adaptiveness vs. obliviousness and randomization vs. determinism Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc University of Quebec en Outaouais Time of Radio Broadcasting

2 Radio network  n nodes with different labels 1,…,N (N=  (n)) communicate via radio network modeled by symmetric graph G  node v knows only it own label and parameter N  communication is in synchronous steps  in every step, node v is either –transmitting, or –receiving

Time of Radio Broadcasting3 Message delivery  Node v receives a message from node w in step i if –node v is receiving in step i –node w is a neighbor of node v in network G and is transmitting in step i –every neighbor z  w of node v in network G is receiving in step i  Otherwise node v receives nothing

Time of Radio Broadcasting4 Broadcasting problem Broadcasting problem: some node, called source, has the message, and transmits it in step 0 Goal: all nodes must know the source message Spontaneous transmission allowed: every node knows global steps, starting step and can transmit since this time Measure of performance: time by the first step when all nodes have the source message

Time of Radio Broadcasting5 Obliviousness and randomization Algorithms may be  adaptive (deterministic or randomized)  oblivious –deterministic - the sequence of transmissions of every node is fixed prior the broadcasting –randomized - the sequence of probabilities of transmission is fixed, for every node, prior the broadcasting

Time of Radio Broadcasting6 Bibliography [ABLP] N. Alon, A. Bar-Noy, N. Linial, D. Peleg: A lower bound for radio broadcast. J. of Computer and System Sciences, [CGLP] B. Chlebus, L. Gasieniec, A. Lingas, A. Pagourtzis: Oblivious gossiping in ad-hoc radio networks. DIALM, [CMS-ft] A. Clementi, A. Monti, R. Silvestri: Round robin is optimal for fault-tolerant broadcasting on wireless networks. ESA, [CMS] A. Clementi, A. Monti, R. Silvestri: Selective families, superimposed codes, and broadcasting on unknown radio networks. SODA, [CGGPR] B. Chlebus, L. Gasieniec, A. Gibbons, A. Pelc, W. Rytter: Deterministic broadcasting in unknown radio networks. Distributed Computing, [KP] D. Kowalski, A. Pelc: Broadcasting in undirected ad hoc radio networks. PODC, 2003, to appear.

Time of Radio Broadcasting7 Goals and results GOAL: understand impact of obliviousness and/or randomization for broadcasting time

Time of Radio Broadcasting8 Adaptive deterministic algorithms  Algorithm broadcasting in time O(n) in [CGGPR]  Lower bound  (n) for n-node networks with constant diameter - correct proof unknown How to choose S,R to get linear broadcasting time of algorithm A on G S,R 0 1n S R Network G S,R R  {n+1,…,2n} S  {1,…, n} layer 0 layer 1 layer 2

Time of Radio Broadcasting9 Lower bound Theorem 1: For every broadcasting algorithm A and every n, there is a network G S,R on  (n) nodes such that broadcasting time of algorithm A on G S,R is  (n). Proof : We construct sets S,R starting from sets S 0 = {1,…,n} and R 0 = {n+1,…,2n}. We proceed construction until step n/2 of algorithm A, to obtain sets S = S n/2 and R = R n/2. Problem : network G is not defined Solution : –introducing abstract object corresponding to the real ones: history and transmitters, and preserving theirs required properties –for constructed network, real and abstract objects are equal

Time of Radio Broadcasting10 Proof of Theorem 1 - objects For every step k  n/2 define (abstract) objects : H k (v) : the history of received messages by the end of step k, for every node v  {0,…,2n} T k : set of nodes v transmitting in step k under given history H k (v) S k  S k-1 : a subset of {1,…,n} being the output of function MODIFY(S k-1,T), where T = {T 1,…,T k-1 }; initially S 0 = {1,…,n} R k  R k-1 : a subset of {n+1,…,2n} being the output of function MODIFY(R k-1,T), where T = {T 1,…,T k-1 }; initially R 0 = {n+1,…,2n}

Time of Radio Broadcasting11 Proof of Theorem 1 - construction Procedure MODIFY(S,T)  set stop:=0  while stop = 0 do –stop:=1 –if there is a set T l  T such that |T l  S | = 1 then choose such a set with smallest index, say T k, such that T k  S = {i} remove node i from S set stop:=0

Time of Radio Broadcasting12 Proof of Theorem 1 - invariant The following invariant is preserved after step k of the construction, according to sets S k and R k and objects: –No single transmitter : for every set T l, l  k, |T l  S k |  1 and |T l  R k |  1. –Removed nodes correspond to disjoint transmitters’ sets : At least n-|S k | sets T l are disjoint with S k, and at least n-|R k | sets T l are disjoint with R k, for l  k. –Large size : |S k |  n-k and |R k |  n-k. –No message in second layer : if v  R k then H k (v) is the empty history.

Time of Radio Broadcasting13 Oblivious randomized algorithms  Algorithm broadcasting in time O(n log 2 n) in [CGLP]  Lower bound  (D+ log 2 n) follows from [ABLP] Algorithm Randomized-Oblivious  count := 1  repeat N 2 /log N times ã for l := 1 to log N do => iteration of stages (a.) each node transmits independently with probability 2 -l (b.) node with label count transmits, count := count+1 mod N

Time of Radio Broadcasting14 Analysis of randomized algorithm Theorem 2: Algorithm Randomized-Oblivious broadcasts in time O(n min{D,log n}) on any n-node network with diameter D. Proof :  D < log n : broadcasting completed during first nD executions of instruction (b.), by round-robin property  D  log n : consider a shortest path v 0,…,v k =v from the source to a node v ; let d i+1 be degree of node v i+1. Claim: v i receives a message from v i+1 during d i+1 consecutive stages with (positive) constant probability. Since  i  k d i  2n we get expected number O(nlog n) of steps

Time of Radio Broadcasting15 Lower bound for randomized algorithms Theorem 3: For every oblivious randomized broadcasting algorithm A and every sufficiently large n, there exists an n-node network G A of diameter 3, such that the algorithm A requires time  (n), with probability at least 1/2, to complete broadcasting on G A. Idea of the proof : Select network G A,v with uniform probability, among v = 1,…,n-2. With probability at least 1/2 node v n-1 receives a source message in algorithm A in time  (n). Network G A,v n-2 1 0vn-1

Time of Radio Broadcasting16 Oblivious deterministic algorithms  Oblivious algorithm in [CGLP] broadcasts in time O(n 3/2 )  Lower bound  (n log D) in [CMS] Observation : Interleaving algorithm from [CGLP] with round-robin algorithm we obtain algorithm broadcasting in time O(n min{D,n 1/2 }). Theorem 4: For all parameters n,D such that 1 < D < n, and for any deterministic oblivious broadcasting scheme A, there exists an n-node network G A of radius D, such that scheme A requires time  (n min{D,n 1/2 }) to broadcast on G A.

Time of Radio Broadcasting17 Strongly-selective families Strongly-selective family [CMS] : A family F of subsets of R is called (|R|,k)-strongly- selective, for k  |R|, if for every subset Z of R such that |Z|  k, and for every element z  Z, there is a set F  F such that Z  F = {z}. Lemma [CMS]: Let F be an (|R|,k)-strongly-selective family (ssf in short). Then (a) if 3  k < (2|R|) 1/2 then | F |  (k 2 log |R|)/(48 log k), (b) if k  (2|R|) 1/2 then | F |  |R|.

Time of Radio Broadcasting18 Proof of lower bound : case D  (n/8) 1/2 Suppose sets X 0, X 1,…, X i constructed by step t of scheme A, i < D/2 Let R i+1 contain remaining nodes, |R i+1 | > n/2 Consider family T t+1,…,T t+n/2 of transmitters in steps t+1,…,t+n/2; it is not (|R i+1 |,n/(2D))-ssf Define X i+1  R i+1 and v i+1  X i+1 s.t. X i+1  T t+j  {v i+1 } |X i+1 |  n/(2D) layer 0 layer 1 layer 2 layer 3 layer D/2+3 layer D/2+4 layer D-1 layer D Remaining nodes Path v0v0 v1v1 v2v2 v3v3 X1X1

Time of Radio Broadcasting19 Proof of lower bound : case D > (n/8) 1/2 Suppose sets X 0, X 1,…, X i constructed by step t of scheme A, i < D’ = n 1/2 /4 Let R i+1 contain remaining nodes, |R i+1 | > n/2 Consider family T t+1,…,T t’ of transmitters in steps t+1,…,t’= t+n/2; it is not (|R i+1 |,n/(2D’))-ssf Define X i+1  R i+1 and v i+1  X i+1 s.t. X i+1  T t+j  {v i+1 } |X i+1 |  n/(2D’) layer 0 layer 1 layer 2 layer 3 layer D’/2+3 layer D’/2+4 layer D-1 layer D Remaining nodes Path v0v0 v1v1 v2v2 v3v3 X1X1

Time of Radio Broadcasting20 Concluding remarks We analyzed impact of obliviousness and randomization for radio broadcasting  randomization is better than determinism for D such that log n << D << n  adaptive algorithms are faster than oblivious for D s.t. 1 << D << n Conjecture :  randomization is better than determinism for D iff 1 << D << n  adaptive algorithms are faster than oblivious for D iff 1 << D