Function Computation in
Simple Broadcast Networks
Presented by Lei Ying
Model
 Network Model
 Broadcast network: a transmission
by one node can be received by each
other node.


Noisy channels: the channel
between any pair of nodes is a
binary symmetric channel with error
probability p.
There are N nodes in the network.
Receiver (Fusion
Center)
Data Node (Sensor)
Model
 Data Model


Each node has one bit (bi)--- “0” or “1.”
The goal of the network is to compute some
functions based on the nodes’ data.
(1) Parity Computation:
(i bi)(mod 2) (Gallager’88)
(2) Threshold Detection:
(i bi) T (Kushilevitz & Mansour’ 98)
Goal
 Design algorithms such that
limN!1Pr(Correct parity computation (Threshold Detection))>1-
 Goal: Find out how many transmissions we need (one transmission
means transmission of one bit).
 Trivial lower bound: Each node has to transmit each bit once, so it
requires N transmissions.
 Trivial upper bound: O(N ln(N)).
Notations: f(n)=O(g(n)) means f(n)· cg(n) for n¸ m
f(n)=(g(n)) means f(n)¸ cg(n) for n¸ m
f(n)=(g(n)) means f(n)=O(g(n)) and f(n)=(g(n)).
Lower and Upper Bounds
 Lower bound: Since the channels are noisy, N
transmissions are not sufficient.
 Upper bound:
Lemma 1: Suppose one bit of data is transmitted m
times over a binary symmetric channel with error
probability p, and the receiver decodes the bit using
majority rule. Then, the probability of decoding error
is no greater than
(4p(1-p))m/2.
Upper Bounds
Proof: Define m independent binary random variables {Ii},
where Ii=0 with probability p and Ii=1 with probability
1-p. Using Chernoff’s bound, we have
Upper Bounds
Upper Bound: If one node broadcasts its bit aln(N) times,
then the fusion center can decode the bit correctly with
probability 1-(4p(1-p))a/2 ln(N).
There are N nodes, using the union bound, we have that the
probability the fusion center obtains all bits correctly is
1-N (4p(1-p))a/2 ln(N) .
Choose a large enough such that (4p(1-p))a/2<e-2, then
1-N (4p(1-p))a/2 ln(N)>1-N/N2! 1.
Upper Bound
So O(N ln(N)) is enough.
Can the number of transmissions be further reduced?
Difficulty: each node has only one bit, so block coding
cannot be used.
Yes!
Idea: Note that it is a broadcast network, when one node
transmits, all other nodes can hear the transmission. So
the other nodes collectively could make a good decision.
Main Results
Parity computation: In [Gallager’88], it develops an
algorithm that requires O(N lnln(N)) transmissions.
Threshold detection: In [Kushilevitz & Mansour’ 98], it
provides an algorithm that requires (N) transmissions.
Parity Computation
 Algorithm:
(1) Partition N nodes into subsets such that each subset has
a1ln(N) nodes.
(2) Each node broadcasts its bit a2lnln(N) times.
(3) Each node computes the parity of its subset, and
broadcasts the parity once.
(4) The fusion center decodes the parities for each subset,
and then obtain the parity of the network.
 Number of transmissions:
a2Nlnln(N)+N.
Parity Computation
Algorithm:
(1) Partition N nodes into
subsets such that each
subset has a1ln(N) nodes.
(2) Each node broadcasts its
bit a2lnln(N) times.
(3) Each node computes the
parity of its subset, and
broadcasts the parity
once.
(4) The fusion center
decodes the parities for
each subset, and then
obtains the parity of the
network.
 Error Probability:
(1) Choose (4p(1-p))a2/2<e-2. For each node, it
obtains the parity of its subset with probability
error less than
a1ln(N) (4p(1-p))a2/2 lnln(N) < a1/ln(N).
Parity Computation
Algorithm:
(1) Partition N nodes into
subsets such that each
subset has a1ln(N) nodes.
(2) Each node broadcasts its
bit a2lnln(N) times.
(3) Each node computes the
parity of its subset, and
broadcasts the parity
once.
(4) The fusion center
decodes the parities for
each subset, and then
obtains the parity of the
network.
 Error Probability:
(1) For each node, it obtains the parity of its
subset with probability error less than
a1ln(N) (4p(1-p))a2/2 lnln(N) < a1/ln(N).
(2) For each subset, the fusion center receives
the parity of that subset from a1ln(N) nodes,
and the error probability of each bit is less
than
(1-a1/ln(N))p+(1-p)a1/ln(N)=p+.
Note that  can be arbitrarily small when N goes to infinity
Parity Computation
Algorithm:
(1) Partition N nodes into
subsets such that each
subset has a1ln(N) nodes.

Error Probability:
(2) The fusion center receives the parity of a subset
from a1ln(N) nodes, and the error probability of each
bit is less than p+.
(2) Each node broadcasts its
bit a2lnln(N) times.
(3) Each node computes the
parity of its subset, and
broadcasts the parity
once.
(4) The fusion center
decodes the parities for
each subset, and then
obtains the parity of the
network.
(3) The parity of each subset can be
obtained with error probability less than
(4(p+)(1-p-))a1/2 ln(N).
There are N/(a1ln(N)) subsets, so the error
probability is
N/(a1ln(N)) (4(p+)(1-p-))a1/2 ln(N)
Choose (4(p+)(1-p-))a1/2 · e-2, the
fusion center obtains the correct parity
with probability at least
1/(a1N ln(N)).
Threshold Detection
 Goal: For each node (not only fusion center), we want to
know
(i bi) T (Kushilevitz & Mansour’ 98)
 Algorithm:
(1) Each node broadcasts its bit M times.
(2) So each node receives MN bits, let Ai be the sum of the bits
node I received, and  i=1 if Ai>f(T); and  i=0 otherwise.
Transmit  i once
(3) Let  i be the majority value of the bits received; transmit  i
once.
(4) Let Fi be the majority value of the bits received. The node
decides the threshold is reached if Fi=1, and is not reached if
Fi=0.
Threshold Detection
 f(T)=?
 Lemma 1: suppose i bi=L, then
(1) {Ai} are i.i.d;
(2) E[Ai]=NMp +2M2 L, where M2=M(1-2p)/2
Proof: E[Ai] = M(L(1-p)+(N-L)p)
= M(L(1-2p)+Np)
= NMp+2M2L.
Threshold Detection
L=T+1
NMp+2M2T+2M2
L=T-1
NMp+2M2T-2M2
NMp+2M2T-4M2
L=T-2
NMp+2M2T
L=T
NMp+2M2T-M2
 Consider the worst case: i bi=T or i bi=T-1.
 Choose f(T)=NMp+2M2T-M2.
Threshold Detection
Lemma 2: Let {Xi} be N i.i.d. binary random variables,
and p=Pr[Xi=1]. Then,
2N
-2
Pr[i Xi<(p-)N]<e
Lemma 3: Let Xi be N independent, binary random
variables, and let =E[i Xi]. Then,
2N
-2
Pr[i Xi<(/N-)N]<e
Threshold Detection
Algorithm:
(1) Each node broadcast its
bit M times.
(2) i=1 if Ai>f(T) and i=0
otherwise. Transmit i
once.
Lemma 4: If L¸ T,
If L T-1,
(3) Let i be the majority
value of the bits
received; transmit i
once.
(4) Let Fi be the majority
value of the bits
received.
After step (2), each node can correctly
detect the threshold with a probability
Threshold Detection
Algorithm:
(1) Each node broadcast
its bit M times.
(2) i=1 if Ai>f(T); and
i=0 otherwise.
Transmit i once.
(3) Let  i be the majority
value of the bits
received; transmit  i
once.
(4) Let Fi be the majority
value of the bits
received.
 Now with probability
, each node has the
correct i.
 Recall that {i} are i.i.d. since {Ai} are i.i.d.
 Lemma 5. At the end of step (2), with probability
2
at least 1-e-c M2/2, at least
nodes has the correct i.
Proof: Suppose L¸ T, i=1 is correct, by Lemma 3
Threshold Detection
Algorithm:
(1) Each node broadcast
its bit M times.
(2) i=1 if Ai>f(T); and
i=0 otherwise.
Transmit i once.
(3) Let  i be the majority
value of the bits
received; transmit  i
once.
(4) Let Fi be the majority
value of the bits
received.
 Lemma 6: Assume that there are at least
nodes have the correct i, then after
2N
-2q
step (3), with probability 1-e
, at
least (1-2q)N nodes has the correct i,
where
Proof: First, consider the probability
that i is correct.
Threshold Detection
Algorithm:
(1) Each node broadcast
its bit M times.
(2) i=1 if Ai>f(T); and
i=0 otherwise.
Transmit i once.
(3) Let  i be the majority
value of the bits
received; transmit  i
once.
(4) Let Fi be the majority
value of the bits
received.
 Suppose ji is the out of the binary symmetric
channel between node j and node i with input i.
Threshold Detection
Algorithm:
(1) Each node broadcast
its bit M times.
 Now i=1 with probability at least 1-q.
(2) i=1 if Ai>f(T); and
i=0 otherwise.
Transmit i once.
 Suppose there are at least (1-2q)N nodes have
(3) Let  i be the majority
value of the bits
received; transmit  i
once.
(4) Let Fi be the majority
value of the bits
received.
Pr(i i>(1-2q)N)= Pr(i i>(1-q-q)N) 1-e-2q
2N
the correct i=1. Suppose ji is the out of the
binary symmetric channel between node j and
node i with input i.
Threshold Detection
Algorithm:
(1) Each node broadcast
its bit M times.
(2) i=1 if Ai>f(T); and
i=0 otherwise.
Transmit i once.
(3) Let  i be the majority
value of the bits
received; transmit  i
once.
(4) Let Fi be the majority
value of the bits
received.
 Use the union bound, the probability that
all Fi are correct is
 Choose large M, we have that the
probability that all nodes have the correct
Fi is at least 1-.
 The number of transmissions is
MN+N+N=(M+2)N=(N)
Thanks
Finding the States of All Nodes
 Partition the N nodes as follows:

Each subset contains a1ln(N).
Each node belongs to a1ln(N) subsets

No two nodes belong to more than one subset.

Each node associates with one subset, and each
subset has only one node associate with it.

Finding the States of All Nodes
 Algorithm:
(1) Each node broadcasts its bit a2lnln(N) times.
(2) Each node computes the parity of it associated subset,
and broadcasts the parity once.
(3) Fusion center determines b~i from step (1). Suppose
node i belongs to subset {Si, k}, and the parities obtained at
the fusion center are {Pi,k}.
Then,
The state of node i is the majority of
.