CPSC 689: Discrete Algorithms
for Mobile and Wireless Systems
Spring 2009
Prof. Jennifer Welch
Lecture 12
 Topic:
 Broadcast in General Networks
 Sources:
 Bar-Yehuda, Goldreich, Itai. Time complexity of
broadcast.
 Bar-Yehuda, Goldreich, Itai. Efficient emulation of
single-hop radio network on multi-hop radio network.
 Kowalski, Pelc. Time of deterministic broadcasting.
 MIT 6.885 Fall 2008 slides
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Network-Wide Broadcast
 Broadcasting a message from a known source
node to all other nodes in a fixed network.
 Network = connected, undirected graph, edge
between neighbors iff they can communicate
directly via single-hop broadcast.
 Common task for sensor networks, other ad hoc
networks.
 Has also been studied extensively by
theoreticians.
 We’ll look at several theoretical papers, see
whether the results have practical significance.
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Model Assumptions
 Unknown network topology---each node knows just local
information, typically its own id and sometimes those of its
neighbors
 Slot-based (synchronous)
 Strong assumptions about collision behavior:
 Receiver definitely does not receive any message
 Receiver cannot distinguish collision vs. idle
 A weaker model:
 Receiver might receive one of the messages correctly.
 Deterministic and randomized models.
 The problem: Obtain good upper and lower bounds for the
amount of time required to complete a broadcast.
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Results
 [Bar-Yehuda, Goldreich, Itai 1]:
 Randomized algorithm, with completion time
O((D + log(n/)) log n) slots, with probability at
least 1 – .
 D = max distance between source node and any
other node.
 n = total number of nodes in the network.
  = arbitrary positive real.
 Deterministic lower bound, (n) slots.
 For the weaker collision model.
 Claim exponential (in n) gap between randomized
and deterministic complexity.
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Results
 [Bar-Yehuda, Goldreich, Itai 2]: Simpler paper using
ideas from [BGI1].

Randomized algorithm to emulate any algorithm for a
single-hop network with collision detection, in a multi-hop
network with no CD.
 Application to leader election.
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More Results
 [Kowalski, Pelc]: Hard, carefully written paper,
deals with deterministic case only.
 Show that [BGI1] lower bound is incorrect, with a
O(log n) algorithm for the graphs in [BGI1]
Cf. errata web page by B, G and I
 Sublinear (in n) time algorithms for all graphs of
small max distance D
time o(n) for graphs with D = o(log log n)
 Lower bound of (n1/4) for graphs with D = 4
again claim exponential gap between randomized and
deterministic complexity.
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[BGI1] Randomized Algorithm
 Review of communication model:
 In each slot, each node chooses whether to
transmit, receive, or be inactive.
 If node u is a receiver in slot k and exactly one of
u’s neighbors in the graph transmits in slot k, then
u receives the message.
 Otherwise (if u is a receiver and 0 or >1 or its
neighbors transmits, or if u is not a receiver), then
u receives nothing ().
 No collision information!
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[BGI1] Randomized Algorithm
 Knowledge of nodes:
 All node programs are identical.
 Nodes know a “close” upper bound N on n.
 Nodes know an upper bound  on maximum node
degree.
  = arbitrary positive real, known to all nodes.
 Randomized model:
 Nodes can make random choices of what to do next.
 The broadcast problem:
 A single message, starting at node 0, must reach all
nodes (with high probability).
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The Decay Subroutine
Decay(k,m): (k ≥ 1)
repeat
transmit m
coin := 0 or 1, with equal probability
k := k - 1
until coin = 0 or k = 0
 At each slot, every node chooses whether to continue
transmitting or not, with probability ½.
 So the number of senders is approximately halved at each
slot.
 In any case, cut off the sequence of tries after k steps.
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Properties of Decay
 Suppose d nodes (say, neighbors of a fixed
node y) execute Decay(k,m) in synch,
starting with slot 1. Define:
 P(k,d) = probability that, at some slot ≤ k,
one of the d nodes transmits alone (and
thus y receives a msg).
 P(,d) = limkP(k,d) = probability that
one of the nodes transmits alone, if
attempts continue forever.
 Then P(k,0) = 0 and P(k,1) = 1.
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Properties of Decay
 Theorem 1 (a): For d  2, P(,d)  2/3.
 Proof:
 Recurrence: P(,d) = i=0d C(d,i)2–d P(,i).
 Sums, over all i, the probability that exactly i
remain after one Decay step, multiplied by the
probability of success if i remain.
 Solve the recurrence.
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Properties of Decay
 Theorem 1 (b): For d  2, for k  2 log d, P(k,d)  ½.
 Proof:
 Consider some small cases (d ≤ 5) “by inspection”.
 For d  6, note that P(,i) ≤ P(k,d) + d 2-k.
 Probability of success ever ≤ probability of success
within the first k slots plus probability that someone
performs more than k slots.
 Final term overestimates the probability that someone
performs more than k slots.
 Solve the recurrence (relies on part (a)).
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[BGI1] Broadcast Algorithm




Decay guarantees that one competitor out of d is isolated within 2
log d slots, with at least a constant probability.
Now use Decay in a network broadcast protocol.
Need to guarantee success probability  1–, not just ½ or 2/3, so
use it repeatedly.
Define:
 k = 2 log ,  an upper bound on degree, which will be a bound on


the number of competitors in uses of Decay within Broadcast.
t = log ( N / ), N a bound on number of nodes.
Idea:


Group slots into size k batches.
Upon first receiving m, each node waits for the beginning of the
next batch of slots, then runs Decay(k,m), t times.
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[BGI1] Broadcast Algorithm
i*k (i+1)*k (i+2)*k (i+3)*k …..
start
Decay(k,m)
m arrives
start
Decay(k,m)
(i+t)*k
(i+2)*k
start
Decay(k,m)
start
Decay(k,m)
execute Decay(k,m) t times
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[BGI1] Broadcast Algorithm
k = 2 log , t = log (N / )
Broadcast(m), for node 0:
do t times:
wait until time  0 mod k
run Decay(k,m)
Broadcast(m), for node u  0:
wait to receive a message, say m;
do t times:
wait until time  0 mod k
run Decay(k,m)
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A Scenario
 Node 0 broadcasts successfully to
both its neighbors, 1 and 2.
 Then 1 and 2 broadcast
simultaneously.
 Message reaches nodes 3 and 5
successfully, but a conflict occurs at
node 4.
1
3
4
0
2
5
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Algorithm Analysis
 Claim: With probability at least 1– , all nodes receive m
within O(D*k*t) = O(D*log()*log(N/)) slots.
 Proof idea (from proof of Lemma 2):
 For each node v, once a neighbor u receives the message,
with probability at least ½, v will receive the message within
one Decay execution by u.
 So with probability at most (1/2)t = (1/2)log(N/) =  / N, v will
fail to receive it within t Decay executions by u.
 So with probability at most N ( / N) = , some node will fail
to receive it within D*t sequential Decay execution times,
or D*k*t slots.
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Algorithm Analysis
 Theorem 4: Improved bound on slots:
O( (log ) (D + max{sqrt(D), sqrt(log(N/)}*log(N/))
= O((log N) (D + log(N/))
= O((log n) (D + log(n/))
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Additional Properties of [BGI1]
Broadcast Protocol
 Only requires a loose bound on n
 Doesn't require knowledge of neighbors
 Constant local computation per time slot for each
node
 Expected number of message transmissions is at
most 2n log (N/)
 Tolerates some topology changes / crash failures, as
long as network remains connected
 No acks are used so links need not be bidirectional
 Decay subroutine can be used to compute a breadthfirst search
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[BGI1] Lower Bound for Deterministic
Algorithms
 (n) slots.
 For the weaker collision model, in which a node may
receive one of the actual messages sent.
 Choice of message vs.  is not under control of the
algorithm, but of an “adversary”.
Algorithm would have to cope with either possibility.
Lower bound could take advantage of either.
 Claim exponential (in n) gap between randomized
and deterministic complexity.
 Seems valid for the weaker collision model, because
their algorithm seems to work in the weaker model
(LTTR).
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Overview of Lower Bound Proof
 Define a special class of graphs Cn
 Define a class of "abstract" broadcast protocols
 Argue that any time lower bound for abstract broadcast
protocols implies an asymptotically equal lower bound
for general broadcast protocols
 Show that any abstract broadcast protocol on
some Cn graph takes (n) rounds
 reduce to a combinatorial problem called "hitting game"
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Special Graphs Cn




Source connected to all Layer 1 nodes.
Sink connected to just the nodes in a set
S  {1,…,n}.
Let GS denote Cn graph based on S.
Problem: Message from 0 should reach
all nodes.



1
0
Since initial broadcast by 0 will reach all
nodes in Layer 1, the real problem is
simply to reach the sink.
Algorithm must ensure that, at some
time, only one node in S should send.
Layer 0
In a lower bound proof, an adversary
will try to prevent this from occurring.
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S
sink,
n+1
n
Layer 1
Layer 2
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Abstract Broadcast Protocols
 In every time slot, all nodes in Layer 1 are transmitters, and
either the source or sink is a receiver (but not both).
 Messages consist of (m, transmitter’s id, S-indicator).

S-indicator = Boolean saying whether the transmitter is in S.
 After each slot, all nodes in Layer 1 know whether someone in
Layer 1 has transmitted alone; if so, they know the contents of
the message.


Immediate feedback.
Strong assumption. OK for lower bound---strengthens the result.
 Broadcast completed once a node in S manages to transmit
alone.


Optimistic---this doesn’t imply that the message has reached the sink,
since this could be a slot where the source is a receiver.
Weakens the requirements. OK for lower bound.
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Relationship Between Abstract and
General Broadcast Protocols
 Lemma 6: If there is a general protocol
that terminates within k slots, on every
graph in Cn, then there is an Abstract
Protocol that terminates within 2k slots.
 I.e., if no abstract protocol can terminate
within m slots, then no general protocol
can terminate within m/2 slots.
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Execution of an Abstract Protocol



Hk = P1,…,Pk, history through k slots.
Each Pi is just the id of the unique node that transmitted alone at
slot i, if any, else .
Each node’s decision about whether to transmit is a function of:






its own id,
its S-indicator, and
the history so far.
Gives the node a lot of information, OK for a lower bound.
Express decision as a predicate .
Notation:



Ti = transmitters in slot i.
Ti1 = Layer 1 nodes that would transmit in slot i if they were in S
(for the given history), = nodes u for which (u,1,Hi-1) = 1.
Ti0 = Layer 1 nodes that would transmit in slot i if they were not in
S, = nodes for which (u,0,Hi-1) = 1.
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The Hitting Game

Game n, played on graphs in Cn, Explorer vs. Referee.





What happens at move i:




Referee picks a subset S  {1,…,n}.
Explorer chooses query sets (moves) M1, M2,…
Explorer tries to “hit” (isolate) an element of S.
Explorer’s subsequent Mi may depend on results of prior moves.
If | Mi  S | = 1, the game is over, Explorer wins.
Otherwise, if | Mi  Sc | = 1, the Referee reveals the unique element
in the intersection to the Explorer and the game continues.
Otherwise, the Referee reveals nothing and the game continues.
Roughly like what happens in the Abstract Protocol:


First condition corresponds to one member of S transmitting, which
means the sink should receive the message.
Second condition corresponds to the possibility that only one node
broadcast (but someone in S could also broadcast).
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Relationship Between Hitting Game
and Abstract Broadcast Protocol
 Lemma 7: If there is an Abstract Protocol
that terminates within k slots, on every
graph in Cn, then there is a 2k-move
winning strategy for the Hitting Game.
 I.e., if there is no m-move winning strategy
for the Hitting Game, then there is no m/2slot Abstract Broadcast Protocol for graphs
in Cn
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Lemma 7, Proof and Error
 Lemma 7: If there is an Abstract Protocol that terminates within k
slots, on every graph in Cn, then there is a 2k-move winning
strategy for the Hitting Game.
 Error occurs here; see [BGI] errata sheet.
 Need the weaker collision model, in which a node may receive a
message.
 Proof sketch:
 From any (worst-case) k-slot Abstract Broadcast Protocol,
construct a 2k-move Hitting Game query sequence.
 Referee answers used to determine outcome of the slot (who
receives what messages).
 Issue: Outcomes must be consistent with what is allowed in the
Abstract Protocol model.
 Requires the weaker model.
 More detail:…
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Reduction from Abstract Broadcast
Protocol to Hitting Game Strategy
 Show (existentially but not constructively) that if there is a
k-round ABP for Cn, then there is a 2k-move winning
strategy for the hitting game.
 Fix any S (subset of nodes in Layer 1 of Cn).
 Use an execution of the k-round ABP on the graph to
construct hitting game strategy.
 Each round of execution produces 2 moves of the hitting
game
 Because of nondeterminism in how collisions are detected,
there can be many executions of same protocol on same
network, so iteratvely specify execution of the protocol as
we construct the strategy
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More on the Reduction
 Let H be the first i-1 rounds of the execution of the
ABP constructed so far
 Use round i to determine moves 2i and 2i+1
 Let T1, set of nodes in S that are supposed to
transmit in round i, be move 2i
 Let T0, set of nodes in Sc that are supposed to
transmit in round i, be move 2i+1
 Specify outcome of the round (which nodes
receive which messages)…
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Specifying Outcome of the Round
 Must conform to the model of computation
 If T1  S is a singleton, say node p, then deliver
the message from p to all recipients (ABP
terminates)
 If T0  Sc is a singleton, say node p, then deliver
message from p to all recipients
 ==> not consistent with stronger model, since it could
be that there are nodes in T1 also transmitting and thus
colliding!
 Otherwise, deliver no messages.
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Finishing Up Reduction
 So execution constructed is "legal"
(conforms to model of execution)
 By assumption, it terminates within k
rounds, i.e., exactly one node in S is
chosen to transmit in a round
 Thus within 2k moves the hitting game has
won.
 Now we can turn our attention solely to the
hitting game…
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(n) Lower Bound on Hitting Game
 Define an “adversary” that constructs a set S that “fools” a given
sequence M1,M2,…of queries.
 Issue: The Hitting Game allows an adaptive sequence of query
sets, whereas the lower bound is proved only for an oblivious
sequence of query sets.
 They argue (in errata note) that the particular adversaries they
construct, which beat all oblivious strategies, will also beat all
adaptive strategies:
 Referee responses give no information to the Explorer.
 Suppose we had an adaptive strategy that did well against
uninformative Referees.
 Then the particular sequence of moves it makes with the
uninformative Referees is an oblivious strategy that does well against
the same Referees.
 Details omitted.
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