Disjoint Multipath Routing in Dual Homing Networks using Colored Trees

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1 Disjoint Multipath Routing in Dual Homing Networks using Colore Trees Preetha Thulasiraman, Srinivasan Ramasubramanian, an Marwan Krunz Department of Electrical an Computer Engineering University of Arizona, Tucson, AZ 2 {pthulasi, srini, krunz}@ece.arizona.eu Abstract Wireless sensor networks (WSNs) employe in monitoring applications require ata collecte by the sensors to be eposite at specific noes, referre to as rains. To improve robustness in ata collection, we consier a ual homing network in which two rains are employe an every noe is require to sen ata to the two rains over link- or noe-isjoint paths. One approach to reuce the number of routing table entries at a noe is to construct two trees, namely re an blue, each roote at a particular rain such that the paths from any noe to the two rains on the trees are link- or noe-isjoint. In this paper, we evelop the first istribute algorithm for constructing colore trees in a ual-homing network whose running time is linear in the number of links. In aition, we show that the average path length may be optimize by employing the generalize low-point concept rather than the traitional low-point concept. I. INTRODUCTION Recent avancements in low-power computing, sensing, an wireless communications have contribute to the emergence of multi-hop wireless sensor networks (WSNs) as a costeffective solution for surveillance/monitoring applications. The ata collecte by sensors in a WSN is require to be eposite at specific noes, referre to as rains. In orer to improve robustness in ata collection, we consier a scenario where two rains are employe an ata from every sensor is require to be eposite at both the rains. As the sensor noes have limite battery energy an computational capabilities, implementation of sophisticate transport layer protocols that guarantee en-to-en reliable transmission is impractical. One approach to achieve the goal of robustness is to employ multipath routing (MPR). MPR operates by transmitting ata over multiple paths. In general, the multiple paths from a source to a estination may have common links (or noes) as long as the share links (or noes) have sufficient resources. To improve the transmission reliability an avoi share-link (or noe) failures, the multiple paths can be selecte to be linkor noe-isjoint. In this case, the MPR approach is referre to as isjoint multipath routing (DMPR). Implementation of MPR an DMPR poses two main challenges []. The first is relate to the computation of loopfree multiple paths. For large networks, a istribute solution that relies only on local information is preferre. Distribute multipath routing algorithms in the literature are evelope in the context of wireless networks. MPR approaches base on Dynamic Source Routing (DSR) [2], [], [] require the estination to select maximally isjoint paths among the receive route requests. MPR approaches base on AODV routing [], [], [], [], [] o not guarantee fining isjoint paths. Protocols such as Directe Diffusion [] that have been evelope specifically for wireless sensor networks o not guarantee isjoint multiple paths an may result in loops after a link/noe failure. The secon challenge of implementing MPR or DMPR techniques is relate to forwaring of ata over the multiple paths. Datagram networks rely on the estination aress in the packet heaer for forwaring packets over one path. To implement MPR or DMPR techniques, every noe must maintain a set of preferre neighbors to reach a estination, such that the paths are loop-free (an isjoint, if neee). The forwaring of packets must be base on estination aress an some aitional information (such as source aress, labels, etc.). The intermeiate noes must be aware of this aitional information or otherwise, it must be carrie in every packet heaer. To reuce the routing table overhea, hence reuce lookup time, a novel multipath routing strategy calle colore trees (CT) was evelope []. Every noe in the network has two preferre neighbors to a estination, namely re an blue. A packet transmitte from a source is marke with one of the two colors. An intermeiate noe that receives the packet forwars it to its preferre neighbor base on the color of the packet. Thus, the routing table at a noe has only two entries for every estination noe. The network may be viewe as two trees, namely re an blue, that are roote at the estination. The two paths from a given source to the root of the two trees are link/noe-isjoint. The colore trees may be constructe by aapting the centralize algorithms evelope for robust multicasting [], [], [], by simply reversing the arcs obtaine in the multicast trees. The first istribute algorithm for constructing colore trees, whose complexity is linear in the number of eges, was evelope in []. A ual homing network (DHN) is traitionally employe in IP-base access networks []. A DHN consists of two eicate routers, referre to as ual homes, through which the noes within the access network are connecte to those outsie the access network. A sensor network with two rains may be viewe as a DHN where the two rains act as the two homes. In orer to overcome a single link (or noe) failure within the network, every noe is require to have a path to the two homes that are link-isjoint (or noe-isjoint). In orer to

2 maintain isjoint routes with minimum routing table overhea, thereby reucing routing table lookup time, we evelop the colore trees to ual homes (CTDH). In this approach, the re tree is roote at one home while the blue tree is roote at the other such that the path from any source to the two homes on the two trees are link-isjoint (or noe-isjoint). Figure shows two trees, one roote at noe an the other roote at, in a ual-homing network with noes an as the ual homes. It may be observe that the paths from any noe to the ual homes on the two trees are noe-isjoint. (a) Re tree roote at. (b) Blue tree roote at. Fig.. An example network with re an blue trees. The path from a source to the two homes (rains) are noe-isjoint on the two trees. Contribution. This paper () proves that 2-ege (or 2- noe) connectivity is a sufficient conition for the existence of a solution to the CTDH problem satisfying link-isjoint (or noe-isjoint) constraint; (2) evelops a istribute solution to the CTDH problem whose complexity is linear in the number of eges; () evaluates the effectiveness of employing generalize low-point concept compare to traitional lowpoint concept; an () evaluates the effectiveness of the trees constructe uner multiple link failures. The rest of the paper is organize as follows: Section II escribes the network moel an problem efinition. Section III evelops the linear-time istribute algorithm for constructing the two colore trees. Section IV presents the performance comparison of the istribute algorithm with traitional an generalize low-point concepts. Our conclusions are presente in Section V. II. PROBLEM STATEMENT AND PRELIMINARIES Consier a network G(N, L) compose of a set of noes N an a set of links L. The links are assume to be bi-irectional. The terminology of arc is use to refer to a irecte link between two noes. An arc from noe i to j is represente as i j. Given ual homes, N, the goal is to construct two trees R an B (referre to as the re an blue trees, respectively) roote at an, respectively, that minimize the average path length from a source to the homes such that the CTDH-LD (CTDH-ND) version of the problem satisfies the link-isjoint (noe-isjoint) path constraint. The isjoint path constraints are state as follows. Let P R s an P B s enote the paths from a noe s to rains an on trees R an B, respectively. Link-isjoint path constraint: s N \ {, } an i, j N i j P R s (i j / P B s ) (j i / P B s ). Noe-isjoint path constraint: s N \ {, } an i N \ {s, } i P R s (i / P B s ). If the two homes are the same, say = =, then we refer to the problem of constructing colore trees roote at a given estination such that the path from any noe to the estination is link-isjoint (noe-isjoint) as the CT-LD (CT-ND) problem. Theorem: A solution to the CTDH-LD (CTDH-ND) problem exists if the network is 2-ege (2-noe) connecte. Proof: Construct a graph G from G with an aitional noe an two biirectional links (, ) an (, ). The graph G remains 2-ege (2-noe) connecte if the graph G is 2-ege (2-noe) connecte. Since G is 2-ege (2-noe) connecte, a solution to the CT-LD (CT-ND) problem exists [] with noe as the estination. As noe has only two links, all the paths on one tree must traverse noe an link as the last noe an link, respectively. Similarly, all the paths on the other tree must traverse noe an link. Removing noe an the links attache to it results in a solution to the CTDH-LD (CTDH-ND) problem, thus proves the theorem. The CTDH-LD (CTDH-ND) problems may be formulate as an integer linear program (ILP) by extening the formulation of the CT-LD (CT-ND) problem evelope in []. The ILP formulation is omitte in this paper ue to space constraints, while the results obtaine from it are use to evaluate the performance of the istribute algorithm evelope here. A. Generalize low-point concept The istribute algorithm for the construction of colore trees to ual homes operates in two phases. The first phase involves istribute epth first search (DFS) numbering of the noes an computing the generalize low point values, istances, an neighbors. The secon phase involves istribute path augmentation for computing the two trees. The DFS numbering an low-point computation helps in ientifying paths for augmentation (uring the secon phase) without backtracking. In orer to reuce the average path length, we employ the generalize low point concept evelope in []. Consier a network in which the noes are numbere in the DFS orer. The low-point value of a noe n is traitionally efine as the lowest DFS-inex of a noe that can be reache from n by using DFS-tree eges an at most one back ege. The low-point path of noe n is the path traverse to reach the low-point noe. The low-point path of a noe n is of the form n i i 2... i k n (k 0) such that: () noe n is the DFS-parent of noe i, (2) noe i j is the DFS-parent A DFS-tree is a tree roote at the rain an the arcs in the tree are irecte away from the rain. A back ege is an ege that connects a higher DFSinex noe to a lower DFS-inex noe. The low-point noe of a noe n is the noe whose DFS-number is the LPV of noe n.

3 of noe i j (2 j k), () the DFS-inex of n is lower than that of n; an () the DFS-inex of n is the lowest among all such possible paths. The algorithm evelope in [] employs the traitional low-point value an path. The generalize low-point value (GLPV) of a noe n is efine as the lowest DFS-inex of a noe that can be reache from noe n by traversing a sequence of noes with increasing DFS-inex with the exception of the last hop. The generalize low-point path of a noe n is of the form n i i 2... i k n (k 0), such that: () the DFS-inex of n is lower than that of i, (2) the DFS-inex of i j is lower than that of i j (2 j k), () the DFS-inex of noe n is lower than that of noe n, an () the DFS-inex of n is the lowest among all such possible paths. The generalize lowpoint neighbor (GLPN) of a noe n is efine as that neighbor of noe n which is on its generalize low-point path. III. LINEAR TIME DISTRIBUTED CONSTRUCTION OF COLORED TREES TO DUAL HOMES The implementation of the istribute algorithm for the CTDH problem is base on the proof of existence of a solution escribe in Section II. Assume that we have the graph G obtaine from the given network G with an aitional noe an two biirectional links (, ) an (, ). We may employ the istribute algorithm evelope for the CT-LD an CT-ND problems with noe acting as the estination, except that noe is a virtual noe. A. Distribute DFS numbering Assume that the noes in G are numbere in the DFS orer starting from noe. The DFS-inex of noe is an without loss of generality, assume that the DFS-inex of is 2. In aition, the (generalize) low-point value of woul be an the (generalize) low-point path shoul traverse noe. As noe oes not exist in the given network, we initialize the DFS-inex of to, the DFS-inex of to 2, an allow to start the DFS numbering phase. The GLPV an GLPN of a noe are compute uring the istribute DFS numbering phase. The algorithm to assign the DFS-inices an compute the GLPV an GLPN is shown in Figure 2. The DFS-inices of all the noes are first initialize to -. We incorporate hop count as a metric to compute the shortest generalize low-point path among those available. Note that the linear-time algorithm evelope in this paper will work with the traitional low-point of a noe, however, the path length optimization cannot be mae as the arcs are force to be on the DFS-tree, except the last hop. The noe numbers in the example network shown in Figure inicate the DFS-inices of the noes when numbere from. The generalize low-point values, istances, neighbors of the noes are shown in Table I. B. Distribute path augmentation The istribute path augmentation is base on the generalize path augmentation technique evelope in []. A generalize version of the same is evelope in [], referre Notation Comment fs[n] DFS-inex of noe n. fsparent[n] DFS-parent of noe n. glpv[n] Generalize low-point value of noe n. glpn[n] Generalize low-point neighbor of noe n. glp[n] Generalize low-point istance of noe n. DFS(parent, n, currfs). if fs[n] > 0 return currfs; 2. fs[n] = currfs; fsparent[n] = parent; currfs = currfs + ;. for every neighbor i parent of n o:.a. currfs = DFS(n, i, currfs);.b. if (fs[i] < fs[n]) an (fs[i] glpv[n]).b.i. glpv[n] = fs[i]; glpn[n] = i; glp[n] = ;.C. else if (fs[i] > fs[n]) an (glpv[i] < glpv[n]).c.i. glpv[n] = glpv[i]; glpn[n] = i;.d. glp[n] = glp[i] + ; else if (fs[i] > fs[n]) an (glpv[i] = glpv[n]) an (glp[i] < glp[n] ).D.i. glpn[n] = i; glp[n] = glp[i] + ;. return currfs; Fig. 2. Algorithm to assign DFS-inices to the noes an compute generalize low-point value an neighbor of a noe. TABLE I GLPV, GLPN, AND GLPD VALUES OF THE NODES IN THE EXAMPLE NETWORK. n glpv(n) glpn(n) glp(n) n glpv(n) glpn(n) glp(n) 2 2 to as the XCT algorithm. The XCT algorithm for the CT-ND or CT-LD problem starts by choosing an arbitrary cycle, (, v,..., v k, ), consisting of at least three noes (k 2). The cycle in one irection is ae to the re tree an the cycle in the other irection is ae to the blue tree. If this cycle oes not contain all of the noes of G, then a subsequent path that starts an ens on that cycle an also passes through at least one noe not on the cycle is chosen for augmentation. The algorithm continues in this manner until all noes have been inclue for augmentation. In orer to ensure link-(or noe-)isjoint path constraints, a partial orering among the noes nees to be maintaine. Due to space constraints, we omit the etails on the partial orering an refer the reaers to [] for an elaborate iscussion. The istribute algorithm evelope in [] implements partial orering by maintaining only local (neighborhoo) information. The overview of the path augmentation process is shown in Figure. For the CTDH problem, the home with DFSinex of 2 initiates the path augmentation proceure. The path search message, SEARCH, receive by a noe, is forware to

4 its neighbor base on the forwaring rules escribe in []. The forwaring rules guarantee that the paths are augmente without backtracking, resulting in a linear time algorithm. The neighbors of a noe are arrange in the increasing orer of their DFS-inices in the neighbor list. Such an arrangement ensures that whenever a path search is initiate from the secon home (noe with DFS-inex 2) through its chil, it follows the low-point path, hence resulting in a path to the primary home. Thus, the key ifference between the working of the algorithm for the CTDH problem from that of the CT problem for one estination, is that the former fins a path between the two homes while the latter fins a cycle. The arrangement guarantees that the paths from any noe to a home will not inclue the other home, unless the other home is an articulation noe 2. Distribute Path Augmentation Algorithm ) Arrange the neighbors in the neighbor list in an increasing orer of their DFS inices. 2) On receiving a TOKEN message, initiate path search along every noe in the neighbor list, one at a time. ) Every noe that receives the SEARCH message forwars it sequentially to every noe in the neighbor list accoring to some forwaring rules. If a new path was augmente through a neighbor, then a flag is set corresponing to that noe. ) Forwar the TOKEN message to every noe if the flag is set for that noe. The neighbor list is traverse in the reverse irection. Every noe finishes its operation an sens a RETURN message back. ) After receiving a RETURN message from all the neighbors to whom the token message was sent, sen RETURN message to the noe that sent the TOKEN message. Fig.. Overview of the steps involve in the istribute algorithm for computing colore trees. We illustrate the working of the algorithm with the example network shown in Figure. Step. Noe 2 starts the path search through noe. Base on the forwaring rules [], the message traverses the generalize low point path until it reaches noe. Noe, being a home, is assume to be a part of the trees, hence sens a SUCCESS message in the reverse irection that the search messages were receive. The links that are ae to the blue tree are shown in Figure (a). Step 2. Noe 2 then initiates a path search through noe. The message traverses the generalize low point path. Figure (b) shows the path ae to the blue tree after this augmentation. Step. Now that noe 2 has attempte to augment paths through all its neighbors, the TOKEN will be sent to noe. Noe attempts the path through noe. The path obtaine is. Figure (c) shows the path ae to the blue tree 2 An articulation noe is one which isconnects the network upon its removal. after this step. Step. Noe passes the token to noe which initiates a search through. The path traverse by the message is. Figure () shows the path ae to the blue tree. The blue an re trees that are generate through this process are shown in Figures () an (e), respectively. IV. PERFORMANCE EVALUATION The linear time istribute algorithm evelope in this paper is evaluate on ranom topologies with 0, 0, 200, an 00 noes. The topologies were constructe using Waxman s moel []. As the solution time of the ILP is prohibitively high for large networks, we compare the results of the istribute algorithm to that of the optimal value for one network topology for each network size. Table II shows the comparison of results obtaine from the istribute algorithm employing traitional low point (TLP) an generalize low point (GLP) concepts to the optimal solution obtaine by solving the ILP (using CPLEX.0 solver []). It is observe that the results obtaine by employing GLP are closer to the optimal than those obtaine by employing TLP. In aition, we observe that the average path length obtaine using GLP is at most % away from the optimal. TABLE II COMPARISON OF RESULTS FROM THE DISTRIBUTED ALGORITHM EMPLOYING TLP AND GLP CONCEPTS TO THE OPTIMAL SOLUTION. Noes Links Average Re Path Length Average Blue Path Length TLP GLP Optimal TLP GLP Optimal For each network size, twenty ifferent topologies with two rains were simulate an the average results are shown in Table III for the CTDH-ND case. It is observe that a significant reuction in the average path lengths is obtaine by employing the generalize low-point concept, which reuces the hop-count on the low-point path. Similar results were obtaine for the CTDH-LD case an are not shown here ue to space constraints. TABLE III COMPARISON OF RESULTS OBTAINED IN RANDOM NETWORK TOPOLOGIES Noes Average of Links USING THE DISTRIBUTED ALGORITHM. Average Re Path Length Average Blue Path Length Average Total Path Length TLP GLP TLP GLP TLP GLP Average Reuction % % % % We evaluate the robustness of the trees constructe using the istribute algorithm uner multiple link failures. Let H an H 2 enote the hop length of the two paths from a noe to

5 (a) (b) (c) () (e) Fig.. An example illustrating the stages of the istribute path augmentation technique. the homes. Given that k links have faile in the network, the probability that both the paths of a noe are affecte, enote by P (H, H 2, k), is given by: P (H, H 2, k) = min(h,k ) i= min(h 2,k ) j= ( H )( H2 i ( L k) )( L H H 2 j k i j The sum of the above probability for all the noes gives the average number of noes that will lose both the paths to the homes uner k-link failures. Table IV shows the average number of noes isconnecte from both the rains for arbitrary two an three link failures. It is observe that employing GLP reuces the number of noes isconnecte from both the homes in the network in comparison to employing TLP. TABLE IV AVERAGE NUMBER OF NODES DISCONNECTED FROM BOTH THE DRAINS of Noes FOR ARBITRARY TWO AND THREE LINK FAILURES. of Links Average of Noes Average of Noes Disconnecte for k=2 Disconnecte for k= TLP GLP TLP GLP V. CONCLUSION This paper evelops a linear-time istribute algorithm for the construction of colore trees in a wireless sensor network employing two rains by moeling the network as a ualhoming network. By allowing the two rains to be roots of the trees, the network is resilient to a single rain failure. This paper also shows that employing the generalize low point (GLP) concept in a DFS tree allows for significant average path length reuction than the traitional low point (TLP) concept. In aition, using GLP rather than TLP allows a network to have greater tolerance for k link failures in terms of the number of noes isconnecte from both the homes. ACKNOWLEDGMENT The research evelope in this paper is supporte by National Science Founation uner grants 02, 00, an EEC-00. ) REFERENCES [] S. Ramasubramanian, M. Harkara, an M. Krunz, Distribute linear time construction of colore trees for isjoint multipath routing, in Proceeings of IFIP Networking, Coimbra, Portugal, May 200, pp. 2. [2] S. Lee an M. Gerla, Split multipath routing with maximally isjoint paths in a hoc networks, in Proceeings of IEEE ICC, 200, pp [] A. Nasipuri an S. R. Das, On-eman multipath routing for mobile a hoc networks, in Proceeings of IEEE International Conference on Computer Communications an Networks, October, pp. 0. [] J. Wu, An extene ynamic source routing scheme in a hoc wireless networks, in Proceeings of th Annual Hawaii International Conference on System Sciences, January 2002, pp. 2. [] M. K. Marina an S. R. Das, On-eman multipath istance vector routing in a hoc networks, in Proceeings of IEEE ICNP, November 200, pp. 2. [] V. D. Park an M. S. Corson, A highly aaptive istribute routing algorithm for mobile wireless networks, in Proceeings of IEEE INFOCOM, April, pp.. [] J. Raju an J. J. Garcia-Luna-Aceves, A new approach to on-eman loop-free multipath routing, in Proceeings of IEEE International Conference on Computer Communications an Networks (ICCCN), October, pp [] A. Valera, W. K. G. Seah, an S. V. Rao, Cooperative packet caching an shortest multipath in mobile ahoc networks, in Proceeings of IEEE INFOCOM, March-April 200, pp [] S. Lee an M. Gerla, AODV-BR: Backup routing in a hoc network, in Proceeings of IEEE WCNC, September 2000, pp.. [] C. Intanagonwiwat, R. Govinan, an D. Estrin, Directe iffusion: A scalable an robust communication paraigm for sensor networks, in Mobile Computing an Networking, August 2000, pp.. [] S. Ramasubramanian, H. Krishnamoorthy, an M. Krunz, Disjoint multipath routing using colore trees, Technical Report, University of Arizona, November 200. [] M. Mear, R.A. Barry, S.G. Finn, an R.G. Gallager, Reunant trees for preplanne recovery in arrbitrary vertex- reunant or ege reunant graphs, IEEE/ACM Transactions on Networking, vol., no., pp. 2, October. [] G. Xue, L. Chen, an K. Thulasiraman, Quality-of-service an qualityof-protection issues in preplanne recovery schemes using reunanttrees, IEEE Journal on Selecte Areas in Communication, vol. 2, no., pp. 2, October 200. [] W. Zhang, G. Xue, J. Tang, an K. Thulasiraman, Linear time construction of reunant trees for recovery schemes enhancing QoP an QoS, in Proceeings of IEEE INFOCOM, Miami, FL, USA, March 200, pp [] J. Wang, V.M. Vokkarane, R. Jothi, X. Qi, B. Raghavachari, an J.P. Jue, Dual homing protection in IP-over-WDM networks, Journal of Lightwave Technology, vol. 2, no., pp., 200. [] B. M. Waxman, Routing of multipoint connections, IEEE Journal of Selecte Areas in Communications, vol., no., pp. 22, December. [] CPLEX Solver,