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

Broadcasting in undirected ad hoc radio networks2 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 networks3 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 networks4 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 –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 networks5 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 networks6 Bibliography [ABLP] N. Alon, A. Bar-Noy, N. Linial, D. Peleg: A lower bound for radio broadcast. J. of Computer and System Sciences, [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, [CMS] A. Clementi, A. Monti, R. Silvestri: Selective families, superimposed codes, and broadcasting on unknown radio networks. SODA, [CGR] M. Chrobak, L. Gasieniec, W. Rytter: Fast broadcasting and gossiping in radio networks. FOCS, [KP] D. Kowalski, A. Pelc: Deterministic broadcasting time in radio networks of unknown topology, FOCS, [KM] E. Kushilevitz, Y. Mansour: An  (Dlog(n/D)) lower bound for broadcast in radio networks. SIAM J. Comp

Broadcasting in undirected ad hoc radio networks7 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 networks8 Randomized algorithms - lower bounds  Lower bound  (Dlog(n/D)) for expected broadcasting time for n-node networks (complete-layered) with diameter D - proved by Kushilevitz and Mansour [KM]  Lower bound  (log 2 n) for broadcasting time for n-node networks with constant diameter proved by Alon et al. [ABLP] even for known network and deterministic algorithms 0 L1L1 L j  {1,…, n} L2L2 L D-1 LDLD Complete- -layered network

Broadcasting in undirected ad hoc radio networks9 Randomized algorithms  Randomized algorithm with O(Dlog n + log 2 n) expected broadcasting time introduced by Bar-Yehuda, Goldreich, Itai [BGI]  Our result: algorithm broadcasting in expected time O(Dlog(n/D) + log 2 n) matching lower bound. Presentation: –Combinatorial tools : universal sequence –Idea of construction –Algorithm and remarks

Broadcasting in undirected ad hoc radio networks10 Universal sequence Remind: N,D are fixed. Definition: An infinite sequence (p i ) 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 p i+1, p i+2, …, p i+3Dx/N contains at least one value 1/x, where x=2 j ;  for every j = log(N/(4 log N))+1, …, log N, the sequence p i+1, p i+2,…, p i+3Dx/(Nlog N) contains at least one value 1/x, where x=2 j. 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 networks11 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 2 j - - 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 probability p i according to the universal sequence

Broadcasting in undirected ad hoc radio networks12 Algorithm source transmits for D:=1 to log N do for i:=1 to a  D do -- executing stage(D,i) if node v received the source message before stage(D,i) then  for k=0 to log(N/D) do transmit with probability 2 -k  transmit with probability p i Expected broadcasting time: O(Dlog(n/D) + log 2 n) Remark: Complete-layered graphs are among most difficult to broadcast by randomized algorithms.

Broadcasting in undirected ad hoc radio networks13 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) + log 2 n)

Broadcasting in undirected ad hoc radio networks14 Deterministic lower bound  For D  n 1/2 : lower bound  (n) claimed in [BGI] and proved by us is [KP-SIROCCO’03] In this case Dlog(n/D) + log 2 n = o(n)  For D > n 1/2 we prove lower bound  (nlog n / log(n/D)) on star-layered graphs 0 L*1L*1 L * j  L j  {D/2+1,…, n} L*3L*3 L * D-3 L D-2 12D/2-1D/2 L1L1 L2L2 L3L3 L4L4 LDLD L D-1 L D-3

Broadcasting in undirected ad hoc radio networks15 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 L 2j-1 we need in the same time:  Keep size |L 2j-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 L 2j-1 to receive a message from node 2(j-1) during (n/D)log D steps after activation of nodes in L 2j-1

Broadcasting in undirected ad hoc radio networks16 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 networks17 Description of algorithm Algorithm Traverse a DFS tree on network G by a token (source starts):  owner of a token transmitsO(1)  owner selects a successor using SELECTO(log n)  owner sends a token to successorO(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 networks18 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