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Broadcasting in UDG Radio Networks with Unknown Topology . Weizmann Liverpool Weizmann Québec Weizmann Liverpool. 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

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Broadcasting in UDG Radio Networks with Unknown Topology

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Broadcasting in udg radio networks with unknown topology l.jpg

Broadcasting in UDG Radio Networks with Unknown Topology

WeizmannLiverpoolWeizmannQuébecWeizmannLiverpool

Yuval Emek, Leszek Gąsieniec, Erez Kantor, Andrzej Pelc,David Peleg, Chang Su,


Udg radio networks l.jpg

  • 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 l.jpg

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 model4 l.jpg

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 l.jpg

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 hoc6 l.jpg

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 l.jpg

Broadcasting

  • a distinguished source node

  • source’s message should be heard by all nodes

  • remote nodes – use graph’s paths

  • connected graphs


Broadcasting8 l.jpg

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 l.jpg

Deterministic model

  • decisions of a node on round tdepends only on:

  • own coordinates

  • t itself

  • messages heard so far


This work l.jpg

  • = 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 work11 l.jpg

This work

  • upper bound

  • lower bound

  • conditionalwake up

  • spontaneouswake up


Previous results l.jpg

Previous results

  • roughly divided into 2 subareas:

  • centralized: complete knowledge, designing fast schedulers

  • distributed: local knowledge, designing fast protocols (this work)


Centralized model l.jpg

  • 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)


Distributed model l.jpg

  • 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!)


Distributed model15 l.jpg

  • 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 l.jpg

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 l.jpg

  • 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


Chain networks18 l.jpg

  • clusters

  • form a clique

Chain networks

  • there is no edge between any and any for |k-i|>1


Chain networks19 l.jpg

  • the message go from directly to

Chain networks

  • from to when only one node from transmit the message


Chain networks20 l.jpg

  • the message go from directly to

Chain networks

  • from to when only one node from transmit the message


Chain networks21 l.jpg

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 l.jpg

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


The adversary l.jpg

  • 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


Game between the algorithm and the adversary l.jpg

  • = 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?


Game between the algorithm and the adversary25 l.jpg

  • 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


Game between the algorithm and the adversary26 l.jpg

  • 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)


Game between the algorithm and the adversary27 l.jpg

  • 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 l.jpg

Lower bound

  • ti = first round on which the nodes of ireceive the message

  • , number of round for delivering the message from ito i+1


Lower bound29 l.jpg

  • execution time:

Lower bound

  • adversary guarantees :

  • for ti<cg2

  • , for i<cg2/log (g)


Conditional wake up lower bound l.jpg

Conditional wake up – lower bound


Conditional wake up lower bound31 l.jpg

  • execution time:

  • rounds

  • rounds

  • rounds

  • rounds

Conditional wake up – lower bound

  • chain network

  • N1

  • N2

  • N3

  • ND/2

  • diameter 2


Conditional wake up lower bound32 l.jpg

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 l.jpg

  • 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 n34 l.jpg

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 l.jpg

Adversary

  • can no longer guarantee that no messages are being heard

  • distinguish silence from collision (stronger model)


Game between the algorithm and the adversary36 l.jpg

Game between the algorithm and the adversary

  • st

  • Adversary:

  • (1) reveal some cells

  • (2) report: collision occur

  • (3) report: silence / collision


Adversarial policy l.jpg

  • 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 l.jpg

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 network39 l.jpg

The concatenate network

  • the message must be delivered via target nodes and auxiliary nodes


The concatenate network40 l.jpg

The concatenate network

  • execution time:


Summary l.jpg

Summary

  • upper bound

  • lower bound

  • conditionalwake up

  • spontaneouswake up


Slide42 l.jpg

END

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