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					ALGORITHM OptPathTrans OPTIMIZED FOR USE WITH LOCATION
 UPDATE VECTORS AND ITS COMPARISON TO MINIMUM HOP
        ROUTING IN MOBILE AD HOC NETWORKS

                           Alexander Roy Geoghegan, Natarajan Meghanathan
                                Jackson State University, Jackson MS, USA
                        roygeoghegan@gmail.com, natarajan.meghanathan@jsums.edu


                                                   ABSTRACT
           In an earlier work, we proposed the OptPathTrans algorithm to minimize the number of
           path transitions during a source-destination (s-d) session in a mobile ad hoc network. The
           sequence of longest-living stable paths determined over the duration of the s-d session is
           called the Stable Mobile Path (SMP). But, the average hop count per static path for SMP
           is significantly larger than the minimum required hop count for a path between the source
           and destination. Also, algorithm OptPathTrans requires complete knowledge about future
           topology changes over the duration of the s-d session. In this paper, we illustrate the
           effectiveness of predicting the future topology changes using the location and mobility
           information of the nodes in the form of Location Update Vectors (LUVs) learnt at the
           time of determining a static stable path of the SMP. The modified algorithm is referred to
           as OptPathTrans-LUV and the sequence of predicted static stable paths (that also actually
           exists) is referred to as SMP-LUV. Simulation results illustrate that the average lifetime
           per static path of SMP-LUV can be as large as 88% of the lifetime per static path
           obtained for SMP. On the other hand, the average hop count per static path of SMP-LUV
           can be as low as 80% of the hop count per static path for SMP.

           Keywords: Stability, Hop Count, Path Lifetime, Simulations, Location Prediction


1   INTRODUCTION                                           topology changes, algorithm OptPathTrans operates
                                                           using the following greedy principle: Whenever a
     A Mobile Ad hoc Network (MANET) is a                  path between the source node s and destination node
dynamic distributed system of arbitrarily moving           d is required at time t, choose the s-d path that exists
resource-constrained wireless nodes. Due to the            for the longest time since t. The above strategy is
limited battery charge per node, each node operates        applied over the duration of the s-d session and the
with a reduced transmission range, thus necessitating      sequence of stable paths obtained is called the Stable
the need for multi-hop routes. In the presence of          Mobile Path (SMP). SMP yields the longest-living
node mobility, routes are susceptible to break over        stable paths, but the average hop count per static path
time and have to be frequently discovered. Route           of the SMP is also significantly higher than the
discoveries are normally accomplished through a            minimum hop count.
flooding-based Route-Request-Reply cycle during                 Algorithm OptPathTrans requires complete
which the source node broadcasts Route Request             knowledge about the future topology changes for the
(RREQ) messages targeted towards the destination.          entire duration of the s-d session. This may not be
The destination learns of the best route to the source     easy to obtain in real-implementation scenarios. In
through the RREQ messages collected and notifies           this paper, we illustrate the effectiveness of
the source about the selected route through a Route        predicting the future topology changes using the
Reply (RREP) packet [1]. Frequent flooding of the          location and mobility information of the nodes learnt
network can significantly exhaust the battery              at the time of determining a stable static path of the
resources of the nodes as well as the network              SMP. The location and mobility information of the
bandwidth [2]. It is highly important that the routes      nodes at the time of determining a static stable path
discovered by a routing protocol are stable and be         are stored in the form of a Location Update Vector
able to sustain node mobility as long as possible.         (LUV) and these LUVs are used to predict the future
     In [3], Meghanathan and Farago proposed a             changes in network topology. The modified
polynomial-time algorithm called OptPathTrans to           algorithm is referred to as OptPathTrans-LUV and
determine the sequence of longest-living stable paths      the sequence of predicted static stable paths (that
over the duration of a source-destination (s-d)            also actually exists) by this algorithm is referred to as
session. Given the complete knowledge of future            SMP-LUV. Such a strategy can be implemented in
practice by letting each node to include its LUV in       successive      flooding-based      broadcast      route
the RREQ message being broadcast during the route         discoveries initiated by LPBR.
discovery process.                                              The rest of the paper is organized as follows:
     Our recently proposed Location Prediction            Section 2 describes the original algorithm
Based Routing (LPBR) protocol [4] also utilizes the       OptPathTrans. Section 3 describes the proposed
LUVs to gather the location and mobility                  OptPathTrans-LUV. Section 4 illustrates the path
information of the nodes. LPBR works as follows:          lifetime and hop count obtained for algorithms
During a flooding-based route discovery, all the          OptPathTrans and OptPathTrans-LUV and compares
nodes in the network, except the destination, include     them with respect to the lifetime and hop count of
their latest location and mobility information in the     minimum hop paths over the same period of
form a LUV record appended to the RREQ message.           simulation time. Section 4 also illustrates the tradeoff
The destination gathers all the RREQ messages and         between path lifetime and hop count for stability-
sends back a Route-Reply (RREP) message on the            based and minimum-hop based routing. Section 5
minimum hop path to the source. If the discovered         concludes the paper.
path fails, the source does not opt for another route
discovery; instead, waits for the destination to          2   ALGORITHM TO DETERMINE THE
predict a new path using the LUV information. The             OPTIMAL    NUMBER        OF PATH
destination locally constructs the global topology            TRANSITIONS (OptPathTrans)
based on the predicted locations for the nodes and
sends a LPBR-RREP message to the source on the            2.1 Terminology
minimum hop predicted path. If the predicted path
exists in reality, the LPBR-RREP message reaches               A mobile graph [6] is defined as the sequence
the source and an expensive flooding-based route          GM = G1G2 … GT of static graphs that represents the
discovery is avoided. This procedure is repeated          network topology changes over some time scale T. In
every time the path used for communication fails.         this research work, we sample the network topology
However, if the predicted path does not exist in          periodically for every 0.25 seconds (i.e., 4 network
reality, an intermediate node (that could not further     snapshots per second), which could in reality be the
forward the LPBR-RREP message to the source)              instants of data packet origination at the source.
sends back a LPBR-RREP-ERROR message to the                    A mobile path [6], defined for a source-
destination. The destination clears the database of       destination (s-d) pair, in a mobile graph GM = G1G2
LUVs and waits for the source to launch a new             … GT is the sequence of paths PM = P1P2 … PT,
flooding-based route discovery. The LPBR protocol         where Pi is a static path between the same s-d pair in
has been observed to significantly reduce the number      Gi = (Vi, Ei), Vi is the set of vertices and Ei is the set
of times the flooding-based route discoveries are         of edges connecting these vertices at time instant ti.
initiated. As LPBR discovers minimum hop paths            That is, each static path Pi can be represented as the
both in the actual as well as the predicted graphs, the   sequence of vertices v0v1 … vl, such that v0 = s and vl
hop count of LPBR is only at most 8% more than            = d and (vj-1,vj) ∈ Ei for j = 1,2, …, l. The timescale
that obtained for the minimum-hop based Dynamic           of tT normally corresponds to the duration of a
Source Routing (DSR) protocol [5].                        session between s and d.
     LPBR and OptPathTrans-LUV are different in                The Stable Mobile Path (SMP) for a given
the following aspects: LPBR is a distributed routing      mobile graph and s-d pair is the sequence of static s-
protocol to discover the sequence of minimum hop          d paths such that the number of route transitions is as
paths such that the number of flooding-based              minimum as possible. A Minimum Hop Mobile Path
broadcast route discoveries is as low as possible.        (MHMP) for a given mobile graph and s-d pair is the
OptPathTrans-LUV is a centralized algorithm to            sequence of minimum hop static s-d paths.
discover stable paths on a sequence of network
graphs predicted using the LUVs. The hop count of         2.2 Algorithm OptPathTrans
SMP-LUV (Stable Mobile Path determined by
OptPathTrans-LUV) would be far larger than that of              Algorithm OptPathTrans operates on the
the minimum hop count, but smaller than that of the       following greedy strategy: Whenever a path is
SMP, as also observed in the simulations in this          required, select a path that will exist for the longest
paper. In between two successive route discoveries,       time. Let GM = G1G2 … GT be the mobile graph
LPBR uses several minimum hop paths, whereas,             generated by sampling the network topology at
OptPathTrans-LUV uses only one stable path.               regular time instants t1, t2, …, tT of an s-d session.
However, as both LPBR and OptPathTrans-LUV are            When an s-d path is required at sampling time instant
based on the LUVs of the nodes gathered at the time       ti, the strategy is to find a mobile sub graph G(i, j) =
of route discovery, the lifetime of the stable paths      Gi ∩ Gi+1 ∩ … ∩ Gj such that there exists at least
determined using OptPathTrans-LUV can be                  one s-d path in G(i, j) and no s-d path exists in G(i,
considered as an upper bound on the time between          j+1). A minimum hop s-d path in G(i, j) is selected.
Such a path exists in each of the static graphs Gi,            obtained from taking intersections of the static
Gi+1, …, Gj. If time instant tj+1 ≤ tT, the above              graphs generated based on the actual locations of the
procedure is repeated by finding the s-d path that can         nodes at time instant ti and the predicted locations of
survive for the maximum amount of time since tj+1.             the nodes at time instants ti +1, ti +2, ..., tj.
A sequence of such maximum lifetime static s-d
paths over the timescale of a mobile graph GM forms            3.2 Location Update Vector (LUV) and Location
the Stabile Mobile s-d path in GM. The pseudo code                 Prediction
of the algorithm is given in Figure 1.
                                                                    The Location Update Vector (LUV) for a node
Input: GM = G1G2 … GT, source s, destination d                 collected at time instant ti contains the following
Output: PS            // Stable Mobile Path                    information: ID of the node, X and Y co-ordinates of
Auxiliary Variables: i, j - representing sampling              the node at ti, the velocity and direction of movement
time instants                                                  of the node with respect to the X-axis at ti. We
Initialization: i=1; j=1; PS = Φ                               assume the LUV of a node is known at every
                                                               sampling time instant ti at which we need to
Begin OptPathTrans                                             determine a new stable path as part of the algorithm
                                                               OptPathTrans-LUV. Using the LUVs of every node
1   while (i ≤ T) do                                           in the network at time instant ti, the predicted mobile
                                                               graph Gpred(i, j) = Gactual(i) n Gpred(i+1) n Gpred(i+2)
2      Find a mobile graph G(i, j) = Gi ∩ Gi+1 ∩ …             n ….. Gpred(j) is constructed.
       ∩ Gj such that there exists at least one s-d                 To construct a predicted static graph at time
      path in G(i, j) and {no s-d path exists in G(i,          instant ti+k (where ti ≤ ti+k ≤ tj), we need to predict the
      j+1) or j = T+1}                                         location (i.e., the X and Y coordinates) of every node
                                                               in the network at time ti+k. This is done using the
3      PS = PS U {minimum hop s-d path in G(i, j) }            LUVs of the nodes at time instant ti. We now explain
                                                               how to predict the location of a node (say node u) at
4      i=j+1                                                   a time instant ti+k based on the LUV gathered from u
                                                               at time ti. Let (Xui, Yui) be the X and Y co-ordinates
5   end while                                                  of node u at time instant ti. Let Angleui and Velocityui
                                                               represent the angle of movement with respect to the
6    return PS                                                 X-axis and the velocity at which u is moving at time
                                                               instant ti. The distance traveled by node u from time
End OptPathTrans                                               instant ti to ti+k would be: Distanceu(ti to ti+k) = (ti+k –
                                                               ti)* Velocityui. We assume each node is initially
Figure 1: Pseudo code for algorithm OptPathTrans               configured with information regarding the network
                                                               boundaries: [0, 0], [Xmax, 0], [Xmax, Ymax] and [0, Ymax].
2.3 Complexity of Algorithm OptPathTrans                            Let (Xui+k, Yui+k) be the predicted location of
                                                               node u at time instant ti+k. The values of Xui+k and
     In a mobile graph GM = G1G2 … GT, the number              Yui+k are given by Xui+Offset-Xui+k and Yui+Offset-
of route transitions can be at most T. A path-finding          Yui+k respectively. The offsets in the X and Y-axes
algorithm will have to be run T times, each time on a          depend on the angle of movement and the distance
graph of n nodes. If we use O(n2) Dijkstra algorithm           traveled by node u from time ti to ti+k. These are
[7], where n is the number of nodes in the network,            calculated as follows:
the worst-case run-time complexity of OptPathTrans                  Offset-Xui+k = Distanceu(ti to ti+k) * cos(Angleui)
is O(n2T).                                                          Offset-Yui+k = Distanceu(ti to ti+k) * sin(Angleui)

3   ALGORITHM OptPathTrans OPTIMIZED                               If (Xui+k < 0), then Xui+k = 0
    FOR USE WITH LOCATION UPDATE                                   If (Xui+k > Xmax), then Xui+k = Xmax
    VECTORS (OptPathTrans-LUV)                                     If (Yui+k < 0), then Yui+k = 0
                                                                   If (Yui+k > Ymax), then Yui+k = Ymax
3.1 Actual and Predicted Mobile Sub Graph
                                                                   As stated above, when a node is predicted to
    Let Gactual(i, j) = Gactual(i) n Gactual(i+1) n            cross over the boundary of the network, we just
Gactual(i+2) n ….. Gactual(j) denote the actual mobile         locate the node to be at the network boundary.
sub graph obtained from taking intersections of the
static graphs generated based on the actual locations          3.3 Algorithm OptPathTrans-LUV
of the nodes at time instants ti, ti +1, ti +2, ..., tj. Let
Gpred(i, j) = Gactual(i) n Gpred(i+1) n Gpred(i+2) n …..            Let t1, t2, …, tT denote the regular sampling time
Gpred(j) denote the predicted mobile sub graph                 instants of an s-d session When an s-d path is
required at sampling time instant ti, the strategy is to    4   SIMULATIONS
find the actual mobile sub graph Gactual(i, j) =
Gactual(i) n Gactual(i+1) n Gactual(i+2) n ….. Gactual(j)        We conduct our simulations in both square and
and the predicted mobile sub graph Gpred(i, j) =            circular network topologies for different conditions
Gactual(i) n Gpred(i+1) n Gpred(i+2) n ….. Gpred(j) such    of network density and node mobility. The area of
that there exists at least one common s-d path in both      both the square and circular network topologies
Gactual(i, j) and Gpred(i, j) and no such common s-d        considered is 1000,000 m2. This corresponds to a
path exists in both Gactual(i, j+1) and Gpred(i, j+1).      square network of side 1000m and a circular network
The minimum hop s-d path that exists in both                of radius 564m. The transmission range of a node in
Gactual(i, j) and Gpred(i, j) is selected. Such a path      both these network topologies is 250m. The network
exists in each of the actual static graphs Gactual(i),      density is varied by conducting the simulations with
Gactual(i+1), Gactual(i+2), …, Gactual(j) and Gpred(i+1),   25 nodes (low), 75 nodes (moderate) and 125 nodes
Gpred(i+2), …, Gpred(j). If tj+1 ≤ tT, the above            (high) – which corresponds to an average
procedure is repeated by finding the s-d path that can      neighborhood density (the number of neighbors per
survive for the maximum amount of time since tj+1           node) of 5, 15 and 25 respectively. We observed a
satisfying the above constraint. The sequence of such       probability of network connectivity of 0.3-0.4, 1 and
predicted static s-d paths over the time scale T is         1 in the low, moderate and high density scenarios
called the Stable Mobile Path – predicted using LUV         respectively.
(represented as SMP-LUV). At each time instant, the              The node mobility model used is the Random
path-finding algorithm has to be run twice, once on         Waypoint model [8], one of the most widely used
the actual mobile sub graph and another time on the         models for simulating mobility in MANETs.
predicted mobile sub graph. If we use O(n2) Dijkstra        According to this model, each node starts moving
algorithm, where n is the number of nodes in the            from an arbitrary location to a randomly selected
network, the worst-case run-time complexity of              destination with a randomly chosen speed in the
OptPathTrans-LUV is O(2n2T) = O(n2T). The pseudo            range [vmin .. vmax]. Once the destination is reached,
code of OptPathTrans-LUV is given below:                    the node stays there for a pause time and then
                                                            continues to move to another randomly selected
                                                            destination with a different speed. We use vmin = 0
Input: GM = G1G2 … GT, source s, destination d              and pause time of a node is 0. The values of vmax
Output: PS-LUV           // Stable Mobile Path – LUV        used are 10, 30 and 50 m/s representing scenarios of
Auxiliary Variables: i, j - representing sampling           low, moderate and high node mobility respectively.
time instants                                                    Note that, two nodes a, b are assumed to have a
Initialization: i=1; j=1; PS-LUV = Φ                        bidirectional link at time t if the Euclidean distance
                                                            between them at time t (derived using the locations
Begin OptPathTrans-LUV                                      of the nodes from the mobility trace file) is less than
                                                            or equal to the wireless transmission range of the
1   while (i ≤ T) do                                        nodes. We obtain a centralized view of the network
                                                            topology by generating mobility trace files for 1000
2      Find a predicted mobile sub graph Gpred(i, j)        seconds. Each data point in Figures 3-6 is an average
       and an actual mobile sub graph Gactual(i, j)         computed over 5 mobility trace files and 5 randomly
       such that there exists at least one common s-d       selected s-d pairs from each of the mobility trace
       path in both Gpred(i, j) and Gactual(i, j) and no    files. The starting time of each s-d session is
       such common s-d path exists in both Gpred(i,         uniformly randomly distributed between 1 to 20
       j+1) and Gactual(i, j+1) or j + 1 = T                seconds. In Figures 3-6, the Minimum Hop Mobile
                                                            Path, the Stable Mobile Path determined by
3      PS-LUV = PS-LUV U {Minimum hop predicted s-          algorithm OptPathTrans and the Stable Mobile Path
       d path that exists in both Gpred(i, j) and           determined by algorithm OptPathTrans-LUV are
       Gaactual(i, j)}                                      identified as MHMP, SMP-Opt and SMP-LUV
                                                            respectively.
4      i=j+1                                                     The performance metrics evaluated are the
                                                            average lifetime per static path for a mobile path and
5    end while                                              the time averaged hop count of the mobile path under
                                                            the conditions described above. The time averaged
6    return PS-LUV                                          hop count of a mobile path is the sum of the products
                                                            of the number of hops per static path and the number
End OptPathTrans-LUV                                        of seconds each static path exists divided by the
                                                            number of static graphs in the mobile graph. For
                                                            example, if a mobile path spanning over 10 static
    Figure 2: Pseudo code for OptPathTrans-LUV              graphs comprises of a 2-hop static path p1, a 3-hop
 Figure 3.1: Average Lifetime per      Figure 3.2: Average Lifetime per   Figure 3.3: Average Lifetime per
      Static Path (25 Nodes)               Static Path (75 Nodes)             Static Path (125 Nodes)
           Figure 3: Average Lifetime per Static Path in the Mobile Path (Square Network Topology)




  Figure 4.1: Average Lifetime per     Figure 4.2: Average Lifetime per    Figure 4.3: Average Lifetime per
       Static Path (25 Nodes)              Static Path (75 Nodes)              Static Path (125 Nodes)
          Figure 4: Average Lifetime per Static Path in the Mobile Path (Circular Network Topology)




  Figure 5.1: Average Hop Count        Figure 5.2: Average Hop Count       Figure 5.3: Average Hop Count
     per Static Path (25 Nodes)          per Static Path (75 Nodes)           per Static Path (125 Nodes)
         Figure 5: Average Hop Count per Static Path in the Mobile Path (Square Network Topology)




  Figure 6.1: Average Hop Count        Figure 6.2: Average Hop Count        Figure 6.3: Average Hop Count
     per Static Path (25 Nodes)          per Static Path (75 Nodes)            per Static Path (125 Nodes)
         Figure 6: Average Hop Count per Static Path in the Mobile Path (Circular Network Topology)

static path p2, and a 2-hop static path p3, with each     approach and OptPathTrans-LUV cannot effectively
existing for 2, 3 and 5 seconds respectively, then the    make use of the increase in network density and
time-averaged hop count of the mobile path would be       determine paths with relatively larger lifetime than
(2*2 + 3*3 + 2*5)/10 = 2.3.                               those determined in low-density networks.
                                                                The average lifetime per static path determined
4.1 Average Path Lifetime                                 by the Minimum hop path algorithm is only 10%-
                                                          40% (i.e., 10 times smaller at the worst case) of the
      For a given level of node mobility, as we           average lifetime per static path for an SMP. On the
increase the network density, the difference in the       other hand, the average lifetime per static path
lifetimes of the paths discovered by algorithms           determined for SMP-LUV is 30%-88% (i.e., 3.5
OptPathTrans and OptPathTrans-LUV increases.              times smaller at the worst case) of the average
OptPathTrans effectively makes use of the increased       lifetime per static path obtained for an SMP. The
availability of the nodes and the knowledge of the        difference in the lifetime per static path for Minimum
locations of the nodes over the entire simulation time    hop routing and OptPathTrans-LUV decreases with
period and determines stable paths with the longest       increase in network density and node mobility. The
lifetime. Both the Minimum-hop based routing              lifetimes per static path obtained for SMP-LUV can
be as large as, three times the lifetime per static path   two metrics. If a routing algorithm has a lower Path
obtained using Minimum hop based routing.                  lifetime ratio, then it implies the routes determined
     In general, at low and high mobility conditions,      by that algorithm are more stable. Similarly, if a
for a given routing algorithm and network density,         routing algorithm has a lower Hop count ratio, then it
the average lifetime per static path incurred in a         implies the routes determined by that algorithm have
circular topology is higher than that incurred in a        a hop count that is closer to that of the minimum.
square topology. On the other hand, in moderate
mobility conditions, the average lifetime per static
path incurred in a square topology is higher than that
incurred in a circular topology.

4.2 Average Hop Count

     The average hop count per static path
determined by algorithms OptPathTrans and
OptPathTrans-LUV is respectively 20%-88% and
11%-66% more than the minimum hop count for
square network topology. On circular network
topologies, the two stable mobile path algorithms
incur relatively higher hop count. The average hop
count per static path determined by algorithms
OptPathTrans         and    OptPathTrans-LUV        is
respectively 20%-150% and 16%-82% more than the
minimum hop count for circular network topology.
OptPathTrans determines static stable paths whose
hop count could be as large as 24% and 37% more
than that determined by OptPathTrans-LUV in
square and circular network topologies respectively.
In general, the hop count of a minimum hop path is
higher for square network topologies than circular
topologies. On the other hand, the hop count of a
stable path is higher for circular network topologies
than square topologies.
     With increase in network density, the average
hop count per minimum hop path decreases. For the
two      stability-based    algorithms     (especially
OptPathTrans), the average hop count per stable path
increases as the network density is increased. This is
attributed to the nature of these two algorithms to
determine       long-living    stable    paths     by
accommodating few more nodes on the path that
would increase the average lifetime of the constituent
links of the path.
                                                              Figure 7: Path Lifetime – Hop Count Tradeoff
4.3 Path Lifetime – Hop Count Tradeoff                                (Square Network Topology)

      Figures 7 and 8 capture the tradeoff between               We observe that OptPathTrans has the highest
path lifetime and hop count for each of the three          Hop count ratio and MHMP has the highest Path
algorithms (Minimum hop based                 routing,     lifetime ratio. This implies that the minimum hop
OptPathTrans, OptPathTrans-LUV) for square and             count paths cannot have the longest lifetime – what
circular network topologies respectively. The Path         we call as the lifetime-hop count tradeoff. On the
lifetime ratio is defined as the ratio of the average      other hand, we observe OptPathTrans-LUV to have a
path lifetime per static path for the SMP determined       relatively lower Path lifetime ratio compared to
by algorithm OptPathTrans to that of the average           MHMP and a relatively lower Hop count ratio
path lifetime per static path for a MHMP or the            compared to OptPathTrans. Thus, OptPathTrans-
SMP-LUV. The Hop count ratio is defined as the             LUV effectively balances the tradeoff between path
ratio of the average hop count of either the SMP or        lifetime and hop count tradeoff as much as possible.
the SMP-LUV to that of the average hop count per                 We observe the Path lifetime ratios of SMP-
static path for a MHMP. Note that the ratios are           LUV are relatively lower for square networks. This
formulated using the optimum values for each of the        implies for square networks, OptPathTrans-LUV
                                                           determines paths with lifetime close to that
determined by OptPathTrans. The Hop count ratios         compared to Minimum hop routing, and also
obtained for SMP-LUV in circular networks are a          generally provides a lower time-averaged hop count
little higher, but are still contained within 1.8. We    than OptPathTrans. OptPathTrans-LUV is able to
also observe that in low density networks, MHMP          accomplish this through effective prediction of the
has almost the same Path lifetime ratio in both square   locations of each node using the Location Update
and circular network topologies. As we increase the      Vectors (LUVs) available for the nodes at the time of
network density, the Path lifetime ratio for MHMP in     determining a new stable path. The effectiveness of
circular networks decreases – implies, the minimum       location prediction is that even if the predicted
hop paths in circular networks are more stable than      locations of the two nodes are different from the
that of square networks in moderate and high density     actual locations of the nodes, there exists a link
networks. Another interesting observation is that the    between two nodes in the predicted graph if they are
hop count of a SMP determined by algorithm               within their transmission range. This helps to
OptPathTrans has a relatively higher Hop count ratio     significantly increase the lifetime of the paths
in circular networks (than square networks),             determined by OptPathTrans-LUV compared to
especially at moderate and high network density.         those of the minimum hop paths, and at the same
                                                         time, the increase in the hop count is well restricted.
                                                         The mobile sub graphs of algorithm OptPathTrans
                                                         span over a larger period of time, and the number of
                                                         links in such mobile sub graphs are relatively lower.
                                                         The stable path (which is a minimum hop path in the
                                                         mobile sub graph) determined between a source and
                                                         destination would have to go through several
                                                         intermediate nodes. On the other hand, the mobile
                                                         sub graphs of algorithm OptPathTrans-LUV have
                                                         relatively more links and the stable path is basically a
                                                         minimum hop path in such a mobile sub graph with
                                                         more links. Thus, OptPathTrans-LUV effectively
                                                         reduces the tradeoff between path lifetime and hop
                                                         count. As future work, we will work on extending
                                                         OptPathTrans-LUV to a distributed routing protocol
                                                         for MANETs.

                                                         6   ACKNOWLEDGMENTS

                                                         The research is supported through the National
                                                         Science Foundation grant (CNS-0851646) entitled:
                                                         REU Site: Undergraduate Research Program in
                                                         Wireless Ad hoc Networks and Sensor Networks,”
                                                         hosted by the Department of Computer Science at
                                                         Jackson State University, USA. The authors also
                                                         acknowledge Dr. Loretta Moore, Dr. Xuejun Liang
                                                         and Mrs. Brenda Johnson (all at Jackson State
                                                         University) for their services to this program.

                                                         7   REFERENCES

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                                                             Networking, pp.151-162, 1999.
5   CONCLUSIONS                                          [3] N. Meghanathan and A. Farago, “On the Stability
                                                             of Paths, Steiner Trees and Connected
   As hypothesized, algorithm OptPathTrans-LUV               Dominating Sets in Mobile Ad Hoc Networks,”
does in fact offers an alternative to both Minimum           Elsevier Ad Hoc Networks, Vol. 6, No. 5, pp.
hop routing as well as OptPathTrans. OptPathTrans-           744 – 769, July 2008.
LUV is able to yield higher average path lifetime        [4] N. Meghanathan, “A Location Prediction Based
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      per Path in Mobile Ad hoc Networks,” The
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[5]   D. B. Johnson, D. A. Maltz and J. Broch, “DSR:
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[6]   A. Farago and V. R. Syrotiuk, “MERIT: A
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[7]   T. H. Cormen, C. E. Leiserson, R. L. Rivest and
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[8]   C. Bettstetter, H. Hartenstein and X. Perez-Costa,
      “Stochastic Properties of the Random-Waypoint
      Mobility Model,” Wireless Networks, vol. 10, no.
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Description: UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.
UbiCC Journal UbiCC Journal Ubiquitous Computing and Communication Journal www.ubicc.org
About UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.