A network topology is a K-FT topology if it can endure K number of link failures, however to find a reliable hardware topology for a set of nodes keeping the total cost of the links within a predefined budget, is a challenging task, especially when the topology is subjective to constraints that the topological network can tolerate K link failures keeping total cost of network within budget. This problem has been addressed in this paper where in a novel algorithm is proposed that uses N X N matrix to represent the cost between the participating nodes, and uses K-FT topology to tackle the fault tolerant problem of Mobile Adhoc Networks. Intention is to achieve optimal resource utilization and fairness among competing end to end flows. A network topology is said to be K-FT if and only if every pair of node is reachable from all other nodes for K link failures. The algorithm has been tested for wide range of node sets and the result obtained there of suggest that the proposed algorithm finds better solutions in comparison to Genetic Algorithm.

1. 1
K- Fault Tolerant in Mobile Adhoc Network under Cost Constraint
Mrs. Sugandha Singh [1], Dr. Navin Rajpal [2], Dr. Ashok Kale Sharma [3]
[1] Information Technology, USIT, GGSIPU, Delhi, India
[2] Information Technology, USIT, GGSIPU, Delhi, India
[3] Computer Science and Engineering, YMCA Engg. College, MDU, Haryana, India
sugandha06@gmail.com[1], navin_rajpal@yahoo.com[2],
ashokkale2@rediffmail.com[3]
Abstract: A network topology is a K-FT topology if
it can endure K number of link failures, however to
find a reliable hardware topology for a set of nodes
keeping the total cost of the links within a
predefined budget, is a challenging task, especially
when the topology is subjective to constraints that
the topological network can tolerate K link failures
keeping total cost of network within budget. This
problem has been addressed in this paper where in a
novel algorithm is proposed that uses N X N matrix
to represent the cost between the participating
nodes, and uses K-FT topology to tackle the fault
tolerant problem of Mobile Adhoc Networks.
Intention is to achieve optimal resource utilization
and fairness among competing end to end flows. A
network topology is said to be K-FT if and only if
every pair of node is reachable from all other nodes
for K link failures. The algorithm has been tested for
wide range of node sets and the result obtained there
of suggest that the proposed algorithm finds better
solutions in comparison to Genetic Algorithm.
Index Terms: Link failure, Fault tolerance, K-FT
topology, Mobile Adhoc network.
I. INTRODUCTION
An important design constraint for any reliable
mobile adhoc network for distributed computing is
that the network must remain connected even on
failure of one or more links. Usually the design of
network layout is subject to the cost constraints as
well as the reliability requirements to cope with the
fault occurred in the network. Compared with wired
networks where flows only contend at the router that
performs flow scheduling, the unique characteristics
of multi-hop wireless networks show that flows also
compete for shared channel if they are within the
interference ranges of each other. This presents the
problem of designing a topology-aware resource
allocation algorithm that is both optimal with
respect to resource utilization and fair across
contending multi-hop flows. Resource allocation in
such networks needs to address both fairness and
overall network performance. Pricing is a
prospective direction to regulate behaviors of
individual nodes while providing incentives for
cooperation. Some pricing strategies for resource
allocation have been developed by taking account of
factors like multiple transmission rates and energy
consumption of nodes [1]. Multi-rate transmission
capability is commonly seen in most wireless
products nowadays, while energy is one of the most
important resources in portable so system takes into
account energy consumptions in the transmitter side,
the receiver side, and those that are non-transmitters
and non-receivers but are interfered by these
activities. So necessity is to more accurately reflect
the real energy constraint in a wireless network.
Due to the decentralized and self-organizing nature
of ad hoc networks, the quest for a fully distributed
and adaptive algorithm further exacerbates the
problem.
This paper concentrates on two constraints that is
cost and fault tolerance, to obtain an outline of a
Mobile Adhoc Network with a possible minimum
cost. For simplicity it has been assumed; that all the
links have the same reliability. Though in reality this
range is in between 0 and 1 (0<rij<1) where rij is the
reliability of the link between node i and node j. In
this paper, an algorithm for designing a Mobile
Adhoc Network with possible minimum cost has
been developed. Through simulation it has been
shown that the proposed algorithm can efficiently
find out an optimal solution for the above defined
problem.
The paper is organized as follows. Section II defines
related work, Section III illustrates the problem. An
overview of the genetic algorithm is given in section
IV. Section V includes the proposed algorithm
henceforth stated as Algorithm-ANS. The
comparative results of the genetic algorithm and
Algorithm-ANS on three sets of synthetic data is

2. 2
given in section VI. Section VII concludes the
paper.
II. RELATED WORK
In previous work, fair packet scheduling mechanism
have been proposed [1], [2], [3] and shown to
perform effectively in providing fair shares among
single-hop flows in wireless ad hoc networks, and in
balancing the trade-off between fairness and
resource utilization. A number of approaches to
topological network optimization have been
developed [4], [5], [6] & [7] are considered the
topological optimization of communication
networks subject to reliability constraints. In [8] the
author presents a decomposition approach to the
problem in which considering the reliability
constraint the total network cost is minimized.
Unfortunately, there exist fundamental differences
between multi-hop wireless adhoc networks and
traditional wired networks, preventing verbatim
applications of the existing pricing theories. In
multihop wireless networks, flows that traverse the
same geographical vicinity contend for the same
wireless channel capacity. This is in sharp contrast
with wireline networks, where only flows that
traverse the same link contend for its capacity.
When it comes to pricing, we may conveniently
associate shadow prices with individual links in
wireline networks to reflect their resource demand
and supply. This is not feasible in wireless networks
with the presence of location dependent contention.
Simulation has been one of the main methods for
analyzing the properties of mobile ad-hoc networks.
It has considered several parameters such as
mobility model, traffic pattern, propagation model,
etc. none of the previously proposed algorithms has
considered end-to-end flows spanning multiple
hops, which reflects the reality in wireless ad hoc
networks. While these mechanisms may be
sufficient for maintaining basic fairness properties
among localized flows, they do not coordinate intra-
flow resource allocations between upstream and
downstream hops of an end-to-end flow, and thus
will not be able to achieve global optimum with
respect to resource utilization and fairness. Due to
the complexities of such intra-flow coordination, we
are naturally led to a price-based strategy, where
prices are computed as signals to reflect relations
between resource demands and supplies, and are
used to coordinate the resource allocations at
multiple hops. Previous research in wireline network
pricing (e.g., [8], [9], [10]) has shown that pricing is
effective as a means to arbitrate resource allocation.
In these research results, a shadow price is
associated with a wireline link to reflect relations
between the traffic load of the link and its
bandwidth capacity. A utility is associated with an
end-to-end flow to reflect its resource requirement.
Transmission rates are chosen to respond to the
aggregated price signals along end-to-end flows
such that the net benefits (the difference between
utility and cost) of flows are maximized. It has been
shown that [11], [12] at equilibrium, such a price-
based strategy of resource allocation may achieve
global optimum, where resource is optimally
utilized. Moreover, by choosing appropriate utilities,
various fairness models can be achieved.
III Problem Statement
a.) Notations
The following notations have been used throughout
the paper
B: Given budget
G: a Graph
K-FT: Fault Tolerant to K link failure.
1≤K≤N-2
N: number of nodes
Nodei : set of Nodes i=1,2,…,N
Ei : number of Edges(among all the edges
incident on Nodei) that are selected while
building the mobile adhoc network.
Costij : Cost of link between node i and
node j where i,j N
lij : link between node i and node j where
i,j N
G(N,L) : Graph with N nodes and L links
rij: reliability of link between Nodei and
Nodej
b.) Assumptions
1. The set of nodes is given and known.
2. Each node has atleast K+1 edges.
3. The link costs are known and are given for
each pair of nodes.
4. No connection is indicated by 0.
5. No self looping is present.
6. All links are bi-directional. i.e. Costij = Costji
for all i,j N.
7. The network layout does not contain any
redundant links. i.e. no two links connect the
same two nodes.
c.) Some Important Definitions

3. 3
A network topology is K-FT iff every pair of nodes
is reachable for any K link failure in the network.
Connected graph: Every node is reachable from all
other nodes. A graph is K-FT iff all the graphs,
which have K less link than graph G are connected.
Degree of a node is the number of links incident on
it.
The problem is to find a network layout for the
given set of nodes such that the link cost is
minimized, subjected to the condition that the layout
is K-FT. Given N and Costij for all i, j then find a K-
FT network topology such that the total link cost is
minimized. The normal description of the problem
is given below:
Minimize
(1)
Where  is the computational complexity.
Subject to
(2)
lij=0 or lij=1 (3)
C[ G(N,L)-{ lij }] = K, (4)
 lij and i, j E
K=1,2….,N-2
The objective function (1) determines the reliable
minimum cost graph. The constraint (2) ensures that
the cost of the links must not surpass the given
budget. The constraint (3) ensures that when the link
exists then value must be greater than or equal to 0.
The constraint equation (4) ensures that the resulting
graph must be connected and satisfy K-FT.
IV Overview of Genetic Algorithm
Genetic Algorithm proposed by Holland [9] has
been successfully applied to many problems. The
basic idea of GA is to begin with some initial
solutions [10], [11]. Each initial solution is then
evaluated to check whether it is a good solution or
not. According to the objective function a fitness
value is assigned to each solution. The crossover of
pair of solutions generates offspring. More fit
solutions are selected for more number of times for
crossover to produce new population, as they
produce more fit solutions. And the least fit
solutions are deleted from the matting population.
All off springs could be mutated with same
probability. The off springs are then evaluated to see
how good they fit into the mating population, thus
replacing their parents to create the next population.
The process is repeated until the termination criteria
are reached. The design of GA consists mainly of 6
tasks:
1. Formulation of the fitness function
2. Representation of a solution point
3. Generation of the initial population
4. Design of genetic operators
5. Determination of the probabilities for the genetic
operators
6. Definition of the termination criteria
V. Proposed Method
The proposed method is for to find out a cost
constraint K-FT topology. The proposed method
chooses K nodes with maximum edge sum and each
such node is considered as pivot. Then the search
for the K+1 nearest adjacent nodes from the given
link cost specification is done. This is done for each
of the newly taken nodes and the process is repeated
to complete the network. Now the networks are built
considering the fact that it should be K-FT. Next the
cost is calculated for it and recorded for further
comparison. The whole process is repeated for all
nodes selected as pivot thus searching for the
network with possible minimum cost. The
maximum edge sum is considered so that the best
possible options of the furthest nodes can be
explored.
Lemma 1: In a K-FT backbone layout, the degree of
all nodes must be at least K+1.An Mobile Adhoc
network with N nodes can have nodes with atmost
N-1 degree. So a failure of N−2 links can be
handled.
Algorithm-ANS
Step 1 Select K nodes with maximum edge sum
from a set of N nodes. The cost matrix Cost where
Costij =cost of the link/edge between node i and
node j is used as the metric of the edge sum.
Edge_Sumi is the sum of the cost of the edges
incident on Nodei. Variable S is used to store the
number of nodes traversed, NT holds the number of
nodes selected or traversed where as Tot_Cost stores
cost of the network and Ei saves the number of

4. 4
edges (among all the edges incident on Nodei) that
are selected while building the network. The
variable Min_Cost is used to compare the cost of
different networks built. The lists TN and CN holds
the nodes traversed and the nodes selected
respectively. Variable S, NT, Tot_Cost , Ei are
initialized to 0. The lists TN and CN are initially
empty. Set
Min_Cost = ∞ (a large value)
Step 2 For each K node with maximum edge sum
Do go to Step2a.
Step 2a Once a node is traversed the node is stored
in a list TN and S is incremented.
Step 2b Find out the K+1 − Ei nearest node
(consider Costij >0. & 0 means no connection). If
the node selected is already in TN the next nearest
node is to be selected unless all nodes are in TN or
Ei of all other nodes is already equal to K+1. For an
edge selected.
Step 2c Once a node is selected it is stored in a list
of chosen node CN unless it is already present. The
total number of nodes already selected/traversed are
noted and stored in a variable NT. Ei of each node is
set to the number of edges associated with it. For an
edge selected between Nodei & Nodej set Costij =
Costji = ∞ (a large value)
Step 2d The cost is accumulated in Tot_Cost. Go to
Step 3.
Step 2e Select K+1 − Ei neighbor(s) such that (K+1
− Ei)× (N- NT) ×1.5/(NT – S) × K nodes that are not
included in CN but are nearest as compared to all
other nodes not in CN gets selected. This formula
ensures that all nodes are selected while building
the network topology. That is an estimate is
obtained from the above expression that how many
new nodes (even without the minimum cost factor)
should be selected while traversing the already
selected nodes, to include all the given nodes in the
newly built network. The variable S stores the
number of traversed nodes and TN holds the list
node traversed. Go to Step 2c.
Step 3 If S != N & Ei < K+1 for atleast a single
node Nodei i=1,2,…,N. i.e. all nodes have not been
traversed or all nodes do not have K+1 edges/links.
The next node Nodei in the list CN is considered
If Ei < K+1
If (NT – S) × K/1.5 > (N – NT) go to Step 2a.
Else go to Step 2e.
Else increment S and include Nodei in the list TN
and go to Step 3. Else go to Step4.
Step 4 The Tot_Cost is compared to Min_cost. If
the later one is less than the former, then the
Tot_Cost is stored in Min_Cost. The newly created
network topology is stored.
Step 5 Go to step 2. Until all K nodes with
maximum edge cost is not checked.
Step 6 The network topology selected is the desired
K-FT Mobile Adhoc network with the cost stored in
Min_Cost.
Step 7 End
Note: To ensure all nodes are included in each
network topology in Step 3 the comparison between
the number of nodes traversed and the nodes not yet
traversed is done. To keep the cost minimum least
number of nodes in the network is allowed to have
more than K+1 connection.
Working of the Proposed Method
The working of the proposed algorithm is explained
with an example. The network topology consists of
6 nodes. The edge cost matrix is:
Here in this example K=2 is selected and
Costij = Costji , Costij = 0 for i = j.
Set Min_Cost = 99999 (a random large value)
Step1 The two nodes with maximum edge cost are
Node4 and Node6. By the formula of
N
Edge_Sumi = ∑ Costij i=1,2,….,N
j=1
C11 C12 C13 C14 C15 C16
C11 00 30 50 70 20 60
C21 30 00 70 60 80 40
C31 50 70 00 40 30 20
C41 70 60 40 00 90 120
C51 20 80 30 90 00 110
C61 60 40 20 120 110 00

5. 5
Edge_Sum1 = 30+50+ 70+20+ 60 = 230
Edge_Sum2 = 30+70+ 60+80+ 40 = 280
Edge_Sum3 = 50+ 70+40+30+20 = 210 (Min)
Edge_Sum4 = 70+60+ 40+90+ 120 = 380 (Max)
Edge_Sum5 = 20+80+ 30+90+ 110 = 330
Edge_Sum6 = 60+ 40+20+120+110 = 350
The Edge Sum4 and Edge Sum6 are the largest edge
sum values. Variable S, NT, Tot_Cost, Ei are
initialized to 0.
Set Min_Cost = 99999 (a random large value)
Step 2 Firstly Node4 is considered.
Step2a The variable S in incremented and the Node4
is added to the list TN (which is initially empty).
Step2b The three (Since K+1=3 and E1 =0) nearest
neighbors of Node4 are Node1, Node2 and Node3.
Step2c Node1, Node2 and Node3 are stored in the list
CN(which is initially empty). NT is set to 4. E1 =1,
E2 = 1, E3 = 1 and E4 = 3 .
Set Cost14 = Cost41 = Cost24 = Cost42=Cost34 = Cost43
= 99999 (a random large value)
Step2d Tot_Cost = Tot_Cost + 70+60+40= 170.
Go to Step3.
Step 3 Since S=1 and N=6 & Ei < K+1 for atleast a
single node Nodei, i=1,2,…,5.
Select Node1(present in CN) as the next traversed
node.
Also E1=1 , hence E1 < K+1
(NT – S) × K/1.5 = 4
(N – NT) = 2. So go to Step 2a.
Step 2a The variable S in incremented and the
Node1 is added to the list TN.
Step 2b The two (Since K+1− E1=2) nearest
neighbors of Node1 are Node2 and Node5.
Step 2c Node5 is added to the list CN. NT is set to
5. Set E1 = 3, E2 = 2 and E5 = 1
Set Cost12 = Cost21 = Cost15 = Cost51 = 99999 (a
random large value)
Step 2d Tot_Cost = Tot_Cost +30+20= 220. Go to
Step3.
Step 3 Since S=2 and N=6 & Ei < K+1 for atleast a
single node Nodei, i=1,2,…,5.
Select Node2 as the next traversed node.
Also E2=2, hence E2 < K+1
(NT – S) × K/1.5 = 4
(N – NT) = 1. So go to Step 2a.
Step 2a The variable S in incremented and the
Node2 is added to the list TN.
Step 2b The one (Since K+1− E2=1) nearest
neighbor of Node2 is Node6 .
Step2c Node6 is added to the list CN. NT is
incremented and set to 6. Set E2 = 3 and E6 = 1.
Set Cost26 = Cost62 = 99999 (a random large value)
Step 2d Tot_Cost = Tot_Cost + 40 = 260. Go to
Step3.
Step 3 Since S=3 and N=6 & Ei < K+1 for atleast
any single value of i. i=1,2,…,5.
Select Node3 as the next traversed node.
Also E3=1 , hence E3 < K+1
(NT – S) × K/1.5 = 4
(N – NT) = 0. So go to Step 2a.
Step 2a The variable S in incremented and the
Node3 is added to the list TN (which is initially
empty)
Step 2b The two (Since K+1− E3=2) nearest
neighbors of Node3 are Node6 and Node5 .
Step2c Set E3 = 3 , E5= 2 and E6 = 2.
Set Cost35 = Cost53 = Cost36 = Cost63 = 99999 (a
random large value)
Step 2d Tot_Cost = Tot_Cost + 30+20= 310. Go to
Step3.
Step 3 Since S=4 and N=6 & Ei < K+1 for atleast a
single node Nodei, i=1,2,…,5.
Select Node5 as the next traversed node.
Also E5=2 , hence E5 < K+1
(NT – S) × K/1.5 = 2.67
(N – NT) = 0. So go to Step 2a.
Step 2a The variable S in incremented and the
Node5 is added to the list TN.

6. 6
Step 2b The one (Since K+1− E5=1) nearest
neighbors of Node5 is Node6 (Since all other nodes
are present in TN).
Step2c Set E5= 3 and E6 = 3. Set Cost56 = Cost65 =
99999 (a random large value)
Step 2d Tot_Cost = Tot_Cost + 110= 420. Go to
Step3.
Step 3 Though S=5 and N=6 but Ei ≥ K+1 for
Nodei. i=1,2,…,5. Go to Step4.
Step 4 The Tot_Cost is compared to Min_Cost
Tot_Cost < Min_Cost
Min_Cost = Tot_Cost =420.
The network topology stored is given as follows:
Figure 1: Network Topology (K-FT) created by
selecting Node4 as the pivot.
Step 5 Go to step 2 until all K nodes with maximum
edge cost is not checked. Similarly starting with
Node6 the Tot_Cost = 390 and the network topology
obtained is as follows.
Step 6 Since the network topology created later has
lesser Tot_Cost so Min_Cost holds the value
390.The network topology selected is the desired K-
FT Mobile Adhoc network with the cost stored in
Min_Cost.
Figure 2: The desired Network Topology (K-FT)
created by selecting Node6 as the pivot
Step 7 End
VI Results
The reliable mobile adhoc network layout built here
is done entirely by simulation. The simulation
program generates various network layout problems;
each is characterized by N number of nodes, and the
edge-cost matrix. The simulation parameters are:
Maximum edge cost = 200
Minimum edge cost = 10
Set of possible values for N = {10, 50, 100}
Value of K lies in the range 1 & N - 2
Edge costs are randomly generated in between 10 to
200 with uniform distribution. The value of K is
also created randomly between 1 and N−2. Three
sets of synthetic data has been used where N=10,50
and 100 respectively. Table 1 shows the
performance of GA approach with respect to the
proposed Algorithm-ANS
Table 1 Comparative Study between the
performance of Algorithm-ANS and GA
Approach
N K Min_Cost
Algorithm-
ANS
GA Approach
Node4
Node1 Node3
Node2Node5
Node6
Node6
Node1 Node3
Node2
Node5
Node4

7. 7
M+
=3 M=6 M=9
10 2 430 585 550 540
50 7 1532 3010 2900 2895
100 10 3350 6180 6060 5985
+
M indicates MAX_POP
VII Conclusion
The Algorithm-ANS performs better than that of the
one based on Genetic Algorithm. But this is possible
at the cost of greater computational time. For a K-
FT mobile adhoc network with N node the time
complexity of Genetic algorithm is O(N2
) and that
of Algorithm-ANS is O(KN2
). If N»K then KN2
almost approximates to N2
, in that case the time
complexity of Algorithm-ANS will be comparable
with that of GA approach. The proposed method
always selects the mobile adhoc network layout with
probable minimum cost which may not always turn
out to be the actual minimum cost.
REFERENCES
[1] Yu-Fen kao and Jen-Hung Huang, ”Price based resource
allocation for wireless adhoc networks with multi-rate
capability and energy constraint”, Computer
Communication Volume 31, Issue 15, Sept 2008, pp
3613-3624.
[2] T. Nandagopal, T.-E. Kim, X. Gao, and V. Bharghavan,
“Achieving MAC Layer Fairness in Wireless Packet
Networks,” in Proc. Of ACM Mobicom, 2000, pp. 87–98.
[3] L. Tassiulas and S. Sarkar, “Maxmin fair scheduling in
wireless networks,” in Proc. of INFOCOM, 2002, pp.
763–772.
[4] H. Luo, S. Lu, and V. Bharghavan, “A New Model For
Packet Scheduling in Multihop Wireless Networks,” in
Proc. of ACM Mobicom, 2000, pp. 76–86.
[5] D.E.Goldberg, “Genetic Algorithms in Search,
Optimization and Machine Learning ”, 1989,
Addison-Wesley.
[6] Loknath Ghosh, Amitava Mukherjee, Debashis
Saha, “Design of 1-FT communication network
under budget constraint” IWDC 2002, LNCS 2571.
pp. 300-311,2002
[7] Baoding Liu, K. Iwamura, “Topological
Optimization model for communication network
with multiple reliability goals”, Computer and
Mathematics with Applications, Volume 39, Issues
7-8, April 2000, Pages 59-69
[8] Y.C. Chopra, B.S. Sohi, R.K. Tiwari and K.K.
Aggarwal, “Topological Layout for maximizing the
terminal reliability in a computer communication
network”, Microelectronics reliability Volume 24,
Issue 5, 1984, pp 911-913
[9] Dorit Hochbaum, “In Approximation Algorithms for
NP-hard Problems”, PWS Publishing , 1996.
[10]J.E.Baker, “ Adaptive Selection Method for genetic
Algorithm ”, Proc. Int’l Conf. Genetic Algorithm,
1985, pp 101- 111
[11]J. H. Holland, “Adaptation in natural and artificial
systems”, 1975, Univ. Michigan Press
[12]T. Koide, S. Shinmori & H. Ishii “ Topological
optimization with a network reliability constraint”,
Discrete Applied Mathematics, Volume 115, Issues
1-3, 15 November 2001, Pages 135-149