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

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

w u v (2) single transmission(1) no transmission (silence) (3) multiple transmission v can receive the message from u v cannot receive the message distributed synchronous model

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

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

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

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

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

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

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

This work conditional wake up lower boundupper bound spontaneous wake up

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

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) from to Alon, Bar-Noy, Linial, Peleg ’91: constant D

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

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

Distributed model Ravishankar, Singh ’94: points randomly placed on a line geometric embedding: Kranakis, Krizanc, Pelc ’01: fault-tolerant Diks, Kranakis, Krizanc, Pelc ’01: points on a line, restricted knowledge radius Dessmark, Pelc ‘07: points in the plane, restricted knowledge radius, consecutive labels #stations may be

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.

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

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

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

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 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 knows the coordinates of the cells S t = cells scheduled to transmit on round t by A a typical instruction: “transmit if occupied”

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

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

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

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

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

t i = first round on which the nodes of  i receive the message, number of round for delivering the message from  i to  i+1 Lower bound

for t i <cg 2, for i<cg 2 /log (g) adversary guarantees : execution time:

Conditional wake up – lower bound

chain network diameter 2 N1N1 N2N2 N3N3 N D/2 rounds execution time:

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.

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

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

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

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

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

The concatenate network the target cell of N i is inside of the transmitting range of the next source node s i+1 the auxiliary cells of N i is outside the transmitting range of the next source node s i+1

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

The concatenate network execution time:

conditional wake up lower boundupper bound spontaneous wake up Summary

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