Broadcasting in UDG Radio Networks with Unknown Topology

Broadcasting in UDG Radio Networks with Unknown Topology WeizmannLiverpoolWeizmannQuébecWeizmannLiverpool Yuval Emek, Leszek Gąsieniec, Erez Kantor, Andrzej Pelc,David Peleg, Chang Su,

stations = points in UDG radio networks • transmitting range = 1 • unit disk graph – UDG • (nodes, edges, paths, …) • distributed synchronous model • in every round: transmit or receive • message heard iff exactly one neighbor transmits • else: silence or collision (same effect)

distributed synchronous model (1) no transmission (silence) (2) single transmission (3) multiple transmission v can receive the message from u v u w vcannotreceive the message

distributed synchronous model (1) no transmission (silence) (3) multiple transmission • collisions cannot be distinguished from silence v u w vcannotreceive the message

Unknown topology (ad hoc) • a unique coordinate system • each node knows its own coordinates • does not know the: • coordinates of any other node • the number of nodes • the diameter

Unknown topology (ad hoc) • known granularity g = • inverse of minimum Euclidean distance • , for every pair of nodes • typically:d is much smaller then 1 and g is much larger than 1

Broadcasting • a distinguished source node • source’s message should be heard by all nodes • remote nodes – use graph’s paths • connected graphs

Broadcasting • two models are considered: • conditional wake up: - nodes are initially idle • wakes up upon hearing a message • spontaneous wake up: • – all nodes are awake from the beginning • execution time = • #rounds until all nodes hear the source’s message

Deterministic model • decisions of a node on round tdepends only on: • own coordinates • t itself • messages heard so far

= diameter of the UDG network (in hops) • = granularity: inverse of min Euclidean distance • s • v This work • execution time depends on two parameters: • not Euclidean diameter

This work • upper bound • lower bound • conditionalwake up • spontaneouswake up

Previous results • roughly divided into 2 subareas: • centralized: complete knowledge, designing fast schedulers • distributed: local knowledge, designing fast protocols (this work)

from • to • Alon, Bar-Noy, Linial, Peleg ’91: constant D Centralized model • Chlamtac, Kutten ’85: formulating the model of radio networks • Chlamtac, Weinstein ‘91 • Gaber, Mansour ‘95 • Elkin, Kortsarz ‘05 • Gasieniec, Peleg, Xin ‘05 • Kowalski, Pelc (to appear)

Bar-Yehuda, Goldreich, Itai ’92: • Kushilevitz, Mansour ’98: • Czumaj, Rytter ’03: Distributed model • unknown topology, no labels, randomized: • first to study distributed broadcasting (also deterministic) • (tight!)

Chlebus, Gasieniec, Gibbons, Pelc, Rytter ’02: • Kowalski, Pelc ’05: Distributed model • unknown topology, knowing own labels, spontaneous wake up, deterministic: • Kowalski, Pelc ’05: unknown topology, knowing own labels, conditional wake up, deterministic

Spontaneous wake up – lower bound • Theorem. deterministic broadcasting algorithm A, and  choice of parameters D,g,  UDG network N of diameter D and granularity g s.t. A requires • rounds to broadcast in N under the spontaneous wake up model.

clusters • k consists of • cells Chain networks • each cell may be occupied with a node or empty • each cluster contain at least one occupied cell • source cell (always occupied) in source cluster 0

clusters • form a clique Chain networks • there is no edge between any and any for |k-i|>1

the message go from directly to Chain networks • from to when only one node from transmit the message

Chain networks • if there exists a node in that heard the message • then all the nodes of must being heard the source message

The broadcasting algorithm A • knows the coordinates of the cells • does not know which cells are occupied and which are empty (except the source) • knows that there is at least one occupied cell in every cluster • a typical instruction: “transmit if occupied” • St = cells scheduled to transmit on round t by A

decisions are made separately for every k and online based on The adversary • goal: slow down the broadcasting algorithm • decides for every cell whether occupied or empty

= number of occupied cells in Game between the algorithm and the adversary • u • St schedule to transmit • adversary decide: • (1) single transmission • (2) silence / collision • algorithm can learn? • what u can learn?

u Game between the algorithm and the adversary • adversary: • (1) reveal these cells (occupied/empty) • (2) report silence / collision • must be consists with previous reports

u Game between the algorithm and the adversary • v • algorithm knows v • St schedule to transmit by the algorithm • algorithm can learn whether: • (1) • (u hear v) • (2) • (u did not hear v)

u Game between the algorithm and the adversary • adversary: • (1) reveal these cells • (2) report silence / collision • (2) report that collision occur • must be consists with previous reports

Lower bound • ti = first round on which the nodes of ireceive the message • , number of round for delivering the message from ito i+1

execution time: Lower bound • adversary guarantees : • for ti<cg2 • , for i<cg2/log (g)

Conditional wake up – lower bound

execution time: • rounds • rounds • rounds • rounds Conditional wake up – lower bound • chain network • N1 • N2 • N3 • ND/2 • diameter 2

Conditional wake up – lower bound • Theorem. deterministic broadcasting algorithm A, and  g,  UDG network N of diameter 2 and granularity g s.t. A requires • rounds to broadcast in N under the conditional wake up model.

blocks • in each block: • 1> • auxiliary cells • opposite each block: • a target cell • 1> The network N • exactly 1 target cell is occupied • g auxiliary cells • target

The network N • target cell is outside of the • transmitting range of any other blocks • there is at least oneoccupied cell in the block that opposite to the occupied target cell • the network is connected • auxiliary • target

Adversary • can no longer guarantee that no messages are being heard • distinguish silence from collision (stronger model)

Game between the algorithm and the adversary • st • Adversary: • (1) reveal some cells • (2) report: collision occur • (3) report: silence / collision

execution continues for • rounds Adversarial policy • dead blocks– all cells are revealed, target cell is empty • on every round we “kill” at most 1 block and reveal at most 1 cell in each “live” block

The concatenate network • the auxiliary cells of Niis outside the transmitting range of the next source node si+1 • the target cell of Niis inside of the transmitting range of the next source node si+1

The concatenate network • the message must be delivered via target nodes and auxiliary nodes

The concatenate network • execution time:

Summary • upper bound • lower bound • conditionalwake up • spontaneouswake up

END • Thank You!!!

Broadcasting in UDG Radio Networks with Unknown Topology