Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc

1. Dariusz Kowalski University of Connecticut & Warsaw University Andrzej Pelc University of Quebec en Outaouais Broadcasting in Undirected Ad hoc Radio Networks

2. Structure of the presentation • Preliminaries • Model of ad-hoc radio network • Broadcasting problem - definition and prior work • Goals and results • Efficient randomized algorithm matching lower bound for randomized algorithms • Complete-layered networks • Lower bound for deterministic algorithms • Efficient deterministic algorithm based on technique of solving collision • Conclusions Broadcasting in undirected ad hoc radio networks

3. 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 Broadcasting in undirected ad hoc radio networks

4. Message delivery • Node v receives a message from node w in step i if • node v : • is receiving in step i • nodew : • is a neighbor of node v in network G, and • is transmitting in step i • node zw : • if z is a neighbor of node v in network G then z is receiving in step i • Otherwise node v receives nothing Broadcasting in undirected ad hoc radio networks

5. Broadcasting problem Broadcasting problem: • some node, called source, has the message, called the source message, and transmits it in step 0 • every node different than source is receiving until it receives the source message (no-spontaneous) Goal: all nodes must know the source message Measure of performance: time by the first step when all nodes have the source message Broadcasting in undirected ad hoc radio networks

6. Bibliography [ABLP] N. Alon, A. Bar-Noy, N. Linial, D. Peleg: A lower bound for radio broadcast. J. of Computer and System Sciences, 1991. [BGI] R. Bar-Yehuda, O. Goldreich, A. Itai: On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization. JCSS, 1992. [CMS] A. Clementi, A. Monti, R. Silvestri: Selective families, superimposed codes, and broadcasting on unknown radio networks. SODA, 2001. [CGR] M. Chrobak, L. Gasieniec, W. Rytter: Fast broadcasting and gossiping in radio networks. FOCS, 2000. [KP] D. Kowalski, A. Pelc: Deterministic broadcasting time in radio networks of unknown topology, FOCS, 2002. [KM] E. Kushilevitz, Y. Mansour: An (Dlog(n/D))lower bound for broadcast in radio networks. SIAM J. Comp. 1998. Broadcasting in undirected ad hoc radio networks

7. Goals and results GOAL:understand better what are the properties of graphs on which deterministic/randomized broadcasting is time-consuming RESULT: more advanced property of graphs, which are hard to broadcast by deterministic algorithms, yields randomization is better Broadcasting in undirected ad hoc radio networks

8. Randomized algorithms - lower bounds 0 Lj {1,…, n} L1 L2 LD-1 LD • Lower bound (Dlog(n/D)) for expected broadcasting time for n-node networks (complete-layered) with diameter D-proved by Kushilevitz and Mansour[KM] Complete- -layered network • Lower bound (log2 n) for broadcasting time for n-node networks with constant diameter proved by Alon et al.[ABLP] even for known network and deterministic algorithms Broadcasting in undirected ad hoc radio networks

9. Randomized algorithms • Randomized algorithm with O(Dlog n + log2n) expected broadcasting time introduced by Bar-Yehuda, Goldreich, Itai [BGI] • Our result: algorithm broadcasting in expected time O(Dlog(n/D) + log2n) matchinglower bound. Presentation: • Combinatorial tools : universal sequence • Idea of construction • Algorithm and remarks Broadcasting in undirected ad hoc radio networks

10. Universal sequence Remind: N,D are fixed. Definition: An infinite sequence (pi)i=1,…, of reals from the interval [0,1] is called universal sequence if the following conditions hold: • for every j = log(N/D)+1, … , log(N/(4 log N)) , the sequence pi+1, pi+2, … , pi+3Dx/N contains at least one value 1/x, where x=2j ; • for every j = log(N/(4 log N))+1, … , log N , the sequence pi+1, pi+2,…, pi+3Dx/(Nlog N) contains at least one value 1/x, where x=2j. Lemma: There exists universal sequence. Proof: Idea of construction of universal sequence: • put values 2-j to nodes of the complete binary tree of N leaves according to some rule • traverse this tree, writing values of visiting nodes Broadcasting in undirected ad hoc radio networks

11. Idea of algorithm Idea of algorithm (assuming known D): • partition into stages, each taking log(N/D) + 2 steps • in steps j of stage, for j = 0,1,…,log(N/D) , we want to assure fast transmission to the node having informed neighbor and of degree close to 2j-- hence we transmit with probability 2-j • in step j = log(N/D) + 1 of stage i we want to assure fast transmission to the node having informed neighbor and of degree greater than N/D-- hence we transmit with probabilitypiaccording to the universal sequence Broadcasting in undirected ad hoc radio networks

12. Algorithm source transmits forD:=1 to log Ndo fori:=1 toaDdo-- executing stage(D,i) if node v received the source message before stage(D,i) then • fork=0 to log(N/D) do transmit with probability 2-k • transmit with probability pi Expected broadcasting time: O(Dlog(n/D) + log2n) Remark: Complete-layered graphs are among most difficult to broadcast by randomized algorithms. Broadcasting in undirected ad hoc radio networks

13. Complete-layered networks QUESTION: are complete-layered networks among most difficult graphs to broadcast by deterministic algorithms? Clementi, Monti, Silvestri in [CMS] claimed that every deterministic algorithm needs time (nlog D) to broadcast on some complete-layered graph of n nodes and diameter D Claim is wrong, and answer for the QUESTION is NOT (unlike for randomized algorithms) We showed [KP-STACS’03] deterministic algorithm broadcasting on complete-layered networks in time O(Dlog(n/D) + log2n) Broadcasting in undirected ad hoc radio networks

14. Deterministic lower bound L1 L2 L3 L4 LD-3 LD-2 LD-1 LD 0 1 2 D/2-1 D/2 L*3 L*1 L*jLj {D/2+1,…, n} L*D-3 • For Dn1/2 : lower bound (n) claimed in [BGI] and proved by us is [KP-SIROCCO’03] In this case Dlog(n/D) + log2n = o(n) • For D > n1/2 we prove lower bound (nlog n / log(n/D)) on star-layered graphs Broadcasting in undirected ad hoc radio networks

15. Idea of selecting worst-case network Why are complete-layered networks bad? • Fast broadcasting using selective-family (see also [CMS]) • Fast broadcasting using leader election in every front layer To construct layer L2j-1 we need in the same time: • Keep size |L2j-1| = O(n/D) • Select set L*2j-1 to assure that node 2j will not receive a message from set L*2j-1 during (n/D)log D steps after activation of nodes in L*2j-1 • Not allow nodes in layer L2j-1 to receive a message from node 2(j-1) during (n/D)log D steps after activation of nodes in L2j-1 Broadcasting in undirected ad hoc radio networks

16. Deterministic algorithm • Best known deterministic algorithm broadcasts in time O(nlog nlog D) [CGR,KP-SIROCCO’03] (it works also for directed networks) • Our result: broadcasting time O(nlog n) Procedure SELECT(p,o,s) [KP] • Using node p and procedure ECHO, node o “asks” if there exists unvisited neighbor in range {1,…,N/2} O(1) • If YES then node o recursively restricts the range of SELECT from {1,…,N} to {1,…,N/2} • If NO then node o recursively restricts the range of SELECT from {1,…,N} to {N/2+1,…,N} Broadcasting in undirected ad hoc radio networks

17. Description of algorithm Algorithm Traverse a DFS tree on network G by a token (source starts): • owner of a token transmits O(1) • owner selects a successor using SELECT O(log n) • owner sends a token to successor O(1) Until token in source and no successor selected in SELECT Length of a DFS-traverse: O(n) Broadcasting time: O(nlog n) Broadcasting in undirected ad hoc radio networks

18. Conclusions We considered problem of broadcasting on radio networks: • Randomization is better than determinism • Complete-layered networks are among most hard networks to broadcast by randomized algorithms, but not by deterministic algorithms Remaining open problem • Closing gap between lower and upper bounds on broadcasting time for deterministic algorithms Broadcasting in undirected ad hoc radio networks