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Communication protocols for packet radio networks

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Title: 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) ?

16
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

30
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

32
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
33
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
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