broadcasting in udg radio networks with unknown topology
Post on 13-Jan-2016
18 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
distributed synchronous model
wu
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
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
Game between the algorithm and the adversary
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
Game between the algorithm and the adversary
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
Game between the algorithm and the adversary
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
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
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
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)
Game between the algorithm and the adversary
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
The concatenate network
1
1
the message must be delivered via target nodes and auxiliary nodes
The concatenate network
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!!!
top related