Extensions in Mobile Ad Hoc Routing Using Variable Precision Rough Sets

V. Mary Anita Rajam, V. Uma Maheswari and Arul SiromoneyDepartment of Computer Science and Engineering,

Anna University, Chennai – 600 025, IndiaContact email: asiro@vsnl.com

Abstract

Mobile ad hoc networks are formed dynamically with-out any infrastructure and each node in the network is re-sponsible for routing information. Rough set theory is amathematical tool to deal with vagueness and uncertainty.Variable Precision Rough Sets (VPRS) is a generalizationof rough sets that allows for a controlled degree of misclas-sification. This paper proposes extensions in mobile ad hocrouting using VPRS. The performance of the proposed mo-bile ad hoc routing protocol is compared with that of anexisting routing protocol.

1. Introduction

Mobile ad hoc networks [4] are a collection of wirelessmobile nodes that can dynamically form a network. There isno centralized administration or standard support services.Each node in an ad hoc network is responsible for routinginformation among them and hence acts as a node as wellas a router. Some of the applications of mobile ad hoc net-works are conferences/meetings, battlefields, instant infras-tructure, remote areas and disaster relief.

Since each node in an ad hoc network is responsible forrouting, each node should maintain necessary informationabout the next hop or the path through which the data has tobe routed to the destination. A number of routing protocols(e.g., Destination Sequenced Distance Vector (DSDV) [9],Dynamic Source Routing (DSR) [2], Ad hoc On DemandDistance Vector (AODV) [10]) have been proposed for adhoc networks. Dynamic Source Routing (DSR) [2] is anon-demand routing protocol, in which each node has a routecache to store the routes that are either used by the node orlearnt by looking into the packets that pass by.

Rough set theory, introduced by Zdzislaw Pawlak [7] isa mathematical tool to deal with vagueness and uncertainty.In rough set theory, elements that exhibit the same char-acteristics are indiscernible and form the elementary sets.Variable Precision Rough Sets (VPRS) [12], proposed by

Ziarko is a generalized model of rough sets, aimed at mod-eling classification problems involving uncertain or impre-cise information. In the VPRS formulation, the lower andupper approximation are interpreted in probabilistic terms,leading to generalized notions of rough set approximations.

The basic notion of an information system in mobile adhoc routing is introduced in an earlier paper [11]. Each mo-bile node maintains an information system that assists inrouting packets. In this paper, the information system ismodified to better capture the links in a route, and a thresh-old (as used in VPRS) is used in the identification of thenext hop. The performance of the proposed routing proto-col is compared with the performance of DSR protocol.

The remainder of this paper is organized as follows. Thefollowing section gives the definitions with respect to roughset theory and VPRS model. The section also explains howVPRS is used in mobile ad hoc routing. Section 3 describesthe proposed routing protocol and section 4 gives the per-formance evaluation of the proposed routing protocol.

2. VPRS and mobile ad hoc routing

2.1. Rough sets

Consider a universe U of elements. An information sys-tem I is defined as I = (U,A, V, ρ) where A is a non-empty,finite set of attributes; V =

⋃a∈A Va is the set of attribute

values of all attributes, where Va is the domain (the set ofpossible values) of attribute a; ρ : U × A → V is aninformation function such that for every element x ∈ U ,ρ(x, a) ∈ Va and is the value of attribute a for element x.I = (U,A, V, ρ), is known as a decision system, when anattribute d ∈ A is specified as the decision attribute. ThenA − {d} is known as the set of condition attributes. Thesedefinitions are based on the definition of Rough Set Infor-mation System in [7, 8, 5].

The information system (decision system) can also beviewed as an information table (decision table), where eachelement x ∈ U corresponds to a row, and each attribute

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a ∈ A corresponds to a column. The value of the entry inthe information table at row x and column a is ρ(x, a).

Let B be a subset of A, and let xi and xj be membersof U . A binary relation R(B), called an indiscernibilityrelation, is defined as

R(B) = {(xi, xj) ∈ U × U | ∀a ∈ B, ρ(xi, a) =ρ(xj , a)}

That is, xi and xj are indiscernible to the set of at-tributes B in U . The equivalence class of an elementx ∈ U consists of all objects y ∈ U such that xRy. Theequivalence classes of the indiscernibility relation R(B)are denoted by [x]B and are called elementary sets.

2.2. Variable Precision Rough Sets

The formal definitions of the VPRS model presented be-low are taken from [1]. Consider the information systemI = (U,C ∪ D,V, ρ) where C denotes the set of condi-tion attributes and D denotes the set of decision attributes.The ordered pair AI = < U,R(C) > is called an approx-imation space based on the condition attributes C. Here,R(C) denotes the equivalence relation (an indiscernibil-ity relation) using C. The equivalence classes of the re-lation R(C) are called the elementary sets in AI , becausethey represent the smallest groups of objects that are dis-tinguishable in terms of the attributes and their values. LetR∗(C) = {X1,X2, . . . , Xn} = {[x1]R, [x2]R, . . . , [xn]R}be the collection of equivalence classes induced by the rela-tion R(C).

A concept is an equivalence class of the relation R(D).Let Y ⊆ U be a subset of objects representing a concept. Aβ-approximation space is a triple < U,R(C), P >, whereP is a probability measure and β is a real number in therange (0.5, 1]. The β-approximation space can be dividedinto the following regions:

β-positive region of the set Y :POSC(Y ) =

⋃P (Y |Xi)≥β{Xi ∈ R∗(C)}

β-negative region of the set Y :NEGC(Y ) =

⋃P (Y |Xi)<β{Xi ∈ R∗(C)}

2.3. Information system in a mobile node

Let M be the set of mobile nodes. A route is a paththrough mobile nodes in M and is denoted as a sequence ofmobile nodes m1m2 . . . mk, mi ∈ M, i = 1, . . . , k. Theroute cache of mobile node m stores routes that m knows.A new route is not added to the route cache if it alreadyexists in the route cache, or is a sub-path of a route alreadyin the route cache. We note that any route in the route cache

is a path starting from m itself, and so m1, the first nodein the route, is m itself. Any route in the route cache is asimple path, where no node repeats, that is, mi �= mj formi,mj in the path, i �= j. We note that k ≤ |M |.

Let each mobile node m ∈ M have an information ta-ble Im = (Um, Am, Vm, ρm) associated with it. Let Am bethe set of all possible links between the nodes. Each con-dition attribute a ∈ Am in the attribute set corresponds toa particular link in the set of all possible links between thenodes. Each condition attribute is a boolean attribute, withVa = {0, 1}, and is set to 1 or 0 depending on whether thatlink is present in the route associated with that element ornot. So, Vm = {0, 1}.

We note that when a route is to be added to the routecache, a row corresponding to this route is always added tothe information table. However, the route is added to theroute cache itself only if it is not identical to a path or a sub-path of any other route already present in the route cache.

Consider an element x ∈ U corresponding to aroute m1m2 . . . mk, mi ∈ M, i = 1, . . . , k. Whena row is added to the information table, the valuesof the condition attributes corresponding to the linksm1m2, m2m3, . . . , mk−1mk are set as 1.

2.4. Decision system in the mobile node

Let the decision system of a mobile node m be{Um, Am∪{d}, Vm, ρm}, where d is the decision attribute.

The decision attribute (d) is a boolean attribute andis positive (+) if the given destination is available inthe associated route and negative (−) otherwise. Letmt ∈ M be a particular destination under considera-tion. Let the following be the routes that have to beadded to the route cache of the current node m: m1m2mt,m1m2mt, m1m2m3mt, m2mt, m1m5m6, m1m5. Thiswill correspond to a universe of elements such as Um ={x1, x2, x3, x4, x5, x6}, with route mm1m2mt associatedwith element x1, mm1m2mt with x2, mm1m2m3mt withx3, mm2mt with x4, mm1m5m6 with x5 and mm1m5

with x6.Then, the value of the decision attribute will be as fol-

lows: ρ(x1, d) =′ +′, ρ(x2, d) =′ +′, ρ(x3, d) =′ +′,ρ(x4, d) =′ +′, ρ(x5, d) =′ −′, ρ(x6, d) =′ −′.

Let A′ be a subset of Am, such that each element in A′

is a link that has the current mobile node m as the tail ofthe link. In other words, A′ has all links to the next hopnodes from the current mobile node m. Let xi and xj bemembers of Um. The indiscernibility relation is defined as

R(A′) = {(xi, xj) ∈ Um × Um | ∀a ∈ A′, ρ(xi, a) =ρ(xj , a)}

That is, among the routes considered by a mobile node

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m, those that have the same next hop are indiscernible andbelong to the same elementary set.

In the example given above, A′ = {mm1,mm2}.We see that the equivalence relation is{{x1, x2, x3, x5, x6}, {x4}}.

Let Y be a subset of Um such that for each element inY , the value of the decision attribute is ′+′. That is, theconcept Y has all the routes (considered by m) that havethe destination mt in it.

Consider an elementary set Xi that has all routes consid-ered by m which have the same next hop mi. Among theseroutes, some may have the destination mt in them. Let pi

denote the ratio of the number of routes with mt in Xi to thetotal number of routes in Xi. The β-positive region consistsof those elementary sets for which the value of pi is greaterthan β.

Many of the elementary sets are too small to be consid-ered as representative of either positive or negative trend.The notion of significant elementary sets [6] is used so thatelementary sets with very few elements (e.g. singletons) areignored. (This is to avoid elementary sets with a single straypositive element falling in the β-positive region.)

3. Routing

In the proposed routing protocol (DSRβ), when a routeis learnt or used by the node, a row is updated in the infor-mation table. Initially, the values of all condition attributesare set to 0. For each link in the route that is learnt or used,the values of the corresponding condition attributes in theinformation table are set to one.

When a next hop is to be found from the information sys-tem, all possible next hops are first considered. If the givendestination is a next hop, then that node itself is chosen asthe next hop. Else, of all the next hops corresponding tothe elementary sets in the β-positive region, the next hopthat is found first, for which the number of entries in theelementary set is not one is chosen as the next hop.

When a source node sends a data packet, it uses the nexthop given by the information table, if available. If it is notpossible to determine the next hop from the information ta-ble, the route in the route cache in that node, if any, is used,else a fresh route discovery is done by initiating a route re-quest. In each of the intermediate nodes through which thedata packet is sent, the next hop is found as described above.

When an intermediate node is forwarding a data packetand it does not find the link to the next hop, it sends aroute error with the information about the deadlink to thesource node. When the source node receives the route er-ror, it deletes in each path of the cache the sub-paths start-ing from the deadlink. In the information tables, for all thelinks in the abovesaid sub-paths, the values of the conditionattributes are cleared to zero. This is done in the informa-

Table 1. ResultsNo. Packet Delivery Ratio(%) Avg. Delay(secs)of DSR DSRβ DSR DSRβ

Nodes β = 0.8 β = 0.825 92.19 92.8 0.9757 0.949130 92.43 92.96 0.7126 0.67935 94.12 94.83 0.3789 0.349840 95.77 97.04 0.1329 0.124845 95.77 96.09 0.2245 0.11250 96.46 96.95 0.1832 0.0751

tion tables in the source node and the intermediate nodesthat forwarded the route error.

DSRβ is similar to DSDV and AODV in that it has in-formation about the next hop through which packets areto be routed to a particular destination. Both AODV andDSRβ have information about next hops only for routesthat have been used or learnt compared to DSDV whichhas entries in routing tables for all nodes in the networksand has an overall picture of the network. AODV maintainsonly a single routing table entry for a particular destination.DSR uses source routing whereas DSRβ routes hop by hop.DSR, DSRβ and AODV have route discovery proceduresfor learning unknown routes. In all four protocols, thereare mechanisms by which the other nodes are made knownabout link failures.

4. Performance Evaluation

The network simulator ns2 [3] is used for the exper-iments. The settings usually used in other experimentalstudies are used here: random waypoint mobility model,rectangular field, constant bit rate traffic sources, LucentTechnologies WaveLAN 802.11. radio model, 2Mbps trans-mission rate, Distributed Coordination Function (DCF)linklayer of the IEEE 802.11 wireless LAN standard. Othersettings include: transmission range of 250 m, field di-mensions 1000 m × 1000 m, maximum speed 5 m/sec,pause time 20 seconds, number of communicating source-destination pairs is 10 and 512 byte data packets. Simu-lations are run for 1000 simulated seconds. The source-destination pairs are spread randomly over the network.

The performance is evaluated using the following met-rics:(i) Packet delivery ratio: The ratio of the data packets deliv-ered to the application layer of the destination to those sentby the application layer of the source node.(ii) Average end-to-end delay: The average delay fromwhen a packet is sent by the source node until it is receivedby the destination node.

For each run, the source destination pairs, the initial po-

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sitions of the nodes and the movement pattern of the nodesare changed randomly. Table 1 shows the packet deliveryratio and end-to-end delay when the number of nodes is var-ied from 25 to 50. A few values of β were tried and β = 0.8was chosen for the detailed study. Each value in the table isthe average of the values got in 15 simulation runs.

Figure 1 shows the packet delivery ratio for both DSRand DSRβ when the number of nodes is varied. It is seenthat DSRβ gives a higher packet delivery ratio than DSR.

Figure 2 shows the average end-to-end delay for bothDSR and DSRβ when the number of nodes is varied. It isseen that the end-to-end delay for DSRβ is less comparedto DSR.

Figure 1. Packet delivery ratio vs. Number ofnodes

Figure 2. Average end-to-end delay vs. Num-ber of nodes

5. Conclusion

This paper presents a study in the use of Variable Preci-sion Rough Sets in Mobile Ad hoc routing. Further work

is in progress to study the effect of changes in the differentexperimental parameters.

This work is a preliminary study on the application of thedirect application of VPRS to MANETs. Further work is inprogress to study possible extensions to Rough Set Theoryand VPRS motivated by this application area of MANETs.

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