Fast Broadcasting and Gossiping in Radio Networks

Fast Broadcasting and Gossiping in Radio Networks Marek Chrobak Leszek Gasieniec Wojciech Rytter

Selectors

Lemma 1

Proof (1/3)

Proof (2/3)

Proof (3/3)

Algorithm (1/2)

Algorithm (2/2) Algorithm DoBroadcast. The algorithm consists of stages, with each stage having log n+1 = O(logn) Steps. The transmission set at the jth step of stage s is Sj,s mod mj. Stage 1: S1,1 , S2,1 , S3,1 , S4,1 ,… , Slogn,1 Stage 2: S1,2 , S2,2 , S3,2 , S4,2 ,… , Slogn,2 . . . Stage s: S1,s mod mj , S2,s mod mj, S3,s mod mj , S4,s mod mj ,… , Slogn,s mod mj

m m m m m m Analysis (1/5) dormant Each change dormant→frontier or frontier→inner contributes a unit to the mesure of progress. frontier 2n-1 => all the nodes are inner inner

Analysis (2/5) Fix some stage s, if we can find s' ≥ s such that in each stage s, s+1, …, s' the amortized progress is Ω(1/logn) ……….. • O(nlogn) stages • progress (2n-1) • time complexity O(nlog2n)

Yj X inner v F frontier dormant Analysis (3/5) 2f-1 ≤ F ≤ 2f j = 1,…, f Let Yj be the set of nodes that received the message for the first time in stages s, s+1,…,s+mj-1 (but were dormant when stage s started).

Yj X inner v F frontier dormant Analysis (4/5) Case 1: There is j for which |Yj| ≥ 2j 2j/O(2jlogn) = Ω(1/logn)

Yj pick j let 2j-1≤ |X| ≤ 2j |Yj| ≤ 2j =>family contains a set Sj,ithat hits X and avoids Yj. This set will occur in one of the stages s, s+1, …,s+mj-1 X inner v F frontier dormant Analysis (5/5) Case 2: For each j we have |Yj| ≤ 2j In this case all nodes if F will become inner after mf stages. => 2f-1/2flogn = Ω(1/logn)

Fast Broadcasting and Gossiping in Radio Networks