Optimal Byzantine Attacks on Distributed
Detection in Tree-based Topologies
Bhavya Kailkhura, Swastik Brahma, Pramod K. Varshney
Department of Electrical Engineering and Computer Science
Syracuse University
Syracuse, NY 13244, USA
{bkailkhu, skbrahma, varshney}@syr.edu
Abstract—This paper considers the problem of optimal Byzantine attacks or data falsification attacks on distributed detection
mechanism in tree-based topologies. First, we show that when
more than a certain fraction of individual node decisions are falsified, the decision fusion scheme becomes completely incapable.
Second, under the assumption that there is a cost associated
with attacking each node (which represent resources invested in
capturing a node or cloning a node in some cases), we address
the problem of minimum cost Byzantine attacks and formulate it
as the bounded knapsack problem (BKP). An algorithm to solve
our problem in polynomial time is presented. Numerical results
provide insights into our solution.
Index Terms—Distributed Detection, Byzantine Attacks, Tree
Topologies, Bounded Knapsack Problem
I. I NTRODUCTION
Distributed detection has been a well studied topic in the
detection theory literature [1][2][3] and has traditionally focused on the parallel network topology, in which nodes directly
transmit their observed data to the Fusion Center (FC). Parallel
topology, while both theoretically and practically important
and analytically tractable, however, may not always reflect the
practical scenario [4]. In practice, to increase the coverage
area, the FC may be outside the communication range of some
of the nodes. Such networks have to form a multi-hop network,
where nodes are organized hierarchically into multiple levels
(tree networks). With intelligent use of resources across levels,
tree networks have the potential to provide a suitable balance
between cost, coverage, functionality, and reliability [5]. Some
examples of tree networks include wireless sensor and military
communication networks. For instance, the IEEE 802.15.4
(Zigbee) specifications [6] and IEEE 802.22b [7] can support
tree-based topologies.
Recently, distributed detection in the presence of Byzantine
attacks has been explored by [8][9], where the problem
of determining the most effective attacking strategy of the
Byzantine sensors was explored. However, both works only
considered the parallel topology. In this paper, we address the
problem of optimal Byzantine attacks (data falsification) on
distributed detection for a tree-based topology in which nodes
at different levels have varying costs of being attacked. The
main contributions of this paper are as follows.
1) We obtain the optimal attacking strategies that minimize
the detection error exponent at the FC in a tree network
and show that when more than a certain fraction of
individual node decisions are falsified, the decision
fusion scheme becomes completely incapable.
2) We formulate the problem of optimal Byzantine attacks
on the tree network as a bounded knapsack problem
(BKP), and present a polynomial time algorithm to solve
it.
3) We provide numerical results to gain insights into the
solution.
The rest of the paper is organized as follows. Section II
introduces our system model, including the Byzantine attack
model. In Section III, we formulate the minimum cost Byzantine attack as BKP and provide an algorithm to solve it. Section
IV provides numerical results. Finally, Section V concludes the
paper.
II. S YSTEM M ODEL
We consider a tree network with a set N = {Nk }K
k=1 of
transceiver nodes with K > 1. Nk is the total number of
nodes at level k and K represents the total number of levels
in the network, with FC at Level 0 or as root of the tree. We
assume that the tree is balanced, i.e., nodes at the same level
have an equal number of immediate
children. The total number
PK
of nodes in the network is k=1 Nk = N . Let B = {Bk }K
k=1
denote the set of Byzantine nodes with |Bk | = Bk . We assume
that FC is not aware of the exact set of Byzantine nodes and
considers each node at level k to be Byzantine with a certain
probability αk . We also denote by βk ∈ [0, 1] the probability
that a decision received at the FC originated from nodes at
level k. We consider the Clairvoyant case in this paper and
assume that the FC knows αk , βk .
A. Distributed detection in a tree topology
We consider a binary hypothesis testing problem with the
two hypotheses H0 (signal is absent) and H1 (signal is
present). Each node i at level k acts as a source in that it
makes a one-bit local decision vk,i ∈ {0, 1} and sends uk,i
(which may not be the same as vk,i ) to its parent node at level
k − 1. It also receives the decisions uk0 ,j of all successors j
at levels k 0 ∈ [k + 1, K], which are forwarded to i by its
immediate children, and forwards them to its parent node
at level k − 1. We also assume error-free communication
channels between children and the parent nodes. We denote
the probabilities of detection and false alarm of node i at level
k by Pd = P (vk,i = 1|H1 ) and Pf a = P (vk,i = 1|H0 ),
B
B
H
H
P (zi = j|H0 ) = [β1 α1 + β2 (α1 + α2 )][Pj,0
(1 − Pf a )+ Pj,1
Pf a ]+ [β1 (1 − α1 )+ β2 (1 − α1 − α2 )][Pj,0
(1 − Pf a )+ Pj,1
Pf a ] (1)
B
B
H
H
P (zi = j|H1 ) = [β1 α1 + β2 (α1 + α2 )][Pj,0
(1 − Pd ) + Pj,1
Pd ] + [β1 (1 − α1 ) + β2 (1 − α1 − α2 )][Pj,0
(1 − Pd ) + Pj,1
Pd ] (2)
respectively, which are assumed to be the same for both honest
and Byzantine nodes. If a node is honest, then it forwards
its own decision and received decisions correctly. However,
a Byzantine node, in order to undermine the network performance, may alter its decision as well as received decisions
from its children prior to transmission. We define the following
H
H
B
B
strategies Pj,1
, Pj,0
and Pj,1
, Pj,0
(j ∈ {0, 1}) for the honest
and Byzantine nodes, respectively:
Honest nodes:
H
H
= 1 − P0,1
= P H (x = 1|y = 1) = 1
(3)
P1,1
H
H
P1,0 = 1 − P0,0 = P H (x = 1|y = 0) = 0
(4)
Byzantine nodes:
B
B
= 1 − P0,1
= P B (x = 1|y = 1)
P1,1
B
B
P1,0 = 1 − P0,0 = P B (x = 1|y = 0)
(5)
(6)
where P (x = a|y = b) is the probability that a node sends
a to its parent when it receives b from its child or its actual
decision is b. Furthermore, we assume that if a node (at any
level) is a Byzantine then none of its ancestors and successors
are Byzantine; otherwise, the effect of a Byzantine due to
other Byzantines on the same path may be nullified (e.g.,
Byzantine ancestor re-flipping the already flipped decisions
of its successor). This means that every path from a leaf
node to
the FC will have at most one Byzantine. Thus, we
PK
have, k=1 αk ≤ 1. Next, for clarity of exposition, we first
formulate the binary hypothesis testing problem at the FC for
only two levels in the network, later we generalize our results
for any arbitrary K. Specifically, for a two level network,
distributions of received decisions at the FC zi , i = 1, .., N ,
under H0 and H1 become (1) and (2) respectively.
B. Byzantine attack model
In this section, a mathematical model for Byzantine attacks
is presented. The Byzantine attacker always wants to degrade
the detection performance at the FC as much as possible. In
this work, we employ the Kullback-Leibler divergence (KLD)
[10] to be the network performance metric that characterizes
detection performance. We know that the KLD between the
distributions q = P (zi = j|H0 ) and p = P (zi = j|H1 ) can
be expressed as
X
P (zi = j|H1 )
P (zi = j|H1 ) log
D(p||q) =
(7)
P (zi = j|H0 )
j∈{0,1}
The Byzantine nodes want to make the KL divergence as small
as possible and we know KLD is always non-negative; so they
want to choose P (zi = j|H0 ) and P (zi = j|H1 ) such that
KLD = 0. This is possible when
P (zi = j|H0 ) = P (zi = j|H1 )
∀j{0, 1}
(8)
Substituting (1) and (2) in (8) and after simplification, we can
express the condition to make the KLD = 0 for K = 2 as
β1 (1 − α1 ) + β2 (1 − α1 − α2 )
H
H
B
B
.(Pj,0
− Pj,1
)
Pj,1
− Pj,0
=
β1 α1 + β2 (α1 + α2 )
(9)
Hence, the Byzantine attacker can degrade detection perB
B
formance by intelligently choosing (Pj,1
, Pj,0
), which are
dependent on α1 , β1 , α2 and β2 . Similarly, for a K-level
network, the condition to make KLD = 0 becomes,
Pk
PK
H
H
B
B
i=1 αi )]
k=1 [βk (1 −
.(Pj,0
− Pj,1
) (10)
Pj,1 − Pj,0 = PK
Pk
[β
(
α
)]]
k
i
k=1
i=1
Following the mathematical analysis similar to Marano et.al
[8], we define
PK
Pk
H
H +
B
B +
k=1 [βk (1 −
i=1 αi )]
.(Pj,0
−Pj,1
)
κj = (Pj,1
−Pj,0
) = P
Pk
K
i=1 αi )]]
k=1 [βk (
(11)
Here (a)+ = a when a ≥ 0 and (a)+ = 0 otherwise. Three
possible
P cases may arise with the above equation:
1)
κj = 1 : In this case, κ is a probability mass function
B
B
and the unique solution (Pj,1
, Pj,0
) that can make KLD = 0
can be obtained by assuming
B
Pj,1
= κj
(12)
PK
Pk
[β
(1
−
α
)]
B
H
H
k=1 k
i=1 i
Pj,0
= κj + P
.(Pj,1
− Pj,0
) (13)
Pk
K
[β
(
α
)]]
k=1 k
i=1 i
P
2)
κj < 1 : In this case, there exist an infinite number
B
B
of attacking probability distributions (Pj,1
, Pj,0
) which can
make KLD = 0. These solutions can be obtained by setting
B
B
Pj,1
= κj found in (11),
P however, Pj,1 is not a probability
mass function because
κj < 1. So we arbitrarily increase
either of the Pj,1 , P
j ∈ {0, 1} such that it becomes a probability
mass function or j∈{0,1} Pj,1 = 1. For this Pj,1 we can find
corresponding Pj,0 as
Pk
PK
B
B
H
H
i=1 αi )]
k=1 [βk (1 −
Pj,0 = Pj,1 + PK
.(Pj,1
− Pj,0
) (14)
Pk
[β
(
α
)]]
k
i
k=1
i=1
P
3)
κj > 1 : In this case, there does not exist any attacking
B
B
probability distribution (P
j,1 , Pj,0 ) that can make KLD = 0.
P
As shown above for
κj ≤ 1 there exist an attacking
B
B
probability distribution (Pj,1
, Pj,0
) that can make KLD = 0.
We call the parameter values for αk and βk that result in
KLD = 0, as blinding parameters.
Following the similar procedure as above, it can be shown that
the blinding condition for a K-level network is
PK
Pk
X
H
H +
k=1 [βk (1 −
i=1 αi )]
(Pj,0
− Pj,1
) ≤ 1 (15)
.
PK
Pk
[β
(
α
)]]
k=1 k
i=1 i
j∈{0,1}
which reduces to
K
k
X
X
αi ))] ≤ 0
(16)
[βk (1 − 2(
k=1
i=1
Bk
We approximate αk and βk as αk = N
and βk = PKNk N ,
k
i
i=1
and rewrite the above results as the following theorem.
Theorem 1: In a tree network with K levels, {Bk }K
k=1
must satisfy
PK Bk PK
N
i=k Ni ≥ 2
k=1 Nk .
to make KLD between the distributions P (zi = j|H0 ) and
P (zi = j|H1 ) equal to zero, and thereby blind the FC.
The set {Bk }K
k=1 , that satisfies the above condition is not
unique. So, in the next problem, we consider that to attack
each node the Byzantine has to incur a particular cost, and
focus on finding the optimal set {Bk }K
k=1 that minimizes the
attacking cost of the Byzantine.
III. O PTIMAL B YZANTINE ATTACK
In this section, we investigate how the Byzantines can
launch an attack optimally considering that there is a cost for
attacking each node. We assume that the costs for attacking
nodes at different levels are different. Specifically, let ck be the
cost of attacking any one node at level k. Also, we assume
ck > ck+1 for k = 1, · · · , K − 1, i.e., it is more costly to
attack nodes that are closer to the FC. Observe that, a node i
at level k covers (in other words, can alter the decisions of)
all its successors and node i itself. Let Pk denote the number
of nodes covered by a node at level k. We refer
to Pk as the
PK
Ni
“profit” of a node at level k. Notice that, Pk = i=k+1
+ 1.
Nk
A. Minimum Cost Byzantine Attack Formulation
Observe that, Theorem 1 implies that to make the FC blind,
50% or more nodes in the network need to be covered by
Byzantines. The problem then is to determine {Bk }K
k=1 such
that at least 50% of the nodes in the network are covered and
the total cost incurred is minimized. Hence, one needs to solve
the following optimization problem
K
X
minimize
c k Bk
{Bk }K
k=1
subject to
k=1
K
X
k=1
Pk Bk ≥
K
X
Nk
k=1
2
0 ≤ Bk ≤ Nk , and integer ∀k
Notice that, the optimization problem presented above is
the covering
formulation of the BKP [11] with profit bound
PK
B = k=1 N2k . BKP, in general, is NP-hard to solve, as established by [12] and [13]. However, tree topologies satisfy some
interesting relationships that can be used to solve the BKP in
polynomial time. We extend and generalize the results of [11]
in solving our problem. Next, we discuss the relationships that
enable our minimum cost Byzantine attack problem to have a
polynomial time solution.
1) Profit and Cost Relationship: Notice that, in a tree
topology, Pk can be written as
Pk = ak ∗ Pk+1 + 1
f or k = 1, ..., K − 1
(17)
where Pk is the profit of attacking a node at level k, Pk+1
is the profit of attacking a node at level k + 1 and ak is the
number of immediate children of a node at level k. ak (for
k = 1 to K − 1) is
equaljto k
Pk

if Pk+1 > 1;
Pkk+1
j
ak =
P
k

Pk+1 − 1 if Pk+1 = 1.
Our polynomial time solvable case of the BKP uses the
following set of relationships between costs
ck :: ak ∗ ck+1 + cK
f or k = 1, ..., K − 1
(18)
where (::) denotes mathematical inequality (i.e., < or >).
B. Analysis of the Optimal Solution
First, we define the concept of dominance in our context
which will be later used to explore some useful properties of
the optimal solution.
Definition 1: We say that a set S1 dominates another set
S2 if
P (S1 ) ≥ P (S2 ) and C(S1 ) ≤ C(S2 )
(19)
where P (Si ) and C(Si ) denote the profit and cost incurred
by using set Si , respectively. If in (19), C(S1 ) < C(S2 ),
S1 strictly dominates S2 and if C(S1 ) = C(S2 ), S1 weakly
dominates S2 . Next, we show that if the profit and cost
satisfy the inequalities mentioned in (17) and (18), the optimal
solution {Bk∗ }K
k=1 exhibits the properties given in the lemma
below.
Lemma 1: Given a tree network architecture with K levels,
∗ K
we have anjoptimal
solution
k
l m {Bk }k=1 with
B1 =
B
or
lP1 m
B
P1
if ck < ak ∗ ck+1 + cK
if ck > ak ∗ ck+1 + cK
BK = PBK
and when ck = ak ∗ ck+1 + cK for any k in above inequalities,
the optimal solution is not unique.
Proof: Let us first consider the case ck < ak ∗ ck+1 +
cK . Let us assume, for the sake
that in the
j ofkcontradiction,
optimal solution B1 is less than PB1 = N21 . Then there must
exist another
j kset of nodes which dominates B1 . Let the set S1
contain PB1 nodes from level 1. Also, let set Si (i > 1)
contain any combination of nodes from jdifferent
levels but
k
B
with nodes from level 1 being less than P1 . For instance,
consider set S2 . Set S2 can be constructed by replacing nodes
from S1 . One possible choice is to replace one level-one node
from set S1 by a2 level two nodes and one node from the
last level, such that P (S1 ) = P (S2 ). Notice that, any set
Si with P (Si ) = P (S1 ) can be constructed using a similar
replacement method. Further, observe that any set with one
node from level k strictly dominates a set with ak nodes from
level k + 1 and one node from level K. So set S1 strictly
dominates every set Si , which contradicts
j our
k assumption that
B
in the optimal solution B1 is less than P1 .
The second case can also be proven similarly based on the
fact that, when ck > ak ∗ ck+1 + cK , a set with ak nodes from
level k + 1 and one node from level K strictly dominates a set
with one node from level k. Also, when ck = ak ∗ ck+1 + cK
for any k, the optimal solution is not unique because some
subsets in S1 will be weakly dominant and C(S1 ) = C(Si )
for some
j ki, where C(Si ) is a feasible solution with B1 less
than PB1 .
C. Algorithm for solving covering formulation of BKP
Based on Lemma 1, we now present a polynomial time
algorithm for solving BKP instances that satisfy our proposed
inequalities. Our algorithm is inspired by [11] [13].
First, we show how to find the optimal objective function
value or the minimum cost budget needed to make KLD = 0
in the case of Byzantine attacks. Then, using these results, we
find the set {BK }K
k=1 that minimizes the cost. When there is
only one levell in mthe network, problem is trivially solved by
setting B1 = PB1 . If there exist more than one level in the
network, then there are two cases to analyze.
If ckj< akk ∗ck+1
B1 can be
l +c
m K then in an optimallsolution,
m
either PB1 or PB1 . If we choose B1 = PB1 , then P1 B1 ≥
l m
B and we already have a feasible solution. So we choose PB1
nodes from level 1 and do not choose any node from the other
levels. Minimum
j case
k is equal
l m objective function value in this
to a = c1 PB1 . Now, if we choose B1 = PB1 then we
j k
pick PB1 nodes from level 1 and treat remaining (K − 1)
instances recursively. Minimum objective function value in this
case is denoted by b. The optimal objective function value for
an instance of BKP is equal to min(a, b), i.e., minimum of
objective function value in both of the above cases.
If ck >
l akm∗ ck+1 + cK then in an optimal solution, BK
will be PBK and we already have a feasible solution. So
l m
we choose PBK nodes from level K and do not choose any
node from other levels. lMinimum
objective function value in
m
this case is equal to cK PBK . Pseudo code of the polynomial
∗ K
algorithm to find {BK
}k=1 that minimizes the cost is given
in Algorithm 1.
Algorithm 1 Minimum cost Byzantine attack
Require: Pk = ak ∗ Pk+1 + pk
f or k = 1, ..., k − 1
1: if K = 1 l
then m
2:
B1 ← PB
1
3: else
4:
if (K > 1 and
l cmk > ak ∗ ck+1 + cK , ∀k) then
5:
BK ← PB
K
6:
Bk ← 0, ∀k \ K
7:
else
8:
if (K > 1 and ck < ak ∗ ck+1 + cK , ∀k) then
9:
for k = 1 lto Km do
10:
λk ← PB
j kk
11:
γk ← PB
k
12:
B ← (B − Pk γk )
13:
end for
14:
BK ← λk
15:
k ←K−1
16:
whileP
k 6= 0 do
17:
if K
i=k+1 Bi ci + γk ck > λk ck then
18:
for i = k + 1 to K do
19:
Bi ← 0
20:
Bk ← λk
21:
end for
22:
else P
23:
if K
i=k+1 Bi ci + γk ck < λk ck then
24:
Bk ← γk
25:
end if
26:
end if
27:
k ←k−1
28:
end while
29:
end if
30:
end if
31: end if
nodes covered by Bk nodes at level k + x and Bk+x is greater
than Nk+x . In a non-overlapping case, an attacker can always
arrange nodes {Bk }K
k=1 such that each path in the network
has at most one Byzantine.
Theorem 2: In a tree topology, the optimal solution
{Bk∗ }K
k=1 that makes the KLD between the distributions
P (zi = j|H0 ) and P (zi = j|H1 ) equal to zero will be nonoverlapping.
Proof: Suppose for the sake of contradiction we assume
that the optimal solution set S = {Bk∗ }K
k=1 of the cost minimization problem is overlapping. Then the condition below is
both necessary and sufficient for a set S to be overlapping.
PK NK ∗
k=1 Nk Bk > NK
Pk−1
Observe that Nk ≥ i=1
Ni + N1 , which is easy to show
in a balanced tree if Nk < Nk+1 or if ak ≥ 2, f or k =
1, ..., K − 1. So
'
&K
K
X
X Ni
Ni
N1
NK ≥
+
≥
2
2
2
i=1
i=1
However, we can always form a set S 0 = {Bk0 }K
k=1 which
P
NK 0
satisfies K
B
=
N
by
removing
nodes
from S =
K
k=1 Nk k
l PK
m
∗ K
0
0
i=1 Ni
{Bk }k=1 . The set S satisfies P (S ) ≥ NK ≥
,
2
0
which makes S a feasible and
solution. It
l PKnon-overlapping
m
i=1 Ni
is feasible because P (S 0 ) ≥
and
non-overlapping
2
because
PK NK ∗
k=1 Nk Bk ≤ NK
which is both a necessary and sufficient for a set S to be
non-overlapping.
It is intuitive that set S 0 will have less cost than S =
{Bk∗ }K
k=1 , which contradicts our assumption that S is an
optimal solution.
D. An Illustrative Example
Let us consider a three-level network with N1 = 3, N2 = 6,
N3 = 12 (Figure 1). Thus, the profit bound B = 10.5. Given
two cost structures (c1 = 19, c2 = 7, c3 = 2) and (c1 = 10,
c2 = 5, c3 = 2), we solve the covering formulation of BKP.
Observe that, the first cost structure satisfies ck > 2∗ck+1 +c3
for k ∈ [1, 2]. So the algorithm chooses B3 = 11 and B1 = 0,
B2 = 0 with minimum cost C = 22.
Fusion Center
Level 1
Level
2
Level
3
Next, in Theorem 2 we show that an optimum solution
will always be non-overlapping. Before discussing Theorem
2, we define overlapping concept in our context. We call Bk
and Bk+x are overlapping, if the summation of number of
Fig. 1.
Tree Topology
For the second set of costs, the cost structure satisfies
ck < 2 ∗ ck+1 + c3 for k ∈ [1, 2]. In the for loop in lines
IV. N UMERICAL R ESULTS
In this section, we present numerical results for analyzing
Byzantine attack in a two-level tree network. We consider a
network with N = 120000 (N1 = 30000, N2 = 90000)
arranged in a balanced tree topology with Pd = 0.8 and Pf a =
0.1.
4
3.5
K L distance
3
4
18
x 10
16
14
MAXIMUM
12
Attacking cost
9 to 13, Algorithm 1 first finds λ1 = 2, γ1 = 1, B = 3.5
(iteration 1), λ2 = 2, γ2 = 1, B = 0.5 (iteration 2),
and λ3 = 1 (iteration 3). Then, in the while loop in lines
16-28, the algorithm compares set S1 with two nodes from
level 2 and set S2 with one node from level 3 and one node
from level 2. In the first iteration of the while loop, since
(C(S1 ) = 10) > (C(S2 ) = 7), the algorithm chooses set S2 .
In the second iteration, set S1 with one node from each level
is compared with set S2 with two nodes from level 1. Because
(C(S1 ) = 17) < (C(S2 ) = 20), the algorithm chooses set S1
(one node each from level 1, 2 and 3), which is our optimal
solution with cost C = 17.
10
8
6
MINIMUM
4
2
0
0
Fig. 3.
0.2
0.4
0.6
Fraction of nodes covered
0.8
1
Attacking cost vs Fraction of nodes covered
V. C ONCLUSION AND F UTURE WORK
In this paper, we considered the problem of optimal Byzantine attacks on distributed detection mechanism in tree networks. We analyzed the performance limit of global detection
performance of the network under Byzantine attacks. We
also obtained the optimal attacking strategies that minimize
the detection error exponent while the Byzantine attacker
incurs minimum cost. We formulated minimum cost Byzantine
attacks as the Bounded Knapsack Problem and presented a
polynomial time algorithm to solve it. In our future work, we
plan to extend this analysis to more general cost relationships
and non-linear cost functions.
2.5
R EFERENCES
2
1.5
1
0.5
0
0
Fig. 2.
0.1
0.2
0.3
Fraction of nodes covered
0.4
0.5
KL distance vs Fraction of nodes covered
Figure 2 shows the fraction of nodes to be covered to make
KLD equal to zero. When 50% of the nodes in the network
are covered, KLD between the two probability distributions
becomes zero and FC becomes blind, which corroborates our
theoretical result presented in Theorem 1.
Next, in Figure 3, we study how attacking cost varies with
the fraction of nodes covered. Here, we consider c1 = 6 and
c2 = 1. As can be seen from the figure, a particular fraction
of covered nodes can correspond to multiple attacking costs.
This is because the same number of nodes can be covered by
choosing different α1 and α2 .
Notice that, as can be seen from Figure 2 (which we also
established in Theorem 1), to make KLD = 0, the Byzantines
need to cover 50% nodes. Further, from Figure 3, it can be
clearly seen that the minimum cost incurred to cover 50% of
the nodes is 6∗104. Thus, we can conclude from Figures 2 and
3 that the minimum cost required to blind the FC (KLD = 0)
is 6 ∗ 104. It can be noted that our algorithm also suggests the
optimal cost C = 6 ∗ 104 . This is because,
l m since c1 > 4c2 ,
our algorithm suggests choosing B2 = PB2 = 6 ∗ 104 nodes
from level 2 and zero nodes from level 1, thereby incurring the
optimal cost C = 6 ∗ 104 . This clearly verifies the optimality
of our algorithm.
[1] P. K. Varshney, Distributed Detection and Data Fusion.
New
York:Springer-Verlag, 1997.
[2] R. Viswanathan and P. Varshney, “Distributed detection with multiple
sensors i. fundamentals,” Proceedings of the IEEE, vol. 85, no. 1, pp.
54 –63, jan 1997.
[3] V. Veeravalli and P. K. Varshney, “Distributed inference in wireless
sensor networks,” Philosophical Transactions of the Royal Society A:
Mathematical, Physical and Engineering Sciences, vol. 370, pp. 100–
117, 2012.
[4] Y. Lin, B. Chen, and P. Varshney, “Decision fusion rules in multihop wireless sensor networks,” Aerospace and Electronic Systems, IEEE
Transactions on, vol. 41, no. 2, pp. 475 – 488, april 2005.
[5] P. Kulkarni, D. Ganesan, P. Shenoy, and Q. Lu, “Senseye: a multitier camera sensor network,” in Proc. 13th annual ACM international
conference on Multimedia, 2005.
[6] Aliance, “Z. zigbee specifications,” Zigbee Standard Organisation, 2008.
[7] IEEE. P802.22b Draft Standard for Wireless Regional Area Networks (WRAN)–Specific requirements Part 22: Cognitive Wireless RAN
Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Policies and Procedures for Operation in the TV Bands
Amendment: Enhancement for Broadband Services and Monitoring
Applications.
[8] S. Marano, V. Matta, and L. Tong, “Distributed detection in the presence
of byzantine attacks,” Signal Processing, IEEE Transactions on, vol. 57,
no. 1, pp. 16 –29, jan. 2009.
[9] A. Rawat, P. Anand, H. Chen, and P. Varshney, “Collaborative spectrum
sensing in the presence of byzantine attacks in cognitive radio networks,”
Signal Processing, IEEE Transactions on, vol. 59, no. 2, pp. 774 –786,
feb. 2011.
[10] S. Kullback, Information Theory and Statistics, 1968.
[11] V. G. Deineko and G. J. Woeginger, “A well-solvable special case of
the bounded knapsack problem,” Operations Research Letters, vol. 39,
pp. 118–120, 2011.
[12] R. Karp, “Reducibility among combinatorial problems,” in: R.E. Miller,
J.W. Thatcher (Eds.), Complexity of Computer Computations, Plenum
Press,New York, pp. 85–104, 1972.
[13] M. Zukerman, T. N. L. Jia, and G. Woeginger, “A polynomially solvable
special case of the unbounded knapsack problem,” Operations Research
Letters, vol. 29, pp. 13–16, 2001.