Eventually Byzantine Agreement on CDS-based

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Ad Hoc Networks 10 (2012) 388–400
Contents lists available at ScienceDirect
Ad Hoc Networks
journal homepage: www.elsevier.com/locate/adhoc
Eventually Byzantine Agreement on CDS-based mobile ad hoc network
Mao-Lun Chiang ⇑
Department of Information and Communication Engineering, Chaoyang University of Technology, 168 Gifeng E. Rd., Wufeng, Taichung County 413, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 10 October 2010
Received in revised form 20 December 2010
Accepted 11 July 2011
Available online 30 July 2011
Keywords:
Eventual Byzantine Agreement
Fault-tolerant
Consensus
Mobile ad hoc network
a b s t r a c t
Reliability is an important research topic in the study of distributed systems. Under many
circumstances, a healthy processor in a distributed system needs to reach a common
agreement before performing some special tasks even if the faults exist. In order to achieve
fault-tolerance in distributed systems, one must deal with the Byzantine Agreement (BA)
problem. Most BA problem require all the healthy processors to obtain an agreement at
the same round, this kind of agreement is called an Immediate Byzantine Agreement
(IBA). Another kind of agreement, Eventual Byzantine Agreement (EBA), allows its participants to reach a common agreement at different rounds when the fact < fp (fact is the number of actual arbitrary faulty processors; fp is the number of tolerate arbitrary faulty
processors). However, the traditional EBA problem is solved in well-defined networks,
but the Mobile Ad hoc NETworks (MANETs) are increasing in popularity. Therefore, EBA
problem is revisited under dual failure mode (processors and transmission media) in the
MANET. The proposed protocol, Early Dual Agreement Protocol (EDAP), can achieve agreement while tolerating the maximum number of faulty processors and transmission media
in a MANET by using the minimum number of message exchanges. Furthermore, our protocol can manage and organize the network efficiently even if the processors move around
the network.
Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction
A distributed computing system consists of a set of
processors, which can communicate with each other by
exchanging messages. In order to provide a reliable computer system, a mechanism to allow a given set of processors to agree on a common value is needed [10,11,14,
15,17]. Some examples of applications emphasizing this
fact are: a commitment problem in a distributed database
system [12], a clock synchronization problem [7,17], and
a landing task controlled by a flight control system [4]. Such
a unanimity problem was first studied by Lamport et al.
[10], and called Byzantine Agreement (BA) [1,3,9,10,14,
15,17]. It requires a number of independent processors to
reach agreement in cases where some of those processors
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E-mail address: [email protected]
1570-8705/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.adhoc.2011.07.005
might be faulty. Furthermore, the goal of BA is making the
healthy processors to reach a common value. A closely related sub-problem, the consensus problem, has been extensively studied [7,17] as well. The consensus problem has k
initial values in a k-processor system and subsequently
achieves a common value even if certain processors fail
[7,11,17]. Therefore, the consensus problem is similar to
the BA problem such as executing k copies BA processes.
Subsequently, the results of previous works [7,14,17]
showed that agreement is impossible in an asynchronous
environment if even one processor has failed and that failure is a crash failure. Therefore, the BA problem is most
applicable to a synchronous network and the bounds on
the processing and transmission delays of healthy components are assumed to be finite [3,9]. Lamport argues for
the consensus problem under the assumption of synchronous behavior BA, showing that 3fp + 1 processors can tolerate fp failures where fp is the maximum number of faulty
processors in a network [10]. To clarify this study, the
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
assumptions of the BA are used to explain the concept of
consensus problem.
Traditionally, BA problem as defined by Lamport et al.
[10], is as follows: (1) There are k (k > 3) processors, of
which at most one-third of the total number of processors
can fail without completely disrupting a workable network. (2) The processors communicate with each other
by message exchange in a fully connected network (or
well-defined network). (3) The message’s sender is always
identifiable by the receiver. (4) A processor is chosen as a
source randomly and broadcasting its initial value to other
processors as well as itself. Agreement is reached if all
healthy processors agree on a common value. Based on
these assumptions, various protocols for the BA problem
have been developed in order to meet the following
requirements [1,3,4,8–10,14,15,17]:
(BA1) Agreement: All healthy processors agree on a
common value v.
(BA2) Validity: If the initial value of the source is vs and
the source is healthy, then all healthy processors shall
agree on the value vs; i.e., v = vs.
In general, some protocols require that each healthy
processor stop in the same round (fp + 1) and achieve
agreement immediately when the system can tolerate
the number of arbitrary processor (fp) faults. This kind of
agreement is called Immediate Byzantine Agreement
(IBA) [3,9]. However, if there is no faulty processor in the
system, or the number of faulty processors is less than fp,
fp + 1 rounds of message exchange are still needed to reach
a common agreement. It is unreasonable and inefficient in
practical work. Therefore, an improvement algorithm is
need to be invoked, the Eventual Byzantine Agreement
(EBA) [9], that allows individual processors to stop during
rounds in which the received messages are sufficient to
reach a common agreement.
However, the network technology grows rapidly, traditional network topology is improved with wireless topology. One such area of improvement is the wireless
topology known as a Mobile Ad hoc NETwork (MANET)
showing in Fig. 1 [2,16]. MANET consists of wireless processors that communicate with each other in the absence
of a fixed infrastructure. Unfortunately, previous litera-
Fig. 1. The topology of MANET in unit-disk graphs.
389
tures [1,10,12,17] mostly focus on fixed networks that
use an infrastructure, such as fully connected networks,
broadcast networks, or well-defined networks. Besides,
the fallible processors and transmission media are not to
be discussed simultaneously in a MANET. The traditional
protocol cannot be used to achieve agreement in MANET
due to its dynamic nature. Therefore, this paper revisits
the EBA problem with respect to dual failure mode on both
processors and transmission media and tolerates the
maximum number of faulty components by using the minimum number of message exchanges in a MANET
environment.
The rest of this paper is organized as follows: Section 2
illustrates the related works of environment and assumptions. The details of the proposed protocol are shown in
Section 3. Section 4 illustrates the examples we have devised in detail. Subsequently, the correctness and complexity of our method is illustrated in Section 5. Finally, the
conclusion is presented in Section 6.
2. Related works
Before the describing the previous protocol [3,9], two
basic assumptions must be defined: the network environment and the failure types of faulty components.
2.1. Network environment
As the network technology continues to grow at a high
rate of speed, Mobile Ad hoc NETwork (MANET) [2,16] has
enjoyed an amazing rise in popularity. MANET, consisting
of wireless processors that communicate with each other
in the absence of a fixed infrastructure, is different from
the traditional network structures. Its topology, as shown
in Fig. 1, can be modeled as a unit-disk graph [2] according
to the range of transmission power of the members of the
network. Besides, the MANET can easy to be used flexibly
and quickly in automated battlefields, disaster relief, and
rescue. There exist several challenges to the MANET due
to its dynamic nature, such as low battery power, limited
bandwidth, and unrestraint mobility. However, traditional
routing protocols included hierarchical routing, link state,
and distance vector [13,16] need to be reinvestigated.
The previous routing path is destroyed when the processors immigrate into or emigrate out of the network at
any time. Therefore, some research [2,13] has proposed a
concept of a virtual backbone for organizing the MANET
and classifying processors as gateway or non-gateway
processors.
In this hierarchical topology, one gateway processor can
handle non-gateway processors in its own group and the
routing process is simplified with respect to gateway processors. Namely, only gateway processors need to maintain
the routing table and the search space is reduced to itself.
Non-gateway processors can also change their status to
sleep mode to conserve their battery power. Therefore,
the virtual backbone is being adapted for organize a
MANET.
The Connected Dominating Set (CDS) is a popular algorithm proposed by Wu [16], to build the virtual backbone
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M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
Fig. 2. The proposed protocol EDAP.
of a MANET. The CDS-based virtual backbone is related to
the concept of dominating in graph theory. A subset of vertices on a graph is a dominating set if every vertex not in
the subset is adjacent to at least one vertex in the subset.
Therefore, the processors of the dominating set should be
connected for building the routing path of transmission.
Subsequently, the algorithm removes all locally redundant
gateway processors during the re-marking phase to obtain a
minimal CDS.
Bharghavan and Das [2], proposed another famous
distributed algorithm to build a virtual backbone. Their
algorithm first finds an approximation to the minimum
dominating set by a greedy algorithm; and then, in the second stage, constructs a spanning forest F. Subsequently,
the third stage expands the spanning forest F into a
spanning tree T and forms a minimum CDS.
Besides, Stojmenovic et al. [13] showed a synchronized
distributed constructions of CDS. Their methods classify
the processors in a CDS into clusterhead processors and
border processors. Processors are divided into clusters
with one of them serving as the clusterhead processor in
each cluster. That processor can also connect to any of
the processors in its cluster directly; but the clusterhead
processors are not adjacent to each other.
Based on the reason above, traditional famous papers
[8,10,14,15,17] cannot solve the EBA problem in a MANET
due to its unrestraint mobility. The EBA problem is revised
in this paper under a MANET environment using a
CDS-based virtual backbone. It is because that we can
facilitate saving the routing information and reducing the
spaces required to search for a route to the gateway
processor in a CDS-based virtual backbone. The gateway
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
processor can govern and maintain its group when processors immigrate into or move out of the network at any
time. Therefore, the CDS-based virtual backbone is suitable
to the real world and is used to configure the MANET in
this paper.
2.2. Failure types
In general, we can divide failure into three types: processor failure, transmission medium failure, and generalized failure [7,8,10,15,17]. The symptoms of processor
failure can be classified into two categories, the dormant
fault and the arbitrary fault. The dormant faults of processors include broken processors (crash faults) and message
misses (omission faults). However, the dormant fault can
easy to be detected and solved. By contrast, the arbitrary
fault is a more serious problem; its behavior is unpredictable and damaging.
The symptoms of transmission medium failure can also
be classified into two categories, the dormant fault and
arbitrary fault. The dormant faults of transmission media
are crashes and stuck-at. The crash fault represents the
transmission medium being broken. However, stuck-at
faults occur when the message received from a certain
transmission medium is always a constant value. Likewise,
with processor faults, the arbitrary fault of transmission
media is also the most damaging failure type of all problems due to its behavior being unrestricted. In this paper,
we solve the EBA problem in generalized failure mode
while accounting for dual failures (including the dormant
fault and arbitrary fault simultaneously) in a MANET.
According to the previous literature [10,14,15,17], the
total number of allowable faulty components (processors/
transmission media) in a k-processor fully connected network is fc (fc: faulty component 6 bk/2c 1), in which the
number of allowable faulty processors is fp (fp 6 b(k 1)/
3c). The remainder is the number of allowable faulty transmission media. Based on those results [14,15,17], fp + 2
rounds of message exchange are required to make all
healthy processors reach agreement. This is because that
the influence of faulty processors and faulty transmission
media produced before fp + 1 rounds of message exchange
can be removed by fp + 2 rounds of message exchange.
Therefore, healthy processors can reach an agreement by
requiring fp + 2 rounds of message exchange if the faulty
components do not exceed bk/2c 1, in which case the
number of allowable faulty processors is b(k 1)/3c.
2.3. Eventual Byzantine Agreement
In general, IBA [3,9] protocols require all healthy processors to reach a common agreement during the same
round. The previous works states that IBA cannot be
achieved for k (k > fp + 1) processors with at most fp faults
within fp or fewer rounds. However, the EBA [3,9] protocols
allow the healthy processors to achieve agreement during
different rounds when they receive enough information. In
EBA protocols, the processors may stop the protocol in
round r < fp (fp 6 b(k 1)/3c) due to the number of actual
faulty components (fact) being less than the maximum
number of fault tolerant fp; thus a lower bound of number
391
of rounds of message exchange is min{fact + 2, fp + 1} for
EBA. However, this paper revisited the EBA problem with
a dual failure type (arbitrary fault and dormant fault) in
processors and transmission media in a MANET environment. The lower bound of our proposed protocol is changed to min{fact + 3, fp + 2} and subsequently the basic
concepts and approaches of the proposed protocol are
shown in Section 3.
2.4. The failure components elimination sequence
In generalized cases, both processors and transmission
media may fail simultaneously. For healthy processors to
reach a common agreement, the influence from dormant
faulty transmission media and processors must be removed first. Subsequently, the influence from arbitrarily
faulty transmission media and processors can be eliminated based on the work of Yan et al. [17]. The details
are illustrated as follows.
Our protocol can remove dormant faulty processors and
dormant faulty transmission media because the receiver
can always identify faulty messages produced by dormant
components such as crash, omissions, and stuck-at faults, if
the Manchester code [6,15,17] is used in encoding before
transmission. This is because Manchester encoding is a
synchronous clock encoding technique; thus, the receiver
can easily distinguish between the dormant faulty components. The values sent from dormant faulty components
are replaced by k in our protocol.
After removing the influence of dormant faulty components, the protocol must collect fp rounds of message exchange; where fp denotes the maximum number of
allowable faulty processors that cannot exceed b(N 1)/
3c. Subsequently, the collected messages are stored into a
data structure, called a message storage tree (ms-tree) is
shown in Fig. 3c. When each processor collects enough
messages from other processors, the effects of arbitrary
faulty transmission media can be eliminated.
In last step, the influence of arbitrarily faulty processor
must be removed. To achieve this goal, two tree structures
will be constructed, the message storage tree (ms-tree) and
the information collection tree (ic-tree: Ti) are shown in
Fig. 3c and Fig. 3d [14,15,17]. The vertex of the ms-tree is
labeled with a list of processor names, and the value received from the source processor is denoted as val(s) at
the root of the ms-tree. The processor name list contains
the names of the processors through which the stored
message was transferred. For example, the statement
val(sac) represents the processor having received the value
sa from processor c which was sent from source processor s
to processor a. Subsequently, an ic-tree is constructed by
following reorganization rules:
(1) The leaves at level fp + 2 of the ms-tree are deleted.
(2) The vertices with repeated processor’s names are
deleted.
According to reorganization rules, the ic-tree can be
constructed to avoid cyclical influences from the faulty
processors. It is because that the messages of faulty processors may be stored repeatedly in the vertices, resulting in
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M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
an incorrect common value caused by taking a simple
majority. Eventually, each processor can apply the VOTE
function (a detailed description is presented in Fig. 3e) to
its ic-tree to compute a common value. Subsequently, the
detail of our proposed protocol is shown as follows.
3. Protocol EDAP
Before to introduce the proposed protocol is shown in
Fig. 2, Early Dual Agreement Protocol (EDAP), the parameters in this synchronous MANET are assumed as follows:
(1) N: The total number of gateway processors in a
MANET.
(2) n: The total number of non-gateway processors in a
MANET.
(3) ni: The total number of processors in group i.
(4) di: The decision value of gateway processor i.
(5) Vk: The vector in processor k.
(6) vs: The initial value of processor i broadcasting to all
other processors.
(7) MAJi: The majority value of processor i.
(8) MATi: The gateway processor i collects all received
vectors from other processors.
(9) k: The value that instead of the received value from a
dormant faulty processor.
(10) c: The connectivity of MANET. Based on the Menger
theorem [5], at least c disjoint paths must exist
between any pairs of processor A and B when the
connectivity of the network is c.
(11) Npa: The number of gateway processors with an
arbitrary fault.
(12) Npd: The number of gateway processors with a dormant fault.
(13) npa: The number of non-gateway processors with an
arbitrary fault.
(14) npd: The number of non-gateway processors with a
dormant fault.
(15) Ta: The number of arbitrary faulty transmission
media among the gateway processors.
(16) Td: The number of dormant faulty transmission
media among the gateway processors.
(17) U: The default value, and U 2 {0, 1}.
(18) fp: The number of allowable faulty gateway
processors.
(19) fact: The actual number of faulty gateway processors.
(20) r: The required rounds of message exchange.
(21) r: The actual number of rounds of message
exchange.
(22) Dvik: The number of values which are equal to
MAJvi; 1 6 k 6 fp + 2.
To achieve the efficient management, the CDS-based
virtual backbone is used to organize a MANET in this paper.
Besides, EDAP requires fp + 2 rounds of message exchange
to reach an agreement and the connectivity of MANET
needs to satisfy the Menger theorem [5]. Based on the
[7,14,15,17], all healthy processors can reach agreement
in a MANET environment where N > (b(N 1)/3c) +
2(Npa + Ta) + Npd + Td and c > 2(Npa + Ta) + Npd + Td. The definitions of parameters are listed as Section 2.2.
EDAP has two parts: group agreement process and consensus agreement process. The main work of the group
agreement process is collecting initial values from nongateway processors. Each gateway processor takes the
majority (MAJ) of collected values as its initial value as
shown in Fig. 3a. Subsequently, each gateway processor
forwards its value to other gateway processors in the consensus agreement process. The consensus agreement process
consists of the message exchange phase and the decision
making phase. The message exchange phase must collect enough messages from gateway processors for all healthy
gateway processors to reach an agreement. When a
healthy gateway processor receives the values from other
gateway processors, these values are stored in its vertex
Vi (1 < i < N). Then, each gateway processor broadcasts its
vertex again and constructs MATi (1 < i < N) when r > 1.
The details of MATi are shown in Fig. 3b.
During the message exchange phase, a tree structure, the
ms-tree [14,15,17] is constructed by taking the local majority on each row k of MATi for each round and the value
The function MAJ(α)
1. The majority value in the set of {val(αj)|1≤j≤(N, n)}
MAJ(α)= 2. The complement value of val(α), denoted as ¬val(α), is chosen otherwise.
Fig. 3a. The function MAJ.
Fig. 3b. The function MATi on processor i.
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M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
received from the source processor, denoted as val(s) at the
root of ms-tree. This is convenient data structure for collecting a majority value from the MATi. The details of the
ms-tree are shown as Fig. 3c. The vertex of an ms-tree is labeled with a list of processor names. The processor name
list contains the names of the processors through which
the stored message has been transferred.
Subsequently, an ic-tree [14,15,17] is constructed by
ic-tree conversion function which is improvement of reorganization rules to remove the vertices with repeated processor names of ms-tree. The cyclical influence from the faulty
processors can be avoided in ic-tree. The detail of ic-tree
shown in Fig. 3d.
In last step of the message exchange phase, each gateway
processor executes the early stopping function to check
whether the protocol can be stopped. This function uses
the concept of [9] to modify to reduce the number of
message exchange under the generalized failure mode in
MANET. During the early stopping function, we first apply
the MAJ function to the ic-tree of each processor i
(1 6 i 6 N) to obtain the MAJvi. Subsequently, the numbers
of Dv ki (1 6 k 6 fp + 2) are computed when the values of the
ic-tree of each processor are equal to MAJvi. Finally, the
protocol can be stopped early if the following constraint
can be satisfied:
Dv ki > ðfp ðr 1ÞÞ þ
Level 1
root
s
val(s)
ðn ðr 1Þ kÞ
2
Level 2
(leaf)
sa
val(sa)
sb
val (sb)
sc
val (sc)
sd
val (sd)
se
val (se)
Fig. 3c. The ms-tree.
Level 3
(sub_leaf)
saa val(saa)
sab val(sab)
sac val(sac)
sad val(sad)
sae val(sae)
sba val (sba)
sbb val (sbb)
sbc val (sbc)
sbd val (sbd)
sbe val (sbe)
sca val (sba)
scb val (scb)
scc val (scc)
scd val (scd)
sce val (sce)
sda val (sda)
sdb val (sdb)
sdc val (sdc)
sdd val (sdd)
sde val (sde)
sea val (sea)
seb val (seb)
sec val (sec)
sed val (sdd)
see val (see)
Finally, the second phase of the consensus agreement
process, the decision making phase, is invoked to make each
healthy gateway processor compute a common value by
applying the voting function VOTE to the root of an ic-tree
is shown as Fig. 3e. Therefore, the proposed protocol EDAP,
can reach agreement when the numbers of messages are
sufficient under dual failure mode (processors and transmission media). Besides, the CDS-based virtual backbone
is used to manage and organize the MANET efficiently even
if the processors move around the network. Therefore,
EDAP protocol is more efficient and reasonable in the
MANET environment than previous protocols [8,10,14]
where
N > (b(N 1)/3c) + 2(Npa + Ta) + Npd + Td
and
c > 2(Npa + Ta) + Npd + Td. Furthermore, the round of message exchange in our protocol is more suitable for a MANET
environment than those of previous results [8,10,14].
4. Example of executing EDAP
In this section, an example is shown to illustrate how
the protocol EDAP is to be executed in practice. There are
24 processors in the original MANET in Fig. 4a–4k. In this
paper, the CDS construction algorithm [13] is used to elect
8 gateway processors as a gateway layer and 16 nongateway processors as a non-gateway layer in a MANET.
The relationship between the gateway processors and
non-gateway processors is shown as Table 1.
Our EDAP protocol requires r = b(8 1)/3c + 2 = 4
rounds to exchange messages and the complete steps are
Level 1
root
s
val(s)
Level 2
(leaf)
sa
val(sa)
sb
val (sb)
sc
val (sc)
sd
val (sd)
se
val (se)
Fig. 3d. The ic-tree (Ti).
Level 3
(sub_leaf)
saa val(saa)
sab val(sab)
sac val(sac)
sad val(sad)
sae val(sae)
sba val (sba)
sbb val (sbb)
sbc val (sbc)
sbd val (sbd)
sbe val (sbe)
sca val (sba)
scb val (scb)
scc val (scc)
scd val (scd)
sce val (sce)
sda val (sda)
sdb val (sdb)
sdc val (sdc)
sdd val (sdd)
sde val (sde)
sea val (sea)
seb val (seb)
sec val (sec)
sed val (sed)
see val (see)
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M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
VOTE(α)=
1. val(α), if α is a leaf.
2. The majority value in the set of {VOTE( αi)|1≤i≤N, and vertex αi
is a child of vertex α}, if such majority values exists.
3. A default value φ is chosen, otherwise.
Fig. 3e. The functions of protocol EDAP. The VOTE function.
V a Vb V c V d V e
1 1 1 0 0
λ λ λ λ λ
0
0 0 0
Vf Vg V h
0 0 0
λ λ λ
0 0 0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
Fig. 4d. The vector received from each gateway processor in the first
round of the consensus agreement process.
s val(s)
Fig. 4a. The original MANET environment.
Level 1
sa val(sa) 0
sb val (sb) λ
sc val (sc) 0
sd val (sd)
se val (se)
sf val (sf)
sg val (sg)
sh val (sh) 0
s val(s)
ic-treed,e, f, g, h
Level 1
sa val(sa) 1
sb val (sb) λ
sc val (sc) 0
sd val (sd)
se val (se)
sf val (sf)
sg val (sg)
sh val (sh) 0
ic-treec
Fig. 4e. The ic-trees of the first round.
Δvc1 =4 < ( f g − (r − 1)) + (n − (r − 1) − λ ) = 5.5 (cannot stop)
2
Δvd1 =5 < ( f g − (r − 1)) + (n − (r − 1) − λ ) = 5.5 (cannot stop)
2
Δve1 =5 < ( f g − (r − 1)) + (n − (r − 1) − λ ) = 5.5 (cannot stop)
2
(
n
−
(
r
−
1) − λ )
Δvf =5 < ( f g − (r − 1)) +
= 5.5 (cannot stop)
2
Δvg1 =5 < ( f g − (r − 1)) + (n − (r − 1) − λ ) = 5.5 (cannot stop)
2
(
n
−
(
r
−
1) − λ )
1
Δvh =5 < ( f g − (r − 1)) +
= 5.5 (cannot stop)
2
1
Fig. 4b. The first round in the group agreement process.
Va Vb Vc Vd Ve Vf Vg
1
0 1 0 0 1
Vh
0
Fig. 4f. Execution of the early stopping function in the first round of
message exchange phase.
Fig. 4c. The results of the group agreement process.
shown as Fig. 4a shows that the processors a, l, k, t, v, and x
are arbitrarily faulty processors, and processors b and n are
dormant faulty processors, respectively. In beginning of
the protocol, we assume x as the arbitrary source processor; it broadcasts initial values to groups Ga, Gb, Gc, Gd, Ge,
Gf, Gg and Gh as 1, 0, 0, 1, 0, 0, 1, and 0, respectively. Subsequently, each gateway processor uses the MAJ function to
obtain a majority value and stores it to its vector. The vector of each gateway processor is shown as Fig. 4c. After the
group agreement process, each gateway processor has an
initial value to broadcast to other gateway processors during the consensus agreement process.
During the consensus agreement process, each gateway
processor must exchange its value with other gateway processors during message exchange phase. The result of
healthy gateway processors executing the first round of
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M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
Level 1
MATd,g
Va
1
λ
1
0
1
1
0
1
Vb
λ
λ
λ
λ
λ
λ
λ
λ
Vc
1
λ
0
Vd
0
λ
0
Ve
0
λ
0
Vf
0
λ
0
Vg
0
λ
0
Vh
0
λ
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
sa
val(sa) 1
s val(s)
sb
val (sb) λ
Fig. 4g. The result of processors d and g executing the second round of
the message exchange phase in the consensus agreement process.
sc
val (sc) 0
MATc,e,f,h
Va
0
λ
1
1
0
1
1
1
Vb
λ
λ
λ
λ
λ
λ
λ
λ
Vc
1
λ
0
Vd
0
λ
0
Ve
0
λ
0
Vf
0
λ
0
Vg
0
λ
0
Vh
0
λ
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
Fig. 4h. The result of processors c, e, f, and h executing the second round
of the message exchange phase in the consensus agreement process.
the consensus agreement process is shown as Fig. 4d. Subsequently, the ic-tree of each gateway processor can be constructed from each vertex as shown in Fig. 4e. Therefore,
the early stopping function can be invoked during the first
round of message exchange, and this is shown as Fig. 4f.
In our example, the agreement value of gateway processors
a and b is unimportant due to the fact that they are an arbitrary gateway processor and a dormant gateway processor,
respectively. According to the early stopping function of the
EDAP protocol, each gateway processor cannot stop early
and the number of messages is insufficient to achieve
agreement in the first round. Therefore, each gateway processor must execute the next round of message exchange.
Each gateway processor continuously broadcasts its value to other gateway processors during the second round of
message exchange. The gateway processors d and g build
the corresponding MATd and MATg in Fig. 4g. The MATc,
MATe, MATf, and MATh are shown as Fig. 4h. The difference
between Fig. 4g and Fig. 4h is caused by the faulty gateway
processor a sending different messages to them. Subsequently, we show the ic-tree of each gateway processor
during the second round of message exchange in Figs. 4i
and 4j. The ic-tree is constructed from the MATi and is
eliminated the repeated vertices by means of the ic-tree
conversion function. In next step, as in the procedure above,
the early stopping function is invoked again to test whether
the procedure can stop or not. The results are shown as
Fig. 4k, the Fig. 4k represents the message exchange of
each gateway processor being able to stop. The collected
sd
val (sd)
se
val (se)
sf
val (sf)
sg
val (sg)
sh
val (sh) 0
Level 2
sab val(sab) λ
sac val(sac) 1
sad val(sad)
sae val(sae) 0
saf val(saf) 0
sag val(sag) 0
sah val(sah) 0
sba val(sba) λ
sbc val(sbc) λ
sbd val(sbd) λ
sbe val(sbe) λ
sbf val(sbf) λ
sbg val(sbg) λ
sbh val(sbh) λ
sca val(sca) 1
scb val(scb) λ
scd val(scd) 0
sce val(sce) 0
scf val(scf) 0
scg val(scg) 0
sch val(sch) 0
sda val(sda) 0
sdb val(sdb) λ
sdc val(sdc) 1
sde val(sde 1
sdf val(sdf) 1
sdg val(sdg) 1
sdh val(sdh) 1
sea val(sea) 1
seb val(seb) λ
sec val(sec) 0
sed val(sed) 0
sef val(sef) 0
seg val(seg) 0
seh val(seh) 0
sfa val(sfa) 1
sfb val(sfb) λ
sfc val(sfc) 0
sfd val(sfd)
sfe val(sfe) 0
sfg val(sfg) 0
sfh val(sfh) 0
sga val(sga) 0
sgb val(sgb) λ
sgc val(sgc) 1
sgd val(sgd)
sge val(sge) 1
sgf val(sgf) 1
sgh val(sgh) 1
sha val(sha) 1
shb val(shb) λ
shc val(shc) 0
shd val(shd)
she val(she) 0
shf val(shf) 0
shg val(shg)0
Fig. 4i. The ic-treed,g.
messages of each gateway processor are enough to achieve
consensus and move into next phase, the decision making
396
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
Level 1
sa
val(sa) 1
s val(s)
sb
val (sb) 0
sc
val (sc) 0
sd
val (sd) λ
se
val (se) λ
sf
val (sf ) λ
sg
val (sg) λ
sh
val (sh) 0
Level 2
sab val(sab) λ
sac val(sac) 1
sad val(sad)
sae val(sae) 0
saf val(saf) 0
sag val(sag) 0
sah val(sah) 0
sba val(sba) λ
sbc val(sbc) λ
sbd val(sbd) λ
sbe val(sbe) λ
sbf val(sbf) λ
sbg val(sbg) λ
sbh val(sbh) λ
sca val(sca) 1
scb val(scb) λ
scd val(scd) 0
sce val(sce) 0
scf val(scf) 0
scg val(scg) 0
sch val(sch) 0
sda val(sda) 1
sdb val(sdb) λ
sdc val(sdc) 0
sde val(sde 1
sdf val(sdf) 1
sdg val(sdg) 1
sdh val(sdh) 1
sea val(sea) 0
seb val(seb) λ
sec val(sec) 0
sed val(sed) 0
sef val(sef) 0
seg val(seg) 0
seh val(seh) 0
sfa val(sfa) 1
sfb val(sfb) λ
sfc val(sfc) 0
sfd val(sfd)
sfe val(sfe) 0
sfg val(sfg) 0
sfh val(sfh) 0
sga val(sga) 1
sgb val(sgb) λ
sgc val(sgc) 1
sgd val(sgd)
sge val(sge) 1
sgf val(sgf) 1
sgh val(sgh) 1
sha val(sha) 1
shb val(shb) λ
shc val(shc) 0
shd val(shd)
she val(she) 0
shf val(shf) 0
shg val(shg)0
Fig. 4j. The ic-treec,e,f,h.
process. This process uses the function VOTE to obtain a
common value, and the common value (‘‘0’’) of gateway
processors can be reached if N > (b(N 1)/3c) + 2(Npa + Ta) + Npd + Td where the connectivity of the network is
c > 2(Npa + Ta) + Npd + Td.
In general, the BA protocol needs four rounds of
message exchange to achieve agreement when faulty components exist. Based on the results above, the EDAP protocol only needs two rounds of message exchange to achieve
consensus. Therefore, the EDAP protocol is more efficient
than previous BA protocols [10,12,17] in a MANET
environment.
5. The correctness and complexity of EDAP
In general, our paper compares with the most famous
paper [8,10,14,15,17] of previous works. Therefore, proofs
for the agreement and validity property are given to prove
the EDAP is optimal solution in this section. Lemmas and
theorems are used to prove the correctness and complexity
of EDAP.
5.1. Correctness of EDAP
In this section, we prove the correctness of EDAP if
N > (b(N 1)/3c) + 2(Npa + Ta) + Npd + Td and c > 2(Npa +
Ta) + Npd + Td where Npa is the number of arbitrary faulty
processors, Ta is the number of arbitrary faulty transmission media, Npd is the number of dormant faulty
processors, and Td is the number of dormant faulty transmission media existing simultaneously. Only fp + 2 rounds
of message exchange are required to make all healthy
processors reach a common agreement.
To prove the correctness of our protocol, a tree structure, ic-tree is used to explain our procedures. The ic-tree
collected enough complete messages to eliminate the
influence of faulty components and solve the cyclical influence from the faulty processors by eliminating the repeated names. The function VOTE also must obtain a
common value from the ic-tree during the consensus agreement process. Therefore, this paper proves the correctness
of our protocol by means of the ic-tree structure.
This paper defined a vertex a as common [4,17] if each
healthy processor computes the same value for a. In other
words, the value stored in vertex a of each healthy processor’s ic-tree is common. When each healthy processor has a
common initial value of the source processor in the root of
its ic-tree, an agreement is reached since the root is common. Thus the agreement, (Agreement) and (Validity),
can be rewritten as:
(Agreement’): Root s is common, and
(Validity’): VOTE(s) = vs for each healthy processor, if
the source processor is healthy. Otherwise, a default
value U should be chosen.
To prove a vertex is common, the term common frontier
[4,17] is defined as follows: When every root-to-leaf path
of the ic-tree contains a common vertex, the collection of
the common vertices forms a common frontier. In other
words, every healthy processor collects the same messages
within the common frontier if a common frontier does
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
397
Level 2 of ic-treec,e,f,,h :
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δva = 5 > ( f g − (r − 1)) +
= 4 (stop) Δva =5 > ( f g − (r − 1)) +
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δvc2 = 5 > ( f g − (r − 1)) +
= 4 (stop) Δvc =5 > ( f g − (r − 1)) +
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δvd2 = 5 > ( f g − (r − 1)) +
= 4 (stop) Δvd =5 > ( f g − (r − 1)) +
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δve2 = 5 > ( f g − (r − 1)) +
= 4 (stop) Δve =6 > ( f g − (r − 1)) +
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
Δvf2 =5 > ( f g − (r − 1)) +
= 4 (stop)
Δvf2 = 5 > ( f g − (r − 1)) +
= 4 (stop)
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δvg2 = 5 > ( f g − (r − 1)) +
= 4 (stop) Δvg =6 > ( f g − (r − 1)) +
2
2
(n − (r − 1) − λ )
(n − (r − 1) − λ )
2
= 4 (stop)
Δvh2 = 5 > ( f g − (r − 1)) +
= 4 (stop) Δvh =5 > ( f g − (r − 1)) +
2
2
Level 2 of ic-treed,g :
2
Fig. 4k. Execution of early stopping function in second round of message exchange phase.
Table 1
The layer relationship.
Group
Gateway layer
Non-gateway layer
Ga
Gb
Gc
Gd
Ge
Gf
Gg
Gh
a
b
c
d
e
f
g
h
h, i
p, q
n, o
l, m
j, k
w, x
u, v
s, t
exist in a healthy processor’s ic-tree. Subsequently, using
the same voting function VOTE to compute the root value
of the ic-tree, every healthy processor can obtain the same
root value because they receive the same input and use
the same computing function. Since EDAP can solve the
consensus/BA problem, the above concepts can be used
to prove the correctness of EDAP.
Before proving the correctness of EDAP, the term correct
vertex is defined as: (1) Correct vertex – vertex ai of a tree
is a correct vertex if processor i is healthy. In other words, a
correct vertex is a place to store the value received from a
healthy processor. (2) True value – for a correct vertex ai in
the tree of a healthy processor i, val(ai) is the true value of
vertex ai. Namely, the stored value is called the true value.
By the definition of a correct vertex, the stored value is
received from the healthy processors and a healthy processor always transmits the same value to other processors.
The repeated vertices of ic-tree are deleted, thus the correct
vertices of such an ic-tree are common. Based on the
definition of correct vertex, a common frontier does exist
in ic-trees. Namely, the root can be proven to be a common
vertex (Agreement’) due to the existence of a common
frontier, regardless of the correctness of a source processor.
Based on reason above, an agreement concerning the root
value is reached.
Subsequently, we check the condition of (Validity’).
Based on (Validity’), we know that when the source processor fails, the (Validity’) is true. This is because the proposi-
tional logic P ? Q means (NOT(P) OR Q), then (NOT(P) OR
Q) or (P ? Q) is true when P is false; where P implies
‘‘the source processor is healthy’’ and (P ? Q) implies
BA2’. Conversely, root s is a correct vertex by the definition
of a correct vertex if the source processor is healthy. If all
correct vertices’ true values can be computed by EDAP,
then the true value of the root can be computed because
the root is also a correct vertex. By definition, the true value of the root is the initial value of the source processor if
the source processor is healthy. Namely, each healthy processor’s root value is the initial value of the source processor; if the source processor is healthy, then Validity’ is true
so long as the source processor is healthy. In short, Agreement’ and Validity’ are both true because no matter
whether the source processor is healthy or failed, the consensus/BA problem is solved.
Lemma 1. The messages sent through dormant faulty
components can be detected by a healthy destination
processor.
Proof. A healthy destination processor can detect the message(s) from dormant faulty components if the protocol
appropriately encodes a transmitted message by using
either the Non-Return-to-Zero code or the Manchester
code [6] before transmission. h
Lemma 2. Healthy processors can receive messages from
healthy processors, if c > 2(Npa + Ta) + Npd + Td.
Proof. A healthy sender processor broadcasts a message
to others and itself. In the worst case, a healthy processor
can receive c Npd Td messages transmitted in each
round of the message exchange because dormant faulty
components can be detected. If c Npd Td > 2(Npa + Ta),
a healthy processor can determine the nature messages
from sender processors by utilizing the majority value
from the values received in each message exchange
round. h
398
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
Theorem 1. A healthy processor can remove the faulty
influences from dormant faulty transmission media and
dormant faulty processors, if c > 2(Npa + Ta) + Npd + Td.
Proof. By Lemmas 1 and 2, the theorem is proved. h
Lemma 3. A healthy destination processor can detect a dormant faulty sender processor by the forwarding technique in
the gateway/non-gateway model.
Proof. If the height of a is 0 and a common frontier (a
itself) exists, then a is common. If the height of a is r,
the children of a are all in common by using induction
hypothesis with the height of the children at r 1, then
the vertex a is common. h
Corollary 1. A root is common if a common frontier exists in
the ic-tree.
Theorem 3. The root of a healthy processor’s ic-tree is
common.
Proof. If the number of value k is greater than or equal to
c b(N 1)/3c, then the sending processor has a dormant
fault. The reason is that there are at most b(N 1)/3c arbitrarily faulty components in the network; hence there are
at most b(N 1)/3c non-k values in the vector Vi. h
Proof. By Lemmas 4, 5, and 6, and Corollary 1, the theorem
is proven. h
Theorem 2. A healthy processor can detect a dormant faulty
processor in the network.
Theorem 4. Protocol EDAP solves the consensus/BA problem
in MANET.
Proof. In the protocol EDAP, there are r rounds of message
exchange during consensus agreement, where Npa 6
b(N 1)/3c and N > 3, so there are at least two rounds of
message exchange during the message exchange phase.
Each healthy processor can receive the message from the
source processor during the first round of message
exchange and receives other processors’ messages during
the second round of message exchange. Each processor
can receive all other processors’ messages that exist in
the network after two rounds of message exchange.
According to Lemma 3, each healthy processor can detect
the dormant faulty processor in the network. h
Proof. To prove the theorem, one has to show that EDAP
meets the constraints (Agreement’) and (Validity’)
Lemma 4. All healthy correct vertices of an ic-tree are
common.
Proof. After reorganization of rules, no repeatable vertices
are extant in an ic-tree. At the level fp or above, the correct
vertex a have at least 2fp + 1 children (N fp P 2fp + 1) of
which at least fp + 1 children are correct. The true value
of these fp + 1 correct vertices is common, and the majority
value of vertex a is also common. The correct vertex a is
common in the ic-tree if the level of a is less than fp + 1.
Thus, all correct vertices of the ic-tree are common. h
Lemma 5. A common frontier exists in an ic-tree.
Proof. There are fp + 1 vertices along each root-to-leaf path
of an ic-tree in which the root is labeled by the source
name, and the others are labeled by a sequence of group
names. Since at most Npa (6b(N 1)/3c) processors can
have failed, at least one vertex is correct along each rootto-leaf path of the ic-tree. By Lemma 3, the correct vertex
is common, and a common frontier exists in each healthy
processor’s ic-tree. h
Lemma 6. Let a be a vertex, where a is common if there is a
common frontier in the subtree rooted at a.
(Agreement’): Root s is common. By Theorem 3,
(Agreement’) is satisfied.
(Validity’): VOTE(s) = v for all healthy processors, if the
initial value of the source is vs, say v = vs.
Since most of the processors are healthy, they transmit
the message to all others. As a result, each of the correct
vertices of the ic-tree is common (Lemma 4), and its true
value is v. By Theorem 3, this root is common. The
computed value VOTE(s) = v is stored in the root for all
healthy processors. (Validity’) is satisfied. h
Lemma 7. Each value of a healthy processor j received from
the healthy source processor h is fixed. The value is fixed in
round min{fact + 3, fp + 2} when there are fact faulty processors
extant.
Proof. Case 1: source and receivers are healthy processors.
In general, the healthy source processor j sends the
initial value to the healthy receiver h in next round. Then
the receiver h sends the same value to its descendant
processors. Subsequently, the value in the descendant path
should be the same. This is because the healthy descendant
receiver does not change the original value that transfers
to its descendant receivers.
Case 2: source and receivers are faulty processors.
The faulty source processor sends different values to
faulty receiving processors; the value might alternate from
round to round. Due to only fact faulty processors extant in
system, the longest faulty path only has length fact.
Therefore, the influences of vertices of descendant path
(Pfact + 1) can be eliminated by the majority function;
subsequently the values of the descendant path are fixed
again based on Case 1. In other situations, these influences
M.-L. Chiang / Ad Hoc Networks 10 (2012) 388–400
can be eliminated when the faulty source processor versus
the healthy receiver processors or the healthy source
processor versus the faulty receiver processors. h
Theorem 5. Protocol EDAP can achieve agreement in
min{fact + 3, fp + 2} rounds.
Proof. Based on the Theorem 4, one can prove that the
protocol EDAP can meet the constraints of (Agreement’)
and (Validity’) in the consensus/BA problem when the
fact = fp. By Lemmas 4–7, and Corollary 1, the values of a
descendant path are fixed to a common value in round
min{fact + 3, fp + 2} and early stopping. Therefore, the constraints (Agreement’) and (Validity’) are also achieved in
round min{fact + 3, fp + 2}. h
5.2. Complexity of EDAP
The complexity of EDAP is judged in terms of: (1) the
maximum number of allowable faulty components; (2)
the minimal number of rounds.
Theorem 6. The maximum number of allowable faulty
components of EDAP is Npa arbitrary faulty processors, Npd
dormant faulty processors, Ta arbitrary faulty transmission
media, and Td dormant faulty transmission media; where is
N > (b(N 1)/3c) + 2(Npa + Ta) + Npd + Td and connectivity of
the network c should be greater than 2(Npa + Ta) + Npd + Td
Proof. Wang et al. [14], indicated that the constraints
of the BA problem for processor faults is only
N > (b(N 1)/3c) + 2Npa + Npd and c > 2Npa + Npd in which
the faulty transmission media was treated as faulty processors. In this study, the fault status of processors and transmission medium of our assumption are treated as the
different fault. Therefore, we must include the transmission medium faults, and so the constraints are rewritten
as N > (b(N 1)/3c) + 2(Npa + Ta) + Npd + Td and c > 2(Npa +
Ta) + Npd + Td. Therefore, the total number of allowable
faulty components of EDAP is Npa arbitrary faulty processors, Npd dormant faulty processors, Ta arbitrary transmission media, and Td dormant faulty transmission media;
which is the maximum if N > (b(N 1)/3c) + 2(Npa +
Ta) + Npd + Td and c > 2(Npa + Ta) + Npd + Td. h
Theorem 7. EDAP requires min{fact + 3, fp + 2} rounds to
solve the EBA problem in a MANET.
Proof. Based on the previous works [8,17], for a fail-safe
network, if the transmission medium is reliable, then
fp + 1 rounds are proven to be the lower boundary for message exchanges. Therefore, the required number of rounds
for solving the generalized BA/consensus problem in
MANET should not be less than fp + 1. In the EBA problem,
the protocol of Krings and Feyer [9] also proves that the
actual number of rounds of message exchange is
min{fact + 2, fp + 1} when the fallible components are processors only in well-defined connected networks without
399
faulty transmission media. However, our protocol EDAP
solves the EBA problem underlying dual failure in generalized failure mode where both faulty processors and faulty
transmission media exist. The protocol of Yan et al. [17],
shows that two rounds of message exchanges is the minimum number of rounds needed to solve the BA problem if
fp = 0. Thus, it is impossible that the number of rounds
required is fp + 1. Otherwise, only one round of message
exchange could solve the BA/consensus problem for the
case fp = 0 in a generalized faulty assumption. This contradicts the results of Yan et al. [17]. Therefore, our protocol
requires an extra round to solve the generalized failure
mode according to results of Yan et al. [17]. The actual
number of rounds of message exchange with EDAP is
min{fact + 3, fp + 2}. However, message passing is required
during the message exchange phase and this is a timeconsuming phase. The EDAP protocol is more efficient at
solving the BA/consensus problem than traditional protocols [1,10,14,15,17] when the EDAP protocol stops
early. h
6. Conclusion
In this study, our protocol EDAP solves the EBA problem
in a MANET underlying dual failure mode in fallible processors and transmission media based on the CDS-based
virtual backbone. Besides, the EDAP can achieve an agreement while tolerating the maximum number of faulty processor by using the minimum number of message
exchange and round. In our virtual backbone, our protocol
can manage and organize the network efficiently even if
the processors move around the network. It is because that
only gateway processors need to maintain the routing table, and the search space is reduced to itself. Therefore,
the rounds of message exchange in our protocol are more
suitable for a MANET environment than those of previous
results [10,14,15,17].
Besides, our protocol can achieve the agreement earlier
than others by using the concept of eventual BA protocol
when the numbers of messages are sufficient. If there are
only two (fact = 2) arbitrary gateway processors, the at most
rounds of message exchange are decreased to 5 (r = min{fact + 3, fp + 2}). Therefore, our protocol is more realistic
and efficient than previous protocols in a MANET due the
fact that the actual number of arbitrarily faulty processors
fact is often smaller than fp.
Acknowledgment
This work was supported in part by the Taiwan National
Science Council under Grants NSC99-2221-E-324-041MY3, NSC97-2221-E-324-007-MY3, NSC99-2221-E324022, and NSC99-2221-E018-018.
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Mao-Lun Chiang received the M.S. degree in
Information Management from Chaoyang
University of Technology and the Ph.D. degree
in Department of Computer Science from
National Chung-Hsing University, Taiwan. He
is an Assistant Professor in the Department of
Information and Communication Engineering
at the Chaoyang University of Technology,
Taiwan. His current research interests include
Ad Hoc, mobile computing, distributed data
processing, and fault tolerant computing.
ID
445830
Title
EventuallyByzantineAgreementonCDS-basedmobileadhocnetwork
http://fulltext.study/journal/480
http://FullText.Study
Pages
13
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