k-Best Paths in Fault Tolerant Bi-directional Heterogeneous Wireless Sensor Networks

Hossam M.A. Fahmy#1, Salma A. Ghoneim#2 Faculty of Engineering-Ain Shams University

Computer Engineering and Systems Department Abbsassiah 11517, Cairo, EGYPT

#1 hossam.fahmy@ieee.org #2 salmaghoneim@gmail.com

Abstract— In this paper a procedure is proposed to construct a heterogeneous wireless Sensor network (WSN) that is both bi-directional and fault tolerant. Fault tolerance, is based on finding the k-best disjoint paths using trellis graph transformation. Recognizing limited energy in WSN’s the proposed algorithm minimizes energy consumption. Simulation study reveals how metrics, such as average end to end delay, transmission overhead and throughput, compare for unidirectional and bi-directional graphs.

Index Terms— Bi-directional connectivity, Fault tolerance, heterogeneous WSN’s, k-best paths, trellis graph.

I. INTRODUCTION WSN’s gather a large number of small battery operated

devices with sensing and communication capabilities. These sensors are deployed to monitor movement, chemicals, temperature and a number of other environmental phenomena. Since sensors have severe power and bandwidth constraints, the protocols that control their operation must be very efficient. Actuators are resource rich and potentially mobile, they are involved in taking decisions and performing appropriate actions on themselves (e.g. controlled movement). Sensors that perform environmental monitoring and measuring (e.g. turn on water sprinklers to stop the fire) are of limited processing and storage capacity.

Sensor and actuator networks are expected to operate autonomously in unattended environments. Due to the limited transmission radius, the routes between two nodes are usually created through several hops. Thus, it may be necessary to relay a packet over multiple nodes to reach the destination. In a connected graph, there is at least one path connecting every two nodes.

Connectivity is one of the most important properties of a large-scale wireless network. The network or graph is k-connected if it is still connected after the removal of any k-1 nodes. This is a desirable characteristic for reliable communication and load balancing. Fault tolerance in wireless ad-hoc and sensor networks increases with increasing k for which the network is k-connected. The timely recognition of k-disconnectivity is important in order to perform some data or service replication or sensor replacement or activation, before network becomes fault intolerant, leading to partitions. However, in dynamic networks, where mobility and frequent

changes occur in sleep and active status of nodes, such central entity may not exist, or the collection of network information may cause huge communication overhead.

Different reasons can cause a network failure manifested in the loss of a link or a node. As many failure causes are beyond the control of the network providers, the design of survivable networks has been of great interest. Survivability can be defined as network design and management procedures to minimize the impact of failures on the network [1,2]. The techniques to maintain survivability can be classified into three categories: network design, prevention, and traffic management and restoration.

In this paper we propose a two phases procedure to find k-best paths in a bi-directional network. First, we introduce an algorithm that ensures a bi-directionally connected network, then a second algorithm based on trellis graph transformation is applied on the obtained network to find out the k-best disjoint paths between any origin-destination pair.

Section 2 surveys the literature, Sections 3 and 4 present the algorithm that composes a bi-directional network and the algorithm for selecting k-best paths, respectively. Then, Section 5 illustrates the simulation results where a comparison is performed between unidirectional and bi-directional graphs according to the proposed performance metrics which include the average end to end delay, transmission overhead and throughput. Section 6 provides the conclusion.

II. RELATED WORK The literature presents the benefits of using heterogeneous

WSN’s, containing devices with different capabilities. It is reported in [8] that when properly deployed, heterogeneity can triple the average delivery rate and provide a five-fold increase in the network lifetime. The work in [9] introduces another type of heterogeneous WSN’s called actor networks, consisting of sensor nodes and actor nodes. The role of actor nodes is to collect sensor data and perform appropriate actions. In [12], the heterogeneous connected set covers (HCSC) problem is introduced, its objective is finding a maximum number of set covers such that each set cover monitors all targets and is connected to at least one supernode. A sensor can participate in multiple set covers, but sum of the energy spent in all sets is constrained by the initial energy resources.

In WSN’s, the existing routing protocols do not support bi-directional routing, uplink and downlink both simultaneously due to the insufficient resources. While tree-based uplink routing is used for collecting data from the sensor nodes,

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downlink routing is used to send any interests to the sensor nodes by flooding. As a result, it causes a lot of overhead in the downlink direction. In [6], an efficient bi-directional routing protocol with a new address notation for WSN’s, is proposed, thereby minimizing the memory requirement on each node.

Majority of the existing work in fault-tolerant topology control studies the k-vertex connectivity, requiring the existence of k-vertex disjoint paths between any two nodes in the network. Such a requirement is more appropriate for ad-hoc wireless networks, where any two nodes can be a source and a destination. In WSN’s, data is transmitted from sensors to the sink(s), so maintaining a specific degree of fault tolerance between any two sensors is not critical. However, it is rather important to have fault tolerant data collection paths between sensors and sink(s).

The work in [4] introduces the k-degree anycast topology and addresses fault-tolerant topology control in a heterogeneous wireless sensor network consisting of several resource rich super-nodes that are used for data relaying, and of a large number of energy-constrained wireless sensor nodes. Three solutions are proposed for the k-ATC problem: a k-approximation algorithm, a greedy centralized algorithm that minimizes the maximum transmission range between all sensors, and a distributed and localized algorithm that incrementally adjusts sensors’ transmission range such that the k-vertex super-node connectivity requirement is met. Extended simulation results are presented to verify these approaches.

In [1,2] graph theoretic techniques are used to address network survivability issues by finding the k-best (disjoint) paths through a trellis graph, as compared with the majority of the literature which addresses the k successively shortest link disjoint paths.

Three algorithms are proposed in [5] with excellent accuracy for local confirmation of global k-connectivity. However, they all have problems confirming k-disconnectivity.

In [3], Fault-tolerant topology control algorithms in wireless ad-hoc networks and sensor networks were considered. Two algorithms are presented, a centralized greedy algorithm, FGSSk, and a localized algorithm, FLSSk. Both algorithms can preserve k-connectivity and are min-max optimal. Although FLSSk outperforms other localized algorithms in random networks in terms of power consumption, it does not give any performance bound on power consumption as many centralized algorithms do [7].

In Sections 3,4 we propose a procedure that finds in its first phase the bi-directional links in a network and applies in its second phase trellis transformation to find k-best paths.

III. PHASE 1: FINDING A BI-DIRECTIONAL NETWORK A heterogeneous WSN graph topology is represented by

(G=V,E,c) where V is the set of vertices, E is the set of edges, c is the transmission power (cost) between any two nodes, and nodes are classified into supernodes and sensor nodes. Graph G as generated by an algorithm ALG is bi-directional, if for any two nodes u,v∈V(GALG), u∈NALG(v) implies v∈NALG(u). For any topology generated by ALG, node u is said to be bi-directionally connected to node v (denoted u v) if there exists

a path (p0 = u, p1,…, pm−1, pm = v) such that pi pi+1, i = 0, 1,… ,m− 1, where pk∈V(GALG), k=0, 1, … ,m. It follows that u v if u p and p v for some p∈V(GALG).

A bi-directionally connected network topology facilitates link level acknowledgment, which is critical to packet transmissions over unreliable wireless media. Bi-directionality is also very important to floor acquisition mechanisms such as the RTS/CTS mechanism in IEEE 802.11.

Fault tolerant bi-directional algorithms depend on the cost c between any two nodes (sensor to sensor, sensor to super, or super to super). If the cost c is less than or equal to the sensor transmission range, which means that the two nodes are in the neighbor set of each other, then the connecting edge can be used as bi-directional edge and is added to the set Eb of bi-directional edges constructed by Algorithm Bi-directional proposed in what follows: Algorithm: Bi-directional Input: graph (G (V, E, c), N, M),

V = {n1, n2,..., nN, nN+1,..., nN+M} E = set of edges between nodes i and j {(ni,nj)|1≤i,j≤N+M and i≠j}, C=c(u,v) represents the power requirement for both nodes u and v to establish a bi-directional communication link between u and v. N = number of sensor nodes M = number of supernodes i,j = indices for the nodes in the graph.

Output: Graph Gb (V, Eb, c) where Eb ={(ni, nj)| dist (ni, nj)≤ Tx } where dist ( ) is the euclidean distance function 1. Tx-sensor = transmission range of sensor node 2. Tx-super = transmission range of supernode 3. Eb= φ 4. for each (ni, nj) E do 5. if (i ≤ N) then /* Sensor to any other nodes (sensor/super) edges */ 6. if (c(ni, nj)≤ Tx-sensor) then 7. Eb = Eb (ni, nj) (nj, ni)); 8. end if; 9. end if; 10. if (i > N) and (j > N ) then /* Super to super

edges */ 11. if (c(ni, nj) Tx-super) then 12. Eb = Eb (ni, nj) (nj, ni) ; 13. end if; 14. end if; 15. end for

The output graph is a bi-directional graph Gb(V, Eb, c) where the bi-directional edges have the same cost c.

In steps 1,2 we determine the maximum transmission range for both sensor nodes and supernodes, in Step 3 we initialize the empty edges set Eb. Steps 4 till 15 are looping to check the bi-directional connectivity between every two pair of nodes (sensor-sensor or sensor- super connection).

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Step 4 starts the connectivity check for each edge between two nodes i,j belonging to the set E of edges. Step 5 selects only sensor nodes, Step 6 checks if the edge cost c(ni,nj) is less than or equal to the sensor transmission range, in order to satisfy bi-directional connectivity such that both edges {(ni,nj),(nj,ni)} can be added to the initially empty bi-directional edges set Eb, as shown in Step 7. Steps 8 and 9 terminate dealing with sensor nodes.

Steps 10 through Step 14 repeat the previous process for supernode to supernode bi-connectivity.

This algorithm complexity is found to be O(n2). Fig.1 shows an example graph, where Tx-sensor=12

and Tx-super=20, after applying our algorithm on this graph we get the graph shown in Fig.2 which contains the new set of edges Eb that satisfy the bi-directional connectivity between graph nodes.

sensor node supernode

Fig. 1 G (V, E, c)

Fig. 2 Gb (V, Eb, c)

IV. PHASE 2: FINDING K-BEST PATHS BY TRELLIS TRANSFORMATION

The k-best path problem lists the k mutually disjoint (exclusive) paths connecting a source-destination pair in a network. Mutually disjoint implies that there is no link, or node dependencies between listed paths. The selection of the k-best disjoint paths can take into account many factors, such as selection of the shortest paths (hence minimizing delay), minimization of the bandwidth allocation (given the bandwidth demanded by customers), and maximization of network throughput. “Best” paths are those paths which are as diverse as possible (i.e., there are k-disjoint paths, if the network topology allows), and therefore will maximize chances of survivability, or ensure a graceful degradation, (i.e. display fault tolerance) in the event of a network fault.

A directed graph G = (V, E) is a structure consisting of a finite set of nodes V={v1, v2, ..., vn} and a finite set of links E={(vi, vj) |vi, vj∈V and vi ≠ vj}, where each link is an ordered pair.

A trellis graph [2] is defined as a directed graph G=(V, E) with nodes and directed links that satisfy the following conditions:

(i) The node set V is partitioned into L (mutually disjoint) subsets V1, V2, ..., VL, such that |Vi| = |Vj| = H, 1 ≤ i, j ≤ L.

(ii) Links that connect nodes of consecutive subsets Vl and Vl+1, i.e., if (vi, vj) ∈ E, then vi ∈Vl and vj ∈Vl+1, 1 ≤ l < L.

The depth of the trellis graph is T. A k-trellis is a trellis graph

with two additional properties: (i) It has two more nodes s∈ V0 and t∈ VL+1, such that

(s,vi)∈ E, for every vi∈ V1 and (vj,t)∈ E, for every vj ∈ VL, 1 ≤ i, j ≤ H.

(ii) The node vi of the set Vl is connected (where possible) with k = 2g+1 nodes {vi-g, ..., vi, ..., vi+g} of the set Vl+1, where 1 ≤ i ≤ H, 1 ≤ t < L and g = 1, 2, ..., (H-1)/2.

The depth of a k-trellis graph is L+2.

Also define a partition L(G,s) of the node set V with respect to source node s to be:

L(G,s)= {AL(s,l) |s∈ V, 0≤ l <Ls, 1≤ Ls < |V|} where AL(s,l), 0≤ l<LS are the adjacency-level sets and Ls is the length in hops of the partition L(G,s) such that:

AL(s,l)={u| d(s,u)=l, 0≤ l ≤ Ls} where d(s,u) denotes the distance in hops between nodes s and u in G. Fig.3 illustrates partitioning for a source node s.

The s-cost φ(x,y) of link (x,y) is defined to be the minimum cost of the path P={s,…,x,y} from s to y through node x.

AL(s,0)={s}, AL(s,1)={A,B,C}, AL(S,2)={D,E,F}, AL(s,3)={J,K}, AL(s,4)={t}

Fig.3 Graph partitioning with respect to s [2]

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The following algorithm is used for finding the k-best paths. Algorithm: k-best paths by trellis transformation Input: Bi-directional graph G (V, E, c, s, t)

V = {v1, v2,..., vn} E = { (vi,vj) | vi,vj V and vi ≠ vj} s source, t destination, N(t) is the neighbor set of nodes of node t; AL(s, l), 0 l Ls, are the adjacency-level sets

Output: Set of k-best paths Pki

1. Partition G (V, E, c) into L(G,s) L(G,s) = {AL(s, l) | 0 l Ls , 1 ≤ Ls< |V|}; AL(s, l)={u | d(s,u)=l, 0 l Ls};

2. Disconnect G (V, E, c) into G’, G” G’=G’(V’, E’, c) | V’=V- {t}, E’=E - {(x,t) |xN(t)}; G“=G“(V“,E“,c)|V“={t} N(t),E“={(x,t)|x N(t)};

3. while there are vertical {

3.1. while ( (x,y) AL(s, l)|x,y V’) do /* Operation P1: Removing a vertical link by adding 0-cost horizontal link via added dummy node x’ */

{ Calculate φ(x,v) | min{φ(x,v)|v AL(s,l-1)}>min{φ(y,u)|uAL(s, l-1)} or (min{φ (x,v) |v AL(s, l-1)}= min{φ(y,u) | u

AL(s, l-1)} and Σ φ(x,v) ≥ Σ φ(y,u); v N(x)∩AL(s,l-1) u N(y)∩AL(s,l-1) move (x) to AL(s, l+1); Add (x’| c(x,x’)=0, φ(x,x’)=min{φ(x’,v)|v AL(s, l-1)}, φ(x,y)= min{φ (y,v) | v AL(s, l-1)}+c(x,y));

} end while;

3.2. while (N(x)∩AL(s, l-1) = φ|x V’, x AL(s, l)) do /* Operation P2: Reshaping the graph after adding a dummy node x’ */

{ move (x) to AL(s, l+1)| φ(x,y)= min{φ(y,v) |v

AL(s, l-1)}+c(x,y); }

end while; } end while;

4. /* Merge again the graphs G’, G’’ */ G trellis=G’ G’’;

5. Add ∞-cost links to complete the trellis graph; 6. Select k-best paths

Pki= {P(s,t) | min{φ(s,t)}, 1≤ i≤ k}; Step 1 partitions the graph into levels based on

the distance in hops between nodes, starting from

the source node. Step 2 splits the graph into two graphs, G’ that contains all nodes and edges without the destination node and its edges, the other graph G” has the destination node and the edges connected to it.

Step 3 loops on the graph G’ to remove the vertical edges and reshape the graph. Step 3.1 applies Operation P1 that adds a dummy node and a 0-cost link. Step 3.2 applies Operation P2 to reshape G’.

Step 4 merges the two graphs G’ and G”. Step 5 adds ∞-cost links to complete the trellis graph. Finally, Step 6 identifies the k-best disjoint paths, they are those with the lowest cost.

This algorithm complexity is O(n3), another implementation is proposed in [1] with complexity O(n2).

It is to be noted that selecting the k-best disjoint paths does not imply the k-shortest paths. In Fig.3, there is one shortest path from s to t whose cost is 6, while the costs of the 2-best (disjoint) paths are 9 and 10; removing the links involved in the shortest path implies that no links connecting to t remain, therefore the algorithm terminates with only one path.

V. SIMULATION RESULTS In our simulations the following metrics are calculated:

1) Average Delay: The time elapsed between sending and receiving packets: _ _ _ _ ∑ _ _ __ _ _

2) Transmission Overhead: _ ∑ _ ∑ _∑ _

3) Throughput: The ratio of the number of packets received to the number of packets sent: ∑ _∑ _

Simulations were performed using the Global Mobile

Information System Simulator (GloMoSim 2.03) [10]. It is a library-based sequential and parallel simulator for wireless networks, its library has been developed using PARSEC (Parallel Simulation Environment for Complex Systems [11]). GloMoSim 2.03 was chosen for its modularity and extensibility.

For unidirectional and bi-directional topologies, Fig.4 compares the average end to end delay, Fig.5 shows the compared transmission overhead, Fig.6 illustrates the throughput, and Fig.7 plots the number of obtained disjoint paths for a number of nodes ranging from 20 up to 200.

VI. CONCLUSION While minimizing the total power consumed by

sensor nodes, an approach is proposed to tackle fault-tolerance in bi-directional heterogeneous WSN’s. A two

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phase procedure with overall complepresented, finding a bi-directional identifying the k-best paths using transformation. Comparing the unidirecdirectional topologies, it was found by the throughput of the bi-directional graphdue to the exclusion of unidirectional linend-to-end delay in a unidirectional grapto the increase in throughput which causThe transmission overhead of the bi-direhigher due to the search for bi-directionalso revealed that the k-connectivity increases as the density of nodes increases.

Fig.4 Average delay

Fig.5 Transmission overhead

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Fig.7 Number of disjo

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