Communication protocols for packet radio networks

1
Communication protocols for packet radio networks
algorithmic approach

• Darek Kowalski

2
Introduction to the model of packet radio network

• Network of stations
• Synchronized
• Possible conflicts of messages arriving at a
  station

3
Motivation

• Motivation coming from
• Local Area Networks (LANs)
• ETHERNET
• Upstream HFC (Hybrid Fibre-Coax)
• Wireless networks (802.11 wireless LAN)
• Sensor networks

4
Main paradigms

• Data-link protocols for shared wire or medium
• Ad hoc mobility issues
• Dynamic communication tasks
• Algorithms behind protocols

5
Roadmap

• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Future directions

6
Model of radio networkstations

• 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

7
Model of radio networktransmissions

• 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.

8
Receiving a messagecases
Assume that the middle node is listening
Silence
Successful transmission
Collision
9
Single-hop multi-hop
Single-hop (multiple-access channel) all nodes
are directly connected
Multi-hop D denotes a diameter
10
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

11
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)

12
Efficiencycomplexity 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

13
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 gt 0
  such that for all n and k inequality f(n,k) gt 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))

14
Roadmap

• Presentation of the model
• Single-hop radio networks
• Static problems
• Model with delays
• Dynamic model
• Multi-hop radio networks
• Future directions

15
Multiple-access channelbasic problems

• Detecting collision
• some set K of k stations want to transmit
• how recognize if k gt 1 ?
• Solving collision (broadcasting, leader
  election)
• some set K of k gt 1 stations want to transmit
• how select one of them to transmit successfully
  (without a collision) ?

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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 gt1) 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

17
Detecting collisiononly 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

18
Solving collisionall 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)

19
Solving collisionall 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)

20
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

21
Solving collisiononly a subset active

• Bar-Yehuda, Goldreich, Ittai PODC87
• 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

22
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

23
Solving collision only a subset active (cont.)

• Metcalfe, Boggs CACM76
• 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

24
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

25
Another problem

• Conflict resolution (m2m)
• some set K of k gt 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 PODC05
• Using randomization add k to the complexity
• time O(k log n) Martel IPL94

26
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

27
Roadmap

• Presentation of the model
• Single-hop radio networks
• Static problems
• Model with delays
• Dynamic model
• Multi-hop radio networks
• Future directions

28
Towards a dynamic model

• Reasonable asynchrony is not efficient
• Chlebus,Rokicki SIROCCO04
• 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 ESA01
• -- temporary leavings -- reasonable model
  wanted!
• 3. Combine two above models -- later

29
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

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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 PODC00
• Wake-up is a generalization of broadcast

31
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

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How it works

• Select k rows

0
0
1
0
1
1
1
0
0
0
0
0
Arbitrary shift
1
0
0
1
1
0
1
1
1
1
1
0
Find columns with single 1
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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

34
Universal synchronizers with small delays exist

• Theorem Chlebus et al. ICALP05
• 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 gt 0.
• This is shown by the probabilistic method
• let each row be a sequence of cn log2 n bits,
  for some c gt 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 gt 0 it is as claimed with a
  positive probability.

35
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 STOC01
• 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

36
Open problems and directions

• Efficient construction of synchronizers
• Closing the logarithmic gap
• More realistic model, communication complexity,

37
Roadmap

• Presentation of the model
• Single-hop radio networks
• Static problems
• Model with delays
• Dynamic model
• Multi-hop radio networks
• Future directions

38
Fully-dynamic broadcastintroduction

• Broadcast requests occur in dynamic fashion -
  generalization of static conflict resolution
• Local buffers must have bounded sizes

39
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)

40
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

41
Dynamic broadcastexample
window w
window w
window w
time
window ?(n)?w 3
42
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

43
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 )

44
Example of three rounds
message received
collision
silence
transmitter
transmitters
new packet
45
Dynamic broadcast

• Deterministic
• combined packets, latency-oriented
• Kowalski PODC05
• dynamic packet broadcast, stability oriented
• Chlebus, Kowalski, Rokicki PODC06
• Randomized
• stochastically modelled packet injection
• Hastad Leighton Rogoff SICOMP96
• Goldberg MacKenzie Paterson Srinivasan JACM00
  
• adversarial queuing for backoff protocols
• Bender Farach-Colton He Kuszmaul Leiserson
  SPAA05

46
Related work

• Static broadcast/conflict resolution on
  multiple-access ch.
• Deterministic
• Komlos Greenberg Trans. Inf.85
• Greenberg Winograd JACM85
• Randomized
• Willard SICOMP86
• Kushilevitz Mansour SICOMP98
• Bender Farach-Colton He Kuszmaul Leiserson
  SPAA05
• Adversarial queuing in wired networks
• Borodin Kleinberg Raghavan Sudan Williamson
  JACM01
• Andrews Awerbuch Fernandez Leighton Liu
  Kleinberg JACM01
• Aiello Kushilevitz Ostrovsky Rosen JCSS00

47
Main properties

• Impossibility
• No protocol is stable against injection rate 1
  for n gt 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

48
Impossibility for injection rate 1

• No protocol is stable against injection rate 1
  for n gt 3
• Proof Consider a fair protocol run by 4
  stations.
• Observation 1 adversarys 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

49
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

50
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

51
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.

52
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)

53
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,

54
Roadmap

• Presentation of the model
• Single-hop radio networks
• Broadcast with unlimited energy
• One-shot broadcast

55
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

56
Unlimited energyresults

• Lower bound
• Graph family of radius 2 ?(log2n)
  AlonBar-NoyLinialPeleg, JCSS91
• Upper bound
• O(D log2n) general graphs ChlamtacWeinstein,
  INFOCOM87
• O(D log n log2n) general graphs KowalskiPelc,
  APPROX04
• O(D log5n) general graphs GaberMansour,
  SODA95
• D O(log4n) general graphs ElkinKortsarz,
  SODA05
• D O(log3n) planar graphs ElkinKortsarz,
  SODA05

57
Unlimited energyresults 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

58
Tree rankingdefinition

• 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?

59
Tree rankingexample
3
3
2
1
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
1
1
1
1
2
2
1
1
1
1
2
2
1
1
1
1
1
1
60
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)

61
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

62
Construction of Gathering Spanning Tree (GST)
The pruning process BFS --gt GST
Nodes here have ranks for good
Direction of the pruning process
63
Pruning processremoving 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
64
Broadcasting in general graphsoverview

• Theorem There exists efficient construction of
  the broadcast schedule requiring time O(D log3n)

65
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 gt 1/4e gt 0.

66
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)

67
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) rounds

68
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)

69
Roadmap

• Presentation of the model
• Broadcast with unlimited energy
• One-shot broadcast

70
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
1
2
3
4
Layer U
Layer L
2,4
1,2
1,3
1,4
2,3
3,4
71
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
1
2
3
4
Layer U
Layer L
2,4
1,2
1,3
1,4
2,3
3,4
72
Networks with radius 2 algorithm cont.
1
4
Witness graph
2
3
source
1
2
3
4
Layer U
Layer L
2,4
1,2
1,3
1,4
2,3
3,4
73
Networks with radius 2 algorithm cont.

• In graph with x nodes and y edges an independent
  set of size x2/(2yx) can be constructed in
  polynomial time
• The corresponding set in the initial graph
  informs at least x2/(2yx) 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

74
General networks

• Approach use ranking and GST
• Problems
• fast and slow transmissions dont 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

75
One-shot broadcast in general graphs

• Theorem There exists efficient construction of
  1-shot broadcast schedule requiring time O(D
  n1/2log n)

76
General networkscont.

• 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

77
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

78
Roadmap

• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Broadcasting
• Oblivious broadcasting and gossiping
• Wake-up
• Future directions

79
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

80
Bibliography

• Chlamtac, Kutten IEEE Trans. on
  Communication85
• introduced the model of radio communication
• Bar-Yehuda, Goldreich, Itai PODC87 randomized
  distributed broadcasting
• Gasieniec, Pelc, Peleg PODC00 introduced
  wake-up for multiple-access chan.
• Chlebus, Gasieniec, Gibbons, Pelc, Rytter
  SODA00
• deterministic distributed broadcasting
• Clementi, Monti, Silvestri SODA01 selective
  families in radio networks
• Indyk SODA02 explicit distributed broadcast
  and wake-up
• Chrobak, Gasieniec, Kowalski SODA04 introduced
  synchronizers, wake-up of multi-hop
  networks,applications to leader election and
  synchronization

81
Broadcasting timeresults

• No collision detection

82
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

How to choose S,R to force linear broadcasting
time on GS,R
83
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 n1,,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

84
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 n1,,2n being the
  output of function MODIFY(Rk-1,T), where
  T T1,,Tk-1 initially R0
  n1,,2n

85
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

86
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

87
Strongerdeterministic 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 PODC03

88
Construction of layer L2j-1

• Keep size L2j-1 O(n/D)
• Select set L2j-1 to assure that node 2j will not
  receive a message from set L2j-1 during (n/D)
  logn/D n steps after activation of nodes in
  L2j-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

89
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 gt n1/2 we prove lower bound ?(n log n /
  log(n/D)) on star-layered graphs randomized
  complexity O(n)

90
Deterministic algorithmrecursive selection

• Procedure SELECT(p,q,s) Kowalski, Pelc
  FOCS02
• 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/21,,N

91
Deterministic algorithm

• Algorithm Kowalski, Pelc PODC03
• 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))

92
Lower boundfor randomized broadcast

• Lower bound ?(D log(n/D)) for expected
  broadcasting time for n-node networks
  (complete-layered) with diameter D
• Kushilevitz, Mansour SICOMP98

Complete- -layered network
93
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 STOC89

94
Randomized algorithms

• Randomized algorithm with O(D log n log2 n)
  expected broadcasting time
• Bar-Yehuda, Goldreich, Itai PODC87
• Optimal algorithm broadcasting in expected time
  O(Dlog(n/D) log2 n) matching the lower bound
• Czumaj, Rytter FOCS03 Kowalski, Pelc PODC03
• Presentation of the result
• Combinatorial tool universal sequence
• Idea of construction
• The algorithm and remarks

95
Universal sequence

• Remind N,D are fixed.
• Definition An infinite sequence (pi)i1,,? of
  reals from the interval 0,1 is called universal
  sequence if the following conditions hold for
  every value x2j ( j log(N/D)1,
  , log N )
• if j log(N/D)1, , log(N/(4 log N)) , the
  sequence pi1, pi2, , pi3Dx/N contains at
  least one value 1/x
• if j log(N/(4 log N))1, , log N , the
  sequence pi1, pi2,, pi3Dx/(Nlog N)
  contains at least one value 1/x

96
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

97
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

98
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

99
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
100
Broadcast in multi-hopconclusions

• Randomization is better than determinism
• Complete-layered networks are among most hard
  networks to broadcast by randomized algorithms,
  but not by deterministic algorithms

101
Roadmap

• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Broadcasting
• Oblivious broadcasting and gossiping
• Wake-up
• Future directions

102
Oblivious deterministic algorithm

• Theorem For all parameters n,D such that 1 lt D
  lt 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 minD,n1/2) to
  broadcast on GA.
• Chlebus, Gasieniec, Ostlin, Robson ICALP01

103
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)-strongly-select
  ive family (ssf in short). Then
• (a) if 3 ? k lt (2R)1/2 then F ? (k2 log
  R)/(48 log k) ,
• (b) if k ? (2R)1/2 then F ? R .

104
Proof of lower bound case D ? (n/8)1/2
v0
layer 0
v1
layer 1

• Suppose sets X0, X1,, Xi constructed
• by step t of scheme A, i lt D/2
• Let Ri1 contain
• remaining nodes, Ri1 gt n/2
• Consider family Tt1,,Ttn/2
• of transmitters in steps t1,,tn/2
• it is not (Ri1,n/(2D))-ssf
• Define Xi1 ? Ri1 and vi1 ? Xi1 s.t.
• Xi1 ? Ttj ? vi1
• Xi1 ? n/(2D)

v2
layer 2
v3
layer 3
X1
layer D/23
layer D/24
Path
layer D-1
Remaining nodes
layer D
105
Proof of lower bound case D gt (n/8)1/2
v0
layer 0
v1

• Suppose sets X0, X1,, Xi constructed
• by step t of scheme A,
• i lt D n1/2/4
• Let Ri1 contain
• remaining nodes, Ri1 gt n/2
• Consider family Tt1,,Tt
• of transmitters in steps t1,,t tn/2
• it is not (Ri1,n/(2D))-ssf
• Define Xi1 ? Ri1 and vi1 ? Xi1 s.t.
• Xi1 ? Ttj ? vi1
• Xi1 ? n/(2D)

layer 1
v2
layer 2
v3
layer 3
X1
layer D/23
layer D/24
Path
layer D-1
Remaining nodes
layer D
106
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

107
Analysis of randomized algorithm

• Theorem Algorithm Randomized-Oblivious
  broadcasts in time O(n minD,log n) on any
  n-node network with diameter D.
• Proof
• D lt log n broadcasting completed during first
  nD executions of instruction (b.), by round-robin
  property
• D ? log n consider a shortest path v0,,vkv
  from the source to a node v let di1 be degree
  of node vi1 .
• Claim vi receives a message from vi1 during
  di1 consecutive stages with (positive) constant
  probability.
• Since ?i ? k di ? 2n we get expected number O(n
  log n) of steps

108
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
• Select network GA,v with uniform
• probability, among v 1,,n-2 .
• With probability at least 1/2 node
• vn-1 receives a source message
• in algorithm A in time ?(n).

1
Network GA,v
0
v
n-1
n-2
109
Roadmap

• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Broadcasting
• Oblivious broadcasting and gossiping
• Wake-up
• Future directions

110
Universal synchronizerswake-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
  vij , for j gt 0

111
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

112
Roadmap

• Presentation of the model
• Single-hop radio networks
• Multi-hop radio networks
• Future directions

113
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

114
Future directions

• Dynamic model extend for further aspects
• latency, dropping packets, multi-hop
• Relations between various communication problems
  in different settings
• Explicit/practical deterministic/pseudo-determinis
  tic algorithms for more complex problems
• Considering another efficiency measures
• communication complexity (energy)

115
Future directions (cont.)

• Does collision detection helps in multi-hop
  networks?
• Selected classes of ad-hoc networks
• geometric (random, disc graphs, BIG graphs, )
• random, small-world,
• Centralized algorithms
• sensor networks
• unknown requests make life harder
• Fault-tolerance, attacks, games, more realistic
  models,

116
Thank you