider delay T U on the path to a node u with distance of at most D Probability that node is not informed If –choose α=2 and β = 2 log (n/  ) yielding –Prob.

Algorithms for Radio Networks 17 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  Resctriction: What if the maximal degree is not known? –Corollary (Proof as an exercise) BGI-Broadcast informs all nodes with probability 1-2  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

Algorithms for Radio Networks 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Lower Bound for Deterministic Radio Broadcasting (I)  Let D be the maximimal distance of a node from the source.  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.  Proof strategy: –Given deterministic algorithm –Construct graph –Prove that the time bound is wrong

Algorithms for Radio Networks 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Lower Bounds for Radio Broadcasting (II)  For S ⊆ {1,..,n} define G S as –V = {0,..,n+1} –E = {{0,i} | i  {1,..n}}  {{i,n+1} | i  S}  Deterministic model: –Each of the n players knows his unique identifier ID(i) from the set {0,..,n+1} –Each node has his own algorithm defined by A(i,t) –A(i,t) = 1, if player i sends in round t –A(i,t) = 0, if player i does not send in round

Algorithms for Radio Networks 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Lower Bounds for Radio Broadcasting (III)  The good guys choose algorithms A  The bad boys (adversary) choose a set S which is the set of players which have an edge to the target node n+1 –We are bad and find S≠  for each algorithm algorithm A (we know A)  Concessions to A –Player i has ID(i)=i (advantage for the algorithm) –A may start in round 0 with A(i,0) = 1 –Then in round 1 all nodes {1,..,n} are informed  Task for A starting from round 1: –Inform player n+1 in at most n-1 rounds

Algorithms for Radio Networks 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hitting Game (I) A(1,t) A(2,t) A(3,t) A(4,t) A(5,t) A(6,t) Given n  (n-1) -matrix A inserts 0 or 1 in each cell B may erase up to m=n-1 rows, i.e. set to 0 A wins, if there is a column with exactly one entry 1 B wins, if every column has only 0s or at least two 1s A wins

Algorithms for Radio Networks 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hitting Game (II) A(1,t) A(2,t) A(3,t) A(4,t) A(5,t) A(6,t) B wins Given n  (n-1) -matrix A inserts 0 or 1 in each cell B may erase up to m=n-1 rows, i.e. set to 0 A wins, if there is a column with exactly one entry 1 B wins, if every column has only 0s or at least two 1s

Algorithms for Radio Networks 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hitting Game (III) Winning Strategy for B B-wins(A,n) begin S ← {1,…,n} while there exists a column in A with exactly one entry 1 do t ← column with exactly one entry 1 in A(1,t),A(2,t),…,A(n,t) i ← row with A(i,t) = 1 S ← S \ {i} A(i,t) ← 0, for all t  {1,..,n-1} od end A(1,t) A(2,t) A(3,t) A(4,t) A(5,t) A(6,t) B wins

Algorithms for Radio Networks 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Hitting Game (III) Winning Strategy for B B-wins(A,n) begin S ← {1,…,n} while there exists a column in A with exactly one entry 1 do t ← column with exactly one entry 1 in A(1,t),A(2,t),…,A(n,t) i ← row with A(i,t) = 1 S ← S \ {i} A(i,t) ← 0, for all t  {1,..,n-1} od end Lemma For each matrix A the adversary B wins. Proof: 1.After termination of while P do.. od we have ¬P i.e. there is now column in A with exactly one entry 1 S≠  while loop is used for at most (n  1) rounds, since each column is treated only at most once and the number of columns is n-1

Algorithms for Radio Networks 25 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Lower Bound for Radio Broadcasting  The good guys choose algorithms A  The bad boys (adversary) choose a set S which is the set of players which have an edge to the target node n+1 –We are bad and find S≠  for each algorithm algorithm A (we know A) –We play „B-wins“ and receive S –The graph is connected since S≠  –In each of the n-1 rounds either none or at least two of the players send

Algorithms for Radio Networks 26 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Determinism ersus 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 ).

27 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Merry Christmas, happy holidays, happy new year,... and all the rest Thanks for your attention and see you next year End of 10th lecture Next lecture:Mi 11 Jan 2006, 4pm, F1.110 Next exercise class: Tu 20 Dec 2005, 1.15 pm, F2.211 or Th 15 Dec 2005, 1.15 pm, F1.110 Next mini examMo 09 Jan 2006, 2pm, FU.511