Broadcasting in UDG Radio Networks with Unknown Topology

Broadcasting in UDG Radio Networks with Unknown Topology

Yuval Emek, Leszek Gąsieniec,

Erez Kantor, Andrzej Pelc,David Peleg, Chang Su,

WeizmannLiverpoolWeizmannQuébecWeizmannLiverpool

UDG radio networks

1stations = points in 2

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

wu

v

(2) single transmission(1) no transmission (silence)

(3) multiple transmission v can receive the message from u

v cannot receive the message


wu

v

(1) no transmission (silence)

(3) multiple transmission

v cannot receive the message

collisions cannot be distinguished from silence


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

1

d

dg /1

typically: d is much smaller then 1 and g is much larger than 1

dvu ||,|| , 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)D

not Euclidean diameter

dg /1= granularity: inverse of min Euclidean distance

s v

This work

DgO gD

2),log(min gDgD

conditionalwake up

lower boundupper bound

spontaneouswake 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 nDO 2log

to nOD 2log

Alon, Bar-Noy, Linial, Peleg ’91: constant D n2log

Distributed model

Bar-Yehuda, Goldreich, Itai ’92: nnDO 2loglog

Kushilevitz, Mansour ’98: DnD /log

unknown topology, no labels, randomized:

first to study distributed broadcasting (also deterministic)

Czumaj, Rytter ’03: nDnDO 2log/log (tight!)

Distributed model

Kowalski, Pelc ’05: unknown topology, knowing own labels, conditional wake up, deterministic

Dn

nn

/log

log

Chlebus, Gasieniec, Gibbons, Pelc, Rytter ’02: nO

unknown topology, knowing own labels, spontaneous wake up, deterministic:

Kowalski, Pelc ’05: n

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

2),log(min gDgD

rounds to broadcast in N under the spontaneous wake up model.

Chain networks

clusters D ,,1

k consists of 2g cells

g

g

1 2 3 D

each cell may be occupied with a node or empty

source cell (always occupied) in source cluster 0

0

each cluster contain at least one occupied cell

Chain networks

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

1 2 3 D

1

10

clusters 1 ii form a clique

iu kv

1 2 3 D

0

the message go from directly to 0 1

2D 1D

Chain networks

from to when only one node from transmit the messagei 1i i

1 2 3 D

0

the message go from directly to 0 1

2D 1D

Chain networks

from to when only one node from transmit the messagei 1i i

1 2 3 D

02D 1D

Chain networks

if there exists a node in that heard the message i

then all the nodes of must being heard the source message

i ...21

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

St = cells scheduled to transmit on round t by A

a typical instruction: “transmit if occupied”

ktt

k SS

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 t

kS

Game between the algorithm and the adversary

1k k 1k

(2) silence / collision

(1) single transmission

tkS

tkS 1

tkS 1

algorithm can learn? what u can learn?

u

St schedule to transmit

adversary decide:

1)(# tks

tkSO# = number of occupied cells in tS

u


1k k 1k

(1) reveal these cells (occupied/empty)

(2) report silence / collision

must be consists with previous reports

1)(# tks

tkS

tkS 1

tkS 1

adversary:

u


1k k 1k

• algorithm knows v

(u hear v)

(2)

(1)

v

algorithm can learn whether:

tkS

tkS 1

}{1 vS tk

0)(# tkS

1)(# tkS (u did not hear v)

St schedule to transmit by the algorithm

u


1k k 1k

(1) reveal these cells (2) report silence / collision

must be consists with previous reports

1)(# tks(2) report that collision occur

tkS

tkS 1

}{1 vS tk adversary:

1i i 1i

ti = first round on which the nodes of i receive the message

, number of round for delivering the message from i to i+1

iii tt 1

Lower bound

Lower bound1i i 1i

)log(giti

for ti<cg2

, for i<cg2/log (g)

)log(1 gtt iiiadversary guarantees :

execution time: 2),log(min gDgD

Conditional wake up – lower bound


chain network

diameter 2

N1 N2 N3 ND/2

g rounds g rounds

g rounds

g rounds

execution time: gD


Theorem. deterministic broadcasting algorithm A, and g, UDG network N of diameter 2 and granularity g s.t. A requires

g

rounds to broadcast in N under the conditional wake up model.

The network N

1

11

g

blocksg

in each block:g auxiliary cells

opposite each block:

a target cell

g auxiliary cells target

exactly 1 target cell is occupied

1>

1>

The network N

1

1

auxiliary target

there is at least one occupied cell in the block that opposite to the occupied target cell

the network is connected

1

1

target cell is outside of thetransmitting range of any other blocks

1

Adversary

can no longer guarantee that no messages are being heard

distinguish silence from collision (stronger model)


Adversary:(1) reveal some cells

(3) report: silence / collision

2)(# ts(2) report: collision occur

st

Adversarial policy

on every round we “kill” at most 1 block and reveal at most 1 cell in each “live” block

execution continues for g rounds

dead blocks – all cells are revealed, target cell is empty

The concatenate network

1

1

the target cell of Ni is inside of the transmitting range of the next source node si+1

the auxiliary cells of Ni is outside the transmitting range of the next source node si+1


1

1

the message must be delivered via target nodes and auxiliary nodes


1

1

g g g

execution time: gD

DgO gD

2),log(min gDgD

conditionalwake up

lower boundupper bound

spontaneouswake up

Summary

END

Thank You!!!

Documents

Broadcasting in UDG Radio Networks with Unknown Topology