Communication protocols for
packet radio networks
algorithmic approach
Darek Kowalski
Introduction to the model of
packet radio network
• Network of stations
• Synchronized
• Possible conflicts
of messages
arriving at a station
Wroclaw 2007
2
Motivation
• Motivation coming from
– Local Area Networks (LANs)
• ETHERNET
– Upstream HFC (Hybrid Fibre-Coax)
– Wireless networks (802.11 wireless LAN)
– Sensor networks
Wroclaw 2007
3
Main paradigms
• Data-link protocols for shared wire or
medium
• Ad hoc & mobility issues
• Dynamic communication tasks
• Algorithms behind protocols
Wroclaw 2007
4
Roadmap
• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Future directions
Wroclaw 2007
5
Model of radio network:
stations
• Collection of n stations (also called nodes)
with known labels
• A station can be either active or passive
• Global clock is provided to all stations
• Synchronous rounds assumed
– fast access and bounded delay for message
delivery
• Stations communicate via network
Wroclaw 2007
6
Model of radio network:
transmissions
• Every station has ability to transmit
• Every station receives a message M if
exactly one station in its range transmits
message M in the current round
• Messages are sent in slots of known length
• Transmission is reliable, which means that
messages are never lost in transit etc.
Wroclaw 2007
7
Receiving a message:
cases
Assume that the middle node is listening:
Silence
Successful transmission
Collision
Wroclaw 2007
8
Single-hop & multi-hop
Single-hop (multiple-access channel):
all nodes are directly connected
Wroclaw 2007
Multi-hop:
D denotes a diameter
9
Basic communication tasks
• Broadcasting:
a node, called a source, has to inform all other nodes
• Gossiping:
each node has to inform all others
• Many to many (M2M) :
activate nodes have to exchange information
Wroclaw 2007
10
Related problems
• Leader Election :
designate one among the active nodes as a leader
(they have unique IDs already assigned)
• Synchronization :
have all the nodes to agree on a round to start
counting
(their local clocks are ticking at the same rate
but may show different round numbers)
Wroclaw 2007
11
Efficiency:
complexity measures
• Time complexity
measured from the first activation to the
termination
• Size of buffers
maximum size during the computation
• Size of messages
maximum size during the computation
Wroclaw 2007
12
Asymptotic notation
Let f(n,k) and g(n,k) be mathematical formulas
depending on variables n,k (some of these
variables may not be represented in formulas).
We use notations:
– f(n,k) = (g(n,k)) if there is a constant c > 0 such that
for all n and k inequality f(n,k) > c ·g(n,k) holds
– f(n,k) = O(g(n,k)) if g(n,k) = (f(n,k))
– f(n,k) = (g(n,k)) if g(n,k) = (f(n,k)) and
f(n,k) = (g(n,k))
Wroclaw 2007
13
Roadmap
• Presentation of the model
• Single-hop radio networks
– Static problems
– Model with delays
– Dynamic model
• Multi-hop radio networks
• Future directions
Wroclaw 2007
14
Multiple-access channel:
basic problems
• Detecting collision:
– some set K of k stations want to transmit;
– how recognize if k > 1 ?
• Solving collision (broadcasting, leader
election):
– some set K of k > 1 stations want to transmit;
– how select one of them to transmit
successfully (without a collision) ?
Wroclaw 2007
15
Detecting collision:
all active
Protocol ECHO(K)
STEP 1: all stations in K transmit concurrently
with the station with the smallest label
STEP 2: all stations in K transmit
Output :
2+ (collision, k >1) : if no message received in step 1
and step 2
1 : if k = 1 then either the same message received in
step 1 and step 2 or message received only in step 2
0 : in all other cases
Wroclaw 2007
16
Detecting collision:
only a subset active
• For every deterministic protocol detecting collision there
is a set K of stations such that this protocol requires time
(k log n / log k)
to detect collision among stations in K.
• There is a randomized protocol DECAY (described later)
detecting collision for every set K of stations in time
O(log n)
with probability at least 1/2
Wroclaw 2007
17
Solving collision:
all active
• Used in “taking-turns” protocols, which are
based on the leader or passing the token
• Recursive protocol BIN-SELECTION based
on procedure ECHO succeeds in time
O(log n)
Wroclaw 2007
18
Solving collision:
all active
Procedure BIN-SELECTION(L)
– M is initialized as a subset of L that contains |L|/2
stations with the smallest label
– if ECHO(M) = 0 then BIN-SELECTION(L\M)
– if ECHO(M) = 2+ then BIN-SELECTION(M)
– if ECHO(M) = 1 then stop (successful step)
Protocol BIN-SELECTION
– L is initialized as the set of all stations, |L| = n
– BIN-SELECTION(L)
Wroclaw 2007
19
Solving collision:
only a subset active (cont.)
• For every deterministic protocol solving collision
there is a set K of stations such that this protocol
requires time
(k log n / log k)
to solve collision among stations in K
• For every set K of stations, protocol DECAY(v)
solves collision among stations in K by round
2 log n with probability at least 1/2
Wroclaw 2007
20
Solving collision:
only a subset active
[Bar-Yehuda, Goldreich, Ittai PODC’87]
Protocol DECAY(v)
counter is initially 0
Repeat
– increase counter by 1
– transmit
– set coin to 0 or 1 with equal probability
until coin = 0 or counter = 2 log n
Wroclaw 2007
21
Solving collision:
only a subset active (cont.)
• Protocol Backoff (e.g., using exponentially
growing random function f(x)  {1,…,2x}) is
used in common, but is less efficient in
“heavy-duty” environment
– used in CSMA protocols
• Protocol RoundRobin is usually slow (O(n))
but it works well in “heavy-duty” setting
– used in TDMA protocol
Wroclaw 2007
22
Solving collision:
only a subset active (cont.)
[Metcalfe, Boggs CACM’76]
Protocol Backoff(v,n,f)
size is initially 1
Repeat
– transmit
– if collision then
• set size to f(size)
• wait during next size rounds
until successful transmission
Wroclaw 2007
23
Solving collision:
only a subset active (cont.)
Protocol RoundRobin(v,n)
transmit in round v
Repeat
wait during next n - 1 rounds
transmit
until termination_condition
Wroclaw 2007
24
Another problem
Conflict resolution (m2m)
– some set K of k > 1 stations wants to transmit;
– how to guarantee that each station from K will
transmit successfully (without a collision) ?
• Deterministically: similar to solving collision
– time O(k log(n/k))
[Kowalski PODC’05]
• Using randomization: add k to the complexity
– time O(k + log n)
[Martel IPL’94]
Wroclaw 2007
25
Open problems and directions
• The lower bound matching O(k log(n/k)) for
the conflict resolution problem
• Study of more realistic models of
interferences, depending on the real signal
power
• Study the communication complexity
Wroclaw 2007
26
Roadmap
• Presentation of the model
• Single-hop radio networks
– Static problems
– Model with delays
– Dynamic model
• Multi-hop radio networks
• Future directions
Wroclaw 2007
27
Towards a dynamic model
“Reasonable” asynchrony is not efficient
[Chlebus,Rokicki SIROCCO’04]
How to describe a dynamic model ?
1. Introduce delays between local clocks (new
entities try to join the system) -- this part
2. Introduce permanent/temporary leavings of units
-- something known for permanent leavings
[Clementi,Monti,Silvestri ESA’01]
-- temporary leavings -- reasonable model wanted!
3. Combine two above models -- later
Wroclaw 2007
28
Introducing delays
• Local clocks ticking at the same rate –
this rate defines global rounds
• Round numbers at nodes are local only –
may differ among nodes
1
2
3
1
1
4
2
2
1
Wroclaw 2007
5
3
3
2
6
4
4
3
29
Wake-up problem
Rules:
• At the beginning at least one node is active
• During the computation, a node becomes active
when it is activated by an adversary or receives a
message
Goal:
• every node becomes active eventually
[Gasieniec, Pelc, Peleg PODC’00]
Wake-up is a generalization of broadcast
Wroclaw 2007
30
Radio synchronizers
Schedule of transmissions for a node is a binary sequence where:
i-th bit is 1 iff transmitting in step i
Arrange schedules as rows of an array
Rows can be shifted arbitrarily, to reflect time of activation
(n,k,m)-synchronizer :
binary array of n rows and m columns, such that
when any set K of at most k rows selected and
each row of K shifted by at most m positions to the right,
there is a column C with exactly one occurrence of 1
in C and among the rows in K
Wroclaw 2007
31
How it works
Select k rows
0
Arbitrary
shift
0
1
1
1
0
0
0
1
1
0
0
1
1
0
1
1
0
1
1
0
0
1
0
Find columns with single 1
Wroclaw 2007
32
Universal radio synchronizers
We generalize synchronizers to universal synchronizers
Let g function from [1..n] into positive integers
(n,g)-universal-synchronizer :
binary array of n rows and m = g(n) columns, such that
for any k such that 1  k  n,
when any set K of at most k rows selected and
each rows of K shifted by at most m positions to the right,
then there is a column C such that C  g(k)
with exactly one occurrence of 1 in C and among the rows in K
Function g is called a delay of the universal synchronizer S
If S is a (n,g)-universal-synchronizer, then for any 1  k  n,
the array S with the first g(k) columns is an (n,k,g(k))-synchronizer
Wroclaw 2007
33
Universal synchronizers with
small delays exist
Theorem :
[Chlebus et al. ICALP’05]
For each n there exists an (n,g)-universal synchronizer
with delay g(k) upper bounded by the function
ck log2 n , for some constant c > 0.
This is shown by the probabilistic method:
let each row be a sequence of cn log2 n bits, for some c > 0,
where i-th bit is 1 with the probability about log n / i ,
independently over the rows and the columns.
Then for some c > 0 it is as claimed with a positive
probability.
Wroclaw 2007
34
Explicit synchronizers
A family F(n,k) of (n,k,m)-synchronizers is explicit
if there is an algorithm to find F(n,k) in time polynomial in n
Theorem :
There is a family F(n,k) of explicit (n,k,m(n,k))-synchronizers,
with m(n,k) = O(k2 polylog n)
Method of construction:
based on explicit dispersers, [Ta-Shma, Umans, Zuckerman STOC’01]
Corollary :
There is an explicit algorithm to wake-up a multiple-access
channel with n stations in time O(k2 polylog n),
where up to k stations may wake up spontaneously
Wroclaw 2007
35
Open problems and directions
• Efficient construction of synchronizers
• Closing the logarithmic gap
• More realistic model, communication
complexity, …
Wroclaw 2007
36
Roadmap
• Presentation of the model
• Single-hop radio networks
– Static problems
– Model with delays
– Dynamic model
• Multi-hop radio networks
• Future directions
Wroclaw 2007
37
Fully-dynamic broadcast:
introduction
• Broadcast requests occur in dynamic
fashion - generalization of static conflict
resolution
• Local buffers must have bounded sizes
Wroclaw 2007
38
Protocol and correctness
• Protocol : a function in which
– inputs are sequences of messages
– outputs are one message (to transmit in the
current step)
• empty message in a buffer means no transmission
(or no message received)
Wroclaw 2007
39
Dynamic broadcast
• Packets are injected by the adversary
• Adversary knows the protocol
• ((n),w)-adversary can inject at most (n)w
packets in each time interval of w rounds
– (n) is an injection rate
– w is a window size
• Fairness of protocol: each packet is
eventually received by all stations
Wroclaw 2007
40
Dynamic broadcast
example
window w
window w
window w
window (n)w = 3
Wroclaw 2007
time
41
Classes of protocols
Deterministic protocols, parameter n is known
• Adaptive protocol:
– sends control bits
– uses full history of the channel until the current round
– can see local queue and use the name of the station
• Full-sensing protocol:
– uses restricted history of transmissions until the current round
– can see local queue and the name of the station
• Acknowledgement-based protocol:
– uses only the number of rounds when attempting to transmit
the current packet, and the name of the station
Wroclaw 2007
42
Kinds of stability
• Stability
– Against the adversary:
all queues are bounded in any execution against the
adversary
– Against the injection rate:
all queues are bounded in any execution against an
adversary with considered injection rate
• Universal stability:
all queues are bounded in any execution against an
adversary with injection rate smaller than 1
• Strong stability
– all queues are bounded by O((n)w )
Wroclaw 2007
43
Example of three rounds
silence
message
received
collision
transmitter
transmitters
new packet
Wroclaw 2007
44
Dynamic broadcast
– Deterministic:
combined packets, latency-oriented
[Kowalski PODC’05]
dynamic packet broadcast, stability oriented
[Chlebus, Kowalski, Rokicki PODC’06]
– Randomized:
stochastically modelled packet injection
[Hastad Leighton Rogoff SICOMP’96]
[Goldberg MacKenzie Paterson Srinivasan JACM’00] …
adversarial queuing for backoff protocols
[Bender Farach-Colton He Kuszmaul Leiserson SPAA’05]
Wroclaw 2007
45
Related work
• Static broadcast/conflict resolution on multiple-access ch.:
– Deterministic:
[Komlos Greenberg Trans. Inf.’85]
[Greenberg Winograd JACM’85]
– Randomized:
[Willard SICOMP’86]
[Kushilevitz Mansour SICOMP’98]
[Bender Farach-Colton He Kuszmaul Leiserson SPAA’05]
• Adversarial queuing in wired networks:
[Borodin Kleinberg Raghavan Sudan Williamson JACM’01]
[Andrews Awerbuch Fernandez Leighton Liu Kleinberg JACM’01]
[Aiello Kushilevitz Ostrovsky Rosen JCSS’00] …
Wroclaw 2007
46
Main properties
• Impossibility:
No protocol is stable against injection rate 1 for
n>3
No protocol is strongly stable against injection
rate (1/log n)
• Separation of protocols:
Universal stability can be achieved by adaptive
or full-sensing protocols, but not by ackn-based
ones
Wroclaw 2007
47
Impossibility
for injection rate 1
No protocol is stable against injection rate 1 for n > 3
Proof: Consider a fair protocol run by 4 stations.
Observation 1: adversary’s profit when silence or collision
Observation 2: adversary can “borrow” a packet from the future
to cause a collision
Case 1: Some two queues are nonempty.
The adversary chooses one of them and keeps injecting a
packet per round into the station.
We leverage fairness to obtain profit
Case 2: Exactly one queue is nonempty.
Adversary can select two stations, then keep injecting packets
into them every second round.
One such a selection yields a profit
Wroclaw 2007
48
Universal stability
Protocol Round-Robin-Withholding
• Station 1 initiates a token
• Station with the token keep transmitting a
packet per round until its queue is empty
• If silence is heard then the next station
takes over by getting the token
Wroclaw 2007
49
Universal stability cont.
Properties of Round-Robin-Withholding :
• full-sensing
• does not need collision detection
• universally stable
• may have large queues - is strongly stable
only for very small injection rate
Wroclaw 2007
50
Ackn-based are not
universally stable
Acknowledgement-based protocols are
not stable for injection rate  2/log n
Proof: Consider first log n - 1 bits in
transmission schedule of every station:
there are two identical ones.
Adversary injects 1 packet into the first
successful station and 2 packets into the
other one, every log n rounds: 3 packets
are injected while only 2 packet are heard.
Wroclaw 2007
51
Strong stability:
upper bounds
Full-sensing protocols:
• with collision detection:
for injection rates at most 1/(2 log n)
• no collision detection:
for injection rates at most 1/(const. log2 n)
Ackn-based protocols:
for injection rates at most 1/(const. n log2 n)
Wroclaw 2007
52
Some open questions
• How can we “separate” adaptive from full-sensing
protocols?
• What are threshold values of injection rates to have
(strong) stability of acknowledgement-based
protocols?
• Does randomization provably help?
• Does latency matter?
– latency: max time of queuing a packet
• More realistic model, communication complexity, …
Wroclaw 2007
53
Roadmap
• Presentation of the model
• Single-hop radio networks
• Broadcast with unlimited energy
• One-shot broadcast
Wroclaw 2007
54
Model
Knowledge:
• Topology of D-hop undirected network is known
Complexity measures:
• Time complexity
measured from the first activation to the termination
• Local energy
upper bound on the number of transmissions per node
Task:
• Broadcasting
Wroclaw 2007
55
Unlimited energy:
results
• Lower bound
– Graph family of radius 2: (log2n)
[AlonBar-NoyLinialPeleg, JCSS’91]
• Upper bound
– O(D log2n) general graphs [ChlamtacWeinstein, INFOCOM’87]
– O(D log n + log2n) general graphs [KowalskiPelc, APPROX’04]
– O(D + log5n) general graphs [GaberMansour, SODA’95]
– D + O(log4n) general graphs [ElkinKortsarz, SODA’05]
– D + O(log3n) planar graphs [ElkinKortsarz, SODA’05]
Wroclaw 2007
56
Unlimited energy:
results cont.
• Gasieniec Pelc Peleg PODC 2005:
– D + O(log3n) deterministic construction
– D + O(log2n) expected time randomized algorithm
• Probably optimal in the view of the lower bound (D + log2n)
– 3D deterministic construction (planar graphs)
• Kowalski Pelc Distributed Computing 2006
– O(D + log2n) deterministic construction
Wroclaw 2007
57
Tree ranking:
definition
• The system of ranks in an arbitrary tree T
– Every leaf v in T has rank(v)=1
– A non-leaf node v with children v1,..,vk determines its rank according to
the rank of its children rank(v1),..,rank(vk), where rmax is the highest
rank among its children
– And if rmax is unique
• then rank(v) = rmax
• else rank(v) = rmax +1
Lemma: in an arbitrary tree of size n the largest rank is
bounded by log n
Wroclaw 2007
58
Tree ranking:
example
3
1
3
2
1
1
1
2
2
1
1
2
1
2
2
1
1
2
1
2
1
1
1
2
1
1
2
1
1
2
1
1
1
1
1
1
1
Wroclaw 2007
59
Broadcasting along trees
• Fast transmissions along paths (fast
communication channels) with nodes of the same
rank
– The messages are passed with a constant slowdown
• Slow transmissions across (bottleneck) border of
two different ranks
– The messages are passed with the slowdown O(log2 n)
• Since every message passes at most log n
bottlenecks the total time of broadcasting in trees
is bounded by D + O(log3 n)
Wroclaw 2007
60
Broadcasting in general graphs
• The broadcast algorithm works in 3 stages:
[1] Build a pre-gathering (BFS) spanning tree TPGT
[2] Perform the pruning of the pre-gathering tree
leading to a gathering spanning tree GST
[3] Broadcast messages along fast and short
connections in ranked tree GST
Wroclaw 2007
61
Construction of Gathering
Spanning Tree (GST)
The pruning
process
Nodes here have
ranks for good
BFS --> GST
Direction of the
pruning process
Wroclaw 2007
62
Pruning process:
removing collisions
• Function Check-Collision(i,j): pair of nodes;
– If  u,v  Fji and (u,parent(v))  E, where u≠v
• then return (u,v);
• else return (“null” );
parent(v)
Level i-1
Level i
u
v
Wroclaw 2007
63
Broadcasting in general graphs:
overview
Chlamtac and Weinstein
broadcast procedure
for bipartite graphs
O(log2n)
O(log2n)
O(log n) slow
transmissions
O(log2n)
O(log2n)
O(log2n)
Theorem: There exists efficient construction of the
3n)
broadcast schedule requiring
time
O(D
+
log
Wroclaw 2007
64
Randomized broadcasting
• In randomized algorithm we replace the mechanism (CW
procedure) of slow transmissions by a probabilistic
procedure RCW
• During execution of RCW each participating node in step 1 ≤
i ≤ log n decides to transmit the message randomly and
uniformly with probability 1/2i
Lemma: From the moment the parent (with higher rank) of a
node v is informed, the node v gets the broadcast message
(success) during each execution of one instance of RCW
with probability p > 1/4e > 0.
Wroclaw 2007
65
Randomized broadcasting
cont.
• Note that on each path from the root of GST to any leaf we
need O(log n) successes during slow transmissions
• Using RCW procedure this can be achieved with a help of
O(log n) instances of RCW with high probability
Lemma: there exists a randomized algorithm that for any graph
of size n broadcasts a message from any node with high
probability in time D + O(log2n)
Theorem: There exists a broadcasting schedule requiring
time O(D + log2n)
Wroclaw 2007
66
Faster deterministic broadcast
Theorem: there is a polynomially constructed
deterministic algorithm broadcasting with time
O(D + log2 n)
Sketch of the proof:
• Design algorithm A with broadcasting time
O(D log n + log2 n)
• Replace each consecutive fast logarithmic segment
by one edge and run algorithm A
• The resulting algorithm broadcasts in
O((D/log n) log n + log2 n) = O(D + log2 n)
Wroclaw 2007
67
rounds
Open questions
• Polynomially constructed algorithm
broadcasting in time D + O(log2n)
• Better approximation in planar/geometric
graphs
• Local energy smaller that O(log2n)
Conjecture: (log n)
Wroclaw 2007
68
Roadmap
• Presentation of the model
• Broadcast with unlimited energy
• One-shot broadcast
Wroclaw 2007
69
Networks with radius 2 :
lower bound
• Binomial graph B(k) has k nodes in the upper layer
and k(k-1)/2 in the lower layer
• Every node but one in the upper layer must
broadcast alone
Theorem: (n1/2) broadcasting time is necessary
source
Layer U
1
2
3
4
{1,3}
Wroclaw 2007
{1,4}
{2,3}
{2,4}
Layer L
{1,2}
{3,4}70
Networks with radius 2 :
algorithm
• Witness graph: nodes are from L, for each node in U we pick
two neighbours in L and connect them
• Independent set in witness graph corresponds to the good
transmission set in initial graph (such that after simultaneous
transmission does not isolate non-informed nodes)
source
Layer U
1
2
3
4
{1,3}
{1,4}
{2,3}
{2,4}
Layer L
{1,2}
Wroclaw 2007
{3,4}
71
Networks with radius 2 :
algorithm cont.
1
4
2
3
Witness graph
source
Layer U
1
2
3
4
{1,3}
{1,4}
{2,3}
{2,4}
Layer L
{1,2}
Wroclaw 2007
{3,4}
72
Networks with radius 2 :
algorithm cont.
• In graph with x nodes and y edges an
independent set of size x2/(2y+x) can be
constructed in polynomial time
• The corresponding set in the initial graph informs
at least x2/(2y+x) nodes in L and does not isolate
the remaining nodes in L
Algorithm: keep finding independent/transmitter
sets and remove them from the graphs
Theorem: Algorithm completes broadcast in time
O(n1/2) on every network of radius 2
Wroclaw 2007
73
General networks
Approach: use ranking and GST
Problems:
– fast and slow transmissions don’t work: we need one type
of transmission
– CW procedure has too many transmissions per node
Solution:
– introduce an additional internal ranking: force that each
node transmits exactly in the round indicated by its rank
– use previous algorithm based on witness graphs and
independent sets, instead of CW, to produce the second
part of the new rank
Wroclaw 2007
74
One-shot broadcast in general
graphs
one-shot
broadcast procedure
for bipartite graphs
O(n1/2)
O(n1/2)
O(log n) changes
of external rank
O(n1/2)
O(n1/2)
O(n1/2)
Theorem: There exists efficient construction of 1-shot
1/2log n)
broadcast schedule requiring
time
O(D
+
n
Wroclaw 2007
75
General networks
cont.
Algorithm:
• Node v transmits in round
d(v) + 3 ( lex(v)n1/2 + lin(v) )
where
–
–
–
–
d(v) is the distance of v from the source
3 comes from avoiding collisions with previous and next layers
lex(v) is the initial rank
lin(v) is the internal rank based on 1-shot protocol run on some
subgraph of the initial graph
Lower Bound: Every one-shot broadcasting algorithm
requires time (D + n1/2) on some network of radius D
Wroclaw 2007
76
Remaining problems
• Improving the polylogarithmic additive
component by logarithmic factor
• One-shot broadcasting in other classes of
networks (geometric, planar, random, …)
• What about k-shot broadcast …
Wroclaw 2007
77
Roadmap
• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
– Broadcasting
– Oblivious broadcasting and gossiping
– Wake-up
• Future directions
Wroclaw 2007
78
Multi-hop ad-hoc networks
• n nodes with different labels 1,…,N (N = (n))
communicate via radio network modelled by a
symmetric graph G
• node v knows only it own label and parameter N
• communication is in synchronous steps
• in every step, node v acts either as
– transmitter, or as
– receiver
Wroclaw 2007
79
Bibliography
[Chlamtac, Kutten IEEE Trans. on Communication’85]
introduced the model of radio communication
[Bar-Yehuda, Goldreich, Itai PODC’87] randomized distributed broadcasting
[Gasieniec, Pelc, Peleg PODC’00] introduced wake-up for multiple-access chan.
[Chlebus, Gasieniec, Gibbons, Pelc, Rytter SODA’00]
deterministic distributed broadcasting
[Clementi, Monti, Silvestri SODA’01] selective families in radio networks
[Indyk SODA’02] explicit distributed broadcast and wake-up
[Chrobak, Gasieniec, Kowalski SODA’04] introduced synchronizers, wake-up of
multi-hop networks,applications to leader election and synchronization
Wroclaw 2007
80
Broadcasting time:
results
No collision detection:
Deterministic Randomized
Algorithms
O(n log D)
O(D log(n/D) + log2n)
Lower Bound (n logn/D n) (D log(n/D) + log2n)
Wroclaw 2007
81
Linear time complexity?
• Algorithm “broadcasting” in time O(n) in [CGGPR]
• Lower bound (n) for n-node networks with
constant diameter - incorrect proof since 1987
Network GS,R
S  {1,…, n}
0
layer 0
1
n
S
layer 1
layer 2
R  {n+1,…,2n}
R
How to choose S,R to force linear broadcasting time on GS,R
Wroclaw 2007
82
Linear lower bound
Theorem: For every broadcasting algorithm A and every n,
there is a network GS,R on (n) nodes such that
broadcasting time of algorithm A on GS,R is (n).
Proof :
We construct sets S,R starting from sets S0 = {1,…,n} and
R0 = {n+1,…,2n}. We proceed construction until step n/2 of
algorithm A, to obtain sets S = Sn/2 and R = Rn/2 .
Problem : network G is not defined
Solution :
– introducing abstract object corresponding to the real ones: history
and transmitters, and preserving theirs required properties
– for constructed network, real and abstract objects are equal
Wroclaw 2007
83
Proof of the lower bound:
objects
For every step k  n/2 define (abstract) objects :
Hk(v) : the history of received messages by the end of step k,
for every node v  {0,…,2n}
Tk :
set of nodes v transmitting in step k under given
history Hk(v)
Sk  Sk-1 : a subset of {1,…,n} being the output of function
MODIFY(Sk-1,T), where T = {T1,…,Tk-1};
initially S0 = {1,…,n}
Rk  Rk-1 : a subset of {n+1,…,2n} being the output of
function MODIFY(Rk-1,T), where T = {T1,…,Tk-1};
initially R0 = {n+1,…,2n}
Wroclaw 2007
84
Proof of the lower bound:
construction
Procedure MODIFY(S,T)
• set stop := 0
• while stop = 0 do
– stop := 1
– if there is a set Tl T such that |Tl  S | = 1
then
• choose such a set with smallest index, say Tk, such
that Tk  S = {i}
• remove node i from S
• set stop := 0
Wroclaw 2007
85
Proof of the lower bound:
invariant
The following invariant is preserved after step k of the
construction, according to sets Sk and Rk and objects:
– No single transmitter :
for every set Tl , l  k, |Tl  Sk|  1 and |Tl  Rk|  1
– Removed nodes correspond to disjoint transmitters’
sets : At least n - |Sk| sets Tl are disjoint with Sk , and
at least n - |Rk| sets Tl are disjoint with Rk , for l  k
– Large size : |Sk|  n - k and |Rk|  n - k
– No message in second layer :
if v  Rk then Hk(v) is the empty history
Wroclaw 2007
86
Stronger
deterministic lower bound
Why complete layered networks cannot be used for our purpose?
• Fast broadcasting using leader election in every front layer
• “Slow” broadcasting using selective-family (see also [CMS])
[Kowalski, Pelc PODC’03]
L1
0
L2
1
L*1
L3
L4
LD-3
2
LD-2
D/2-1
LD-1
LD
D/2
L*3
L*j  Lj  {D/2+1,…, n} L*D-3
Wroclaw 2007
87
Construction of layer L2j-1
• Keep size |L2j-1| = O(n/D)
• Select set L*2j-1 to assure that node 2j will not receive a message from
set L*2j-1 during (n/D) logn/D n steps after activation of nodes in L*2j-1
• Not allow nodes in layer L2j-1 to choose a leader, using coordination
node 2(j-1) , during (n/D) logn/D n steps after activation of nodes in L2j-1
L1
0
L2
1
L*1
L3
L4
LD-3
2
LD-2
D/2-1
LD-1
LD
D/2
L*3
L*j  Lj  {D/2+1,…, n} L*D-3
Wroclaw 2007
88
Randomization is better than
determinism
• For D  n1/2 : apply previous lower bound (n);
In this case randomization helps D log(n/D) + log2 n = o(n)
• For D > n1/2 we prove lower bound (n log n / log(n/D))
on star-layered graphs; randomized complexity O(n)
L1
L2
L3
L4
LD-3 LD-2 LD-1 LD
0
1
L*1
2
D/2-1
D/2
L*3
L*j  Lj  {D/2+1,…, n} L*D-3
Wroclaw 2007
89
Deterministic algorithm:
recursive selection
Procedure SELECT(p,q,s) [Kowalski, Pelc FOCS’02]
• Using node p and procedure ECHO, node q
“asks” if there exists unvisited neighbour in range
{1,…,N/2}
• If YES then node q recursively restricts the range
of SELECT from {1,…,N} to {1,…,N/2}
• If NO then node q recursively restricts the range
of SELECT from {1,…,N} to {N/2+1,…,N}
Wroclaw 2007
90
Deterministic algorithm
Algorithm
[Kowalski, Pelc PODC’03]
Traverse a DFS tree on network G by a token
(the source starts):
• owner of a token transmits
O(1)
• owner selects a successor using SELECT O(log n)
• owner sends a token to successor
O(1)
Until token in source and no successor selected in
SELECT
Time complexity: O(n log n) (improved to O(n log n))
Wroclaw 2007
91
Lower bound
for randomized broadcast
• Lower bound (D log(n/D)) for expected
broadcasting time for n-node networks
(complete-layered) with diameter D
[Kushilevitz, Mansour SICOMP’98]
Complete-layered
network
0
L1
L2
Lj  {1,…, n}
Wroclaw 2007
LD-1
LD
92
Another lower bound
• Lower bound (log2 n) for broadcasting time for
n-node networks with constant diameter
holds even for known network topology and
randomized algorithms
[Alon, Bar-Noy, Linial, Peleg STOC’89]
Wroclaw 2007
93
Randomized algorithms
• Randomized algorithm with O(D log n + log2 n)
expected broadcasting time
[Bar-Yehuda, Goldreich, Itai PODC’87]
• Optimal algorithm broadcasting in expected time
O(Dlog(n/D) + log2 n) matching the lower bound
[Czumaj, Rytter FOCS’03 & Kowalski, Pelc PODC’03]
Presentation of the result:
– Combinatorial tool : universal sequence
– Idea of construction
– The algorithm and remarks
Wroclaw 2007
94
Universal sequence
Remind: N,D are fixed.
Definition: An infinite sequence (pi)i=1,…, of reals from the
interval [0,1] is called universal sequence if the following
conditions hold for every value x=2j
( j = log(N/D)+1, … , log N ) :
• if j = log(N/D)+1, … , log(N/(4 log N)) , the sequence
pi+1, pi+2, …, pi+3Dx/N contains at least one value 1/x
• if j = log(N/(4 log N))+1, … , log N , the sequence
pi+1, pi+2,…, pi+3Dx/(Nlog N) contains at least one value 1/x
Wroclaw 2007
95
Universal sequence exists
Lemma: There exists a universal sequence.
Proof: Idea of construction of universal sequence:
– put values 2-j to nodes of the complete binary tree of N
leaves according to some rule
– traverse this tree, writing values of visiting nodes
Wroclaw 2007
96
Ideas behind the algorithm
Ideas of algorithm (assuming known D):
• partition into stages, each taking log(N/D) + 2 steps
• in steps j of stage, for j = 0,1,…,log(N/D) , we want
to assure fast transmission to the node having
informed neighbour and of degree close to 2j
- hence we transmit with probability 2-j
• in step j = log(N/D) + 1 of stage i we want to
assure fast transmission to the node having
informed neighbour and of degree greater than N/D
- hence we transmit with probability pi
according to the universal sequence
Wroclaw 2007
97
Randomized algorithm
source transmits
for D := 1 to log N do
-- D unknown
for i := 1 to const D do
-- executing stage(D,i)
if node v received the source message before
stage(D,i) then
for k = 0 to log(N/D) do transmit with probability 2-k
transmit with probability pi
Wroclaw 2007
98
Complexity
Expected broadcasting time: O(D log(n/D) + log2 n)
Remark: Complete-layered graphs are among
most difficult to broadcast by randomized
algorithms
Complete-layered
network
0
L1
L2
Lj  {1,…, n}
Wroclaw 2007
LD-1
LD
99
Broadcast in multi-hop:
conclusions
• Randomization is better than determinism
• Complete-layered networks are among most hard
networks to broadcast by randomized algorithms,
but not by deterministic algorithms
Wroclaw 2007
100
Roadmap
• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
– Broadcasting
– Oblivious broadcasting and gossiping
– Wake-up
• Future directions
Wroclaw 2007
101
Oblivious deterministic
algorithm
Theorem: For all parameters n,D such that
1 < D < n, and for any deterministic
oblivious broadcasting scheme A, there
exists an n-node network GA of radius D,
such that scheme A requires time
(n min{D,n1/2}) to broadcast on GA.
[Chlebus, Gasieniec, Ostlin, Robson ICALP’01]
Wroclaw 2007
102
Strongly-selective families
Strongly-selective family [CMS] :
A family F of subsets of R is called
(|R|,k)-strongly-selective, for k  |R|, if
for every subset Z of R such that |Z|  k, and
for every element z  Z,
there is a set F  F such that Z  F = {z}.
Lemma [CMS]: Let F be an (|R|,k)-stronglyselective family (ssf in short). Then
(a) if 3  k < (2|R|)1/2 then |F|  (k2 log |R|)/(48 log k) ,
(b) if k  (2|R|)1/2 then |F|  |R| .
Wroclaw 2007
103
Proof of lower bound:
case D  (n/8)1/2 v layer 0
0
Suppose sets X0, X1,…, Xi constructed
by step t of scheme A, i < D/2
v1 layer 1
Let Ri+1 contain
remaining nodes, |Ri+1| > n/2
v3 layer 3
X1
Consider family Tt+1,…,Tt+n/2
of transmitters in steps t+1,…,t+n/2;
it is not (|Ri+1|,n/(2D))-ssf
Define Xi+1  Ri+1 and vi+1  Xi+1 s.t.
Xi+1  Tt+j  {vi+1}
|Xi+1|  n/(2D)
v2 layer 2
layer D/2+3
Path
layer D/2+4
layer D-1
Remaining nodes
Wroclaw 2007
layer D
104
Proof of lower bound:
case D > (n/8)1/2 v layer 0
0
Suppose sets X0, X1,…, Xi constructed
by step t of scheme A,
i < D’ = n1/2/4
v1 layer 1
Let Ri+1 contain
remaining nodes, |Ri+1| > n/2
v3 layer 3
X1
Consider family Tt+1,…,Tt’
of transmitters in steps t+1,…,t’= t+n/2;
it is not (|Ri+1|,n/(2D’))-ssf
Define Xi+1  Ri+1 and vi+1  Xi+1 s.t.
Xi+1  Tt+j  {vi+1}
|Xi+1|  n/(2D’)
v2 layer 2
Remaining nodes
Wroclaw 2007
layer D’/2+3
Path
layer D’/2+4
layer D-1
layer105D
Oblivious randomized
algorithm
Algorithm Randomized-Oblivious
• count := 1
• repeat n2/log n times
 for l := 1 to log n do
-- iteration of stages
(a.) each node transmits independently with
probability 2-l
(b.) node with label count transmits,
count := count + 1 mod n
Wroclaw 2007
106
Analysis of randomized
algorithm
Theorem: Algorithm Randomized-Oblivious broadcasts in time
O(n min{D,log n}) on any n-node network with diameter D.
Proof :
• D < log n : broadcasting completed during first nD
executions of instruction (b.), by round-robin property
• D  log n : consider a shortest path v0,…,vk=v from the
source to a node v ; let di+1 be degree of node vi+1 .
Claim: vi receives a message from vi+1 during di+1 consecutive
stages with (positive) constant probability.
Since i  k di  2n we get expected number O(n log n) of steps
Wroclaw 2007
107
Lower bound
for randomized algorithms
Theorem: For every oblivious randomized broadcasting
algorithm A and every sufficiently large n, there exists an
n-node network GA of diameter 3, such that the algorithm
A requires time (n), with probability at least 1/2, to
complete broadcasting on GA.
Idea of the proof :
1
Select network GA,v with uniform
Network GA,v
probability, among v = 1,…,n-2 .
0
v
n-1
With probability at least 1/2 node
vn-1 receives a source message
in algorithm A in time (n).
n-2
Wroclaw 2007
108
Roadmap
• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
– Broadcasting
– Oblivious broadcasting and gossiping
– Wake-up
• Future directions
Wroclaw 2007
109
Universal synchronizers
wake-up fast
Take a node v, and a shortest path v0, v1, ... , vL = v
from the first active node v0 to node v
Let d(vi ) denote the number of nodes that may
attempt to wake-up vi, but not any further node
vi+j , for j > 0
Wroclaw 2007
110
Universal synchronizers
wake-up fast
By definition of universal synchronizer, node v is
activated by step
1  i  L g(d(vi )) = O(1  i  L d(vi ) log2 n )
Since 1  i  L d(vi )  n, we obtain that
1  i  L g(d(vi )) = O( n log2 n )
which is a bound on wake-up
Wroclaw 2007
111
Roadmap
• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Future directions
Wroclaw 2007
112
Selected remarks
• Some results also hold for directed graphs (e.g.,
randomized broadcasting, wake-up), other do not
(e.g., oblivious broadcasting/gossiping), some we
do not know (e.g., adaptive gossiping)
• If the labels of the nodes are taken from large
interval then some results change, usually by a
logarithmic factor taken from the length of the
range of the interval
Wroclaw 2007
113
Future directions
1. Dynamic model: extend for further aspects
•
latency, dropping packets, multi-hop
2. Relations between various communication
problems in different settings
3. Explicit/practical deterministic/pseudodeterministic algorithms for more complex
problems
4. Considering another efficiency measures
•
communication complexity (energy)
Wroclaw 2007
114
Future directions (cont.)
5. Does collision detection helps in multi-hop
networks?
6. Selected classes of ad-hoc networks
•
•
geometric (random, disc graphs, BIG graphs, …)
random, small-world, …
7. Centralized algorithms
•
•
sensor networks
unknown requests make life harder
8. Fault-tolerance, attacks, games, more realistic
models, …
Wroclaw 2007
115
Thank you
Wroclaw 2007
116