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An Interference-Aware Channel Assignment Scheme for Wireless Mesh Networks Arunabha Sen and Sudheendra Murthy

Samrat Ganguly and Sudeept Bhatnagar

Dept. of Computer Science & Engineering Arizona State University Tempe, Arizona 85281 Email: {asen, sudhi}@asu.edu

Broadband Computing Division NEC Laboratories America, Inc. Princeton, NJ 08540, USA Email: {samrat, sudeept}@nec-labs.com,

Abstract— Multichannel communication in a Wireless Mesh Network with routers having multiple radio interfaces significantly enhances the network capacity. Efficient channel assignment and routing is critical for realization of optimal throughput in such networks. In this paper, we investigate the problem of finding the largest number of links that can be activated simultaneously in a Wireless Mesh Network subject to interference, radio and connectivity constraints. Our goal is to activate all such links and we present an interference aware channel assignment algorithm that realizes this goal. We show that the Link Interference Graph created by utilizing a frequently used interference model gives rise to a special class of graphs, known as Overlapping Double-Disk (ODD) graphs. We prove that the Maximum Independent Set computation problem is NP-complete for this special class of graphs. We provide a Polynomial Time Approximation Scheme (PTAS) for computation of the Maximum Independent Set of an ODD graph. We use this PTAS to develop a channel assignment algorithm for a multiradio multichannel Wireless Mesh Network. We evaluate the performance of our channel assignment algorithm by comparing it with the optimal solution obtained by solving an integer linear program. Experimental results demonstrate that our channel assignment algorithm produces near optimal solution in almost all instances of the problem.

I. I NTRODUCTION Multichannel communication in a wireless mesh environment with routers having multiple radio interfaces significantly enhances the network capacity [18], [20]. Efficient channel assignment and routing schemes are crucial for realization of optimal throughput in such networks. It is well known that commonly used 802.11a/b/g protocols support multiple channels. In the Wireless Mesh Network (WMN) considered in this paper, each node is equipped with multiple radios (Network Interface Cards) and each radio can be tuned to a channel. A pair on nodes (a, b) can communicate with each other if they are within the transmission range of one another and at least one radio in a shares a common channel with a radio in b. Clearly, assignment of channels on the links of the WMN plays a critical role in determining the achievable throughput. We investigate the following question: Given the (i) locations of the wireless mesh routers, (ii) transmission and interference ranges of the transmitters, (iii) the number of channels available on each link and (iv) the number radio interfaces available at each router, what is the largest number

of links that can be activated simultaneously subject to interference and radio constraints so that the resulting network is connected? Our goal is to activate all such links and we present an interference-aware channel assignment algorithm that realizes this goal. Our channel assignment scheme is traffic unaware in the sense that the channels are assigned without taking into account traffic pattern or the paths to be taken for establishing connections between source-destination node pairs. Changes and adjustments in the assignments are made only when new routers are added to the network or some existing routers are disabled due to failure. Such traffic unaware channel assignment schemes have been used by other researchers [3]. Raniwala et. al on the other hand have proposed channel assignment algorithms that require prior knowledge about traffic patterns and communication paths [18]. In this paper, we show that the Link Interference Graph constructed with the widely used interference model gives rise to a special class of graphs known as Overlapping DoubleDisk (ODD) graphs. We prove that the Maximum Independent Set (MIS) computation problem is NP-complete, even for this special class of graphs. The contributions of this paper are summarized below. • •

•

•

Novel characterization of the Link Interference Graphs as Overlapping Double-Disk graphs. Development of a Polynomial Time Approximation Scheme (PTAS) for computation of MIS of an ODD graph. Development of a channel assignment algorithm with an objective of activating the largest number of links subject to interference and radio constraints. Comprehensive performance evaluation of the heuristic solution in comparison with the optimal solution obtained by solving an integer linear program.

The rest of the paper is organized as follows. Section II provides a summary of related work in this area. Section III establishes the network model and formulates the problem. Section IV discusses the interference model. Section V presents the PTAS and the performance guarantee proofs. Section VI presents the optimal and heuristic solution for the channel assignment problem. Section VI also describes the

simulation environment and results and section VII wraps up the paper with conclusion. II. R ELATED W ORK The advantages of deploying multiple channels and multiple radios in wireless mesh networks is obvious and several researchers in the last few years have been studying various aspects of such deployment [1], [3], [4], [8], [10]–[21]. Most of the work in this area can be classified into four different groups, (i) network capacity and related issues, (ii) channel assignment, (iii) routing and (iv) joint routing and channel assignment. Issues related to capacity of wireless networks and impact of multiple radios and interfaces have been studied by several researchers [8], [11], [13], [14]. Modeling the capacity of multiple channels and interfaces and understanding the benefits was studied by authors in [3], [11], [13], [21]. The authors in [12] provide a capacity model to understand the impact of the ratio between number of radios and channels on the system performance in the asymptotic case. The authors in [11] provide capacity model for multiple channels, using which feasibility of a rate matrix can be verified. The authors of [3], [21] provide integer linear program formulation for throughput optimization in a mesh network. They study the impact of interfaces and channels on the overall throughput. Issues related to routing and its impact on capacity maximization have been studied by [4], [10], [20]. Draves et. al. in [4] proposed a metric WCETT, which is suitable for mesh networks with multiple channels. The network considers the channel interference and bandwidth apart from ETX measure. The authors in [20] consider the problem of creating a survival topology in a multichannel scenario and proposes a bandwidth aware routing algorithm. Issues related to channel assignment have been studied in [3], [15], [16]. The authors in [15] extend the notion of Conflict (or Link Interference) Graph to multiple radio environment and use this multiradio conflict graph to model interference between the routers. The authors in [16] study the impact of 802.11 in long distance point-to-point environment using directional antennas. They showed that the problem of minimizing the mismatch between link capacities desired by the network operator and that achieved under a channel allocation is NP-complete. Several researchers have investigated the routing and channel assignment problem simultaneously [1], [12], [17]–[19]. In [18], the authors have proposed a framework for centralized channel assignment scheme for maximizing the throughput. In [17], the authors proposed an adaptive channel assignment based on the congestion on a given link. Bandwidth aware routing for wireless networks with interference constraints was studied in [9]. Although various aspects of channel assignment in a multichannel multiradio wireless mesh networks have been investigated, to the best of our knowledge prior efforts did not characterize the nature of the Link Interference (or Conflict)

Graph and exploit such characterization for the development of efficient channel assignment algorithm. III. P ROBLEM D EFINITION In this section, we formally describe our system model and notations. The system model described here is consistent with the models considered by other researchers [1], [21]. Most of the researchers model a WMN as an undirected graph. Since the wireless routers are distributed over a geometric region (often a two dimensional plane), we consider the network geometry in our model and exploit it to design efficient algorithms. To the best of our knowledge, prior research in the area did not use any geometric information in their proposed solutions. At the first level of abstraction, we view the routers of a WMN as some points (pi , . . . , pn ) (specified by their x, y coordinates) on a two dimensional plane. Associated with each point pj , 1 ≤ j ≤ n is a transmission range, RT and an interference range, RI , RI > RT (we assume that all routers have identical transmission and interference ranges). At the second level of abstraction, we construct a graph G = (V, E), in which each node represents a point on the plane (router) and there is an edge from node vi to node vj if the Euclidean distance between the corresponding points pi to pj is less than or equal to the transmission range RT . This assumption implies that each router has a circular coverage area with the center of the circle at the location of the router. The circular coverage area associated with point pi (and node vi ) will be referred to as the disk associated with the point pi (and node vi ). The graph G = (V, E) will be referred to as the Potential Communication Graph (PCG). A link between any two nodes in this graph indicates that this pair of nodes can communicate with each other if their transmitters and receivers are assigned the same channel. It may be noted that even though these nodes are within the communication range of each other, they may not be able to communicate with each other unless the same channel is assigned to both of them. A Potential Communication Graph is shown in figure 1. The unit disk graphs [2] are a special class of graphs, where each node represents a point in the plane and two nodes have an edge between them if the unit radius disks associated with the points intersect. It may be noted that the Potential Communication Graph is a unit disk graph. It was noted earlier that our channel assignment algorithm does not consider the traffic pattern in the network. In the absence of the load information between source-destination pairs in the network, a good channel assignment strategy would be to do channel assignment in such a way that the resulting communication graph can support as many simultaneous active links. The problem considered in this paper is as follows: Given • L, the location of the wireless routers • RT , the transmission range • RI , the interference range • N , the number of available channels and • K, the number of available radios at each of the routers,

Fig. 1. Potential Communication Graph of 6 mesh routers

Fig. 2. Overlapping Double-Disks (ODD) of links (a, b), (c, d) and (e, f ). ODDs of links (b, c) and (c, e) not shown for clarity.

the problem is to assign channels such that the number of links that can be activated simultaneously is maximized subject to radio, interference constraints and the resulting graph is connected. We view this network design problem as a variation of maximum K-colorable subgraph problem. Specifically, we want to assign colors (channels) to the links of the PCG graph in such a way that (i) two links that interfere with each other are not assigned the same color, (ii) the number of colors used in the process does not exceed the number of available channels N and (iii) for each node v of the PCG, the number of colors used to color the edges incident on v, does not exceed the number of radios K. Problem 1: Given a graph G = (V, E) and integers, R and K, such that R ≤ K and a set of S incompatible pairs of edges, S = {(ei1 , ei2 ), (ei3 , ei4 ), . . . , (eip−1 , ep )}. What is the largest number of edges that can be colored with K colors so that (i) no two incompatible edges have the same color and (ii) no vertex in G has edges incident on it colored with more than R colors? A special case of problem 1, when R = K and the set of incompatible edges S comprises of only those pairs of edges that have a common endpoint, then the problem can be stated as follows. Problem 2: Given a graph G = (V, E), and an integer K. What is the largest number of edges that can be colored using at most K colors so that no two adjacent edges have the same color? The node coloring version of problem 2 (given below) is known as maximum K-colorable subgraph problem in the literature [22]. Problem 3: Given a graph G = (V, E), an integer K. What is the largest number of nodes that can be colored using at most K colors, so that no two adjacent nodes have the same color?

Fig. 3.

Link Interference Graph

IV. I NTERFERENCE M ODEL : OVERLAPPING D OUBLE -D ISK G RAPHS In this section, we focus our attention on the interference graph between the links of the PCG. We follow the interference model used in [1]. Simultaneous transmission on a common channel on two distinct edges e1 and e2 of PCG connecting nodes (u1 , v1 ) and (u2 , v2 ) respectively are said to interfere with each other if minimum {d(u1 , u2 ), d(u1 , v2 ), d(u2 , v1 ), d(v1 , v2 )} ≤ RI , where d(ui , vj ) indicates the Euclidean distance between the nodes ui and vj and RI indicates the interference range. The Link Interference Graph LIG is constructed as follows: Corresponding to every link in PCG, there is a node in LIG and two nodes in LIG have an edge between them only if the corresponding links interfere with each other. The notion of LIG is not new and has been used in [1], [21]. However, in all previous studies LIGs were treated as any arbitrary graphs. We show that LIGs are not arbitrary graphs and belong to a special class, which we refer to as Overlapping Double-Disk (ODD) graphs. Given the locations (p1 , . . . , pn ) of the routers on a two dimensional plane, we draw a line connecting points pa and pb to indicate the link la,b between the routers, if the distance between pa and pb less than or equal to RT . Similarly, we draw a line connecting points pc and pd to indicate the link lc,d between the routers if the distance between pc and pd less than or equal to RT . In order to determine if the links la,b and lc,d interfere with each other, we do the following: We draw a circle with centers at the points pa , pb , pc , pd with radius RI /2. Since d(pa , pb ) ≤ RT and RI ≥ RT , the circles with centers at pa and pb will overlap, and create a figure of the form shown in 2. The same thing will happen for the circles with centers at pc and pd . We refer to this figure as Overlapping Double-Disks (ODD). The mid-point of the line joining the centers of the two disks will be referred to as the center of the double disks. The links la,b and lc,d will interfere with each other if and only if the corresponding ODDs intersect. Since the LIG is the intersection graph of ODDs, we will refer to the LIG as an ODD graph shown in figure 3. Next, we show that the class of ODD graphs is a proper

subset of the set of all graphs. In other words, not all graphs are ODD graphs. We establish our claim with the help of an example. Consider the star graph with 12 nodes {v0 , v1 , . . . , v11 } with v0 at the center and the 11 edges {(v0 , v1 ), (v0 , v2 ), . . . , (v0 , v11 )} as shown in figure 4.

Fig. 4. Star graph with 12 nodes

Fig. 5. Not all graphs are ODD graphs

We claim that this is not an ODD graph. The ODD corresponding to node v0 intersects with the ODDs associated with nodes v1 through v11 . However, the nodes v1 through v11 are not adjacent to each other and as such the ODDs associated with them do not intersect with each other. Suppose that in figure 5 the ODD in the center corresponds to the node v0 . Since v1 is adjacent v0 , the ODD corresponding to v1 will intersect the ODD corresponding to the node v0 . Similarly, since v2 is adjacent v0 , the ODD corresponding to v2 will intersect the ODD corresponding to the node v0 . However, since v2 is not adjacent v1 , the ODD corresponding to v2 will not intersect the ODD corresponding to the node v1 . Continuing such argument, we need to have the ODDs corresponding to the nodes v1 through v11 intersect the ODD at the center (i.e., the ODD corresponding to node v0 ) but not intersect each other. However, from figure 5 it is clear that existence of 11 such ODDs that intersect with the central ODD but do not intersect with each other is impossible. This is true because after 7 ODDs in figure 5 (11 in the worst case), there is no room to place another ODD that intersects with the central ODD, but does not intersect with any other. Accordingly, a star graph with 12 nodes cannot be an ODD graph. The geometric proof establishing 11 to be the maximum number of ODDs is not provided here as it is outside the context of this paper. V. PTAS FOR THE MIS P ROBLEM IN ODD G RAPHS In the previous section, we established that the LIGs are ODD graphs and ODD graphs form a proper subset of the set of all graphs. If a problem is NP-complete for the general class of graphs, it may or may not be NP-complete for a special class of graphs. The chromatic number problem is a case in point. It is NP-complete for the general graph but trivial for trees. The Maximum Independent Set (MIS) problem is NP-complete for the general graph. However, from that result alone we cannot claim that the MIS problem for the LIGs are NP-complete as the LIGs are ODD graphs and ODD graphs are a special class of graphs. Next, we show that the MIS problem for LIGs is NP-complete. Theorem 1: The MIS problem for LIGs are NP-complete.

Proof: We have already established that the LIGs are ODD graphs. Consider a special case of the ODD graph, where the two disks overlap completely (i.e., we see only a single disk). In this special case, the ODD graphs reduces to unit disk graphs [2]. Since the MIS Problem for the unit disk graphs is NP-complete [2] and unit disk graphs are restricted version of the the ODD graphs, we can claim that the MIS problem for the ODD graphs is also NP-complete. Since the MIS problem for the ODD graphs is NP-complete, we need an approximation algorithm to solve the problem. An approximation algorithm for a maximization problem Π is said to provide a performance guarantee σ (σ ≤ 1), if for all instances I of Π, ratio between the approximate solution produced by the algorithm to the optimal solution is at least as large as σ. A Polynomial Time Approximation Scheme for a maximization problem Π is a polynomial time approximation algorithm that given any instance I of Π and a specified ǫ returns a solution that is at least as large as (1 − ǫ) times the optimal solution. In the following, we present a PTAS to solve the channel assignment problem. We use the shifting strategy proposed in [6] and used by Hunt et. al [7] to solve the MIS problem of unit disk graphs. Erlebach et. al. [5] also used a similar strategy to develop PTAS for intersection graphs of disks with different diameters. We first describe the basic idea of the PTAS for MIS of the ODD graphs. Given a set of n ODDs distributed in a two dimensional plane A, we first divide the area into strips of certain width. All the intervals are top closed and bottom open. Given an ǫ > 0, we calculate the smallest integer k such that k/(k + 1) ≥ 1 − ǫ. We partition the ODDs into r disjoint sets by removing the ODDs in the horizontal strips congruent to i mod(k + 1), 0 ≤ i ≤ k. It may be noted that different values of i will give rise to different partitions. Two such partitions are shown in figures 6 and 7. The approximate algorithm for computation of a MIS of a ODD graph is given next. Algorithm 1 PTAS: MIS of ODDGraphs(L, A, ǫ) Input: Locations L of n ODDs distributed in a two dimensional plane A, ǫ < 0.5 Output: An Independent Set that is at least as large (1 − ǫ) times the MIS. 1: Find the smallest integer k/(k + 1) ≥ 1 − ǫ 2: Divide the plane A into horizontal strips of width 4. 3: for i = 0 to k do 4: Partition the set of ODDs into r disjoint sets Gi,1 , Gi,2 , . . . , Gi,r by removing all the ODDs in every horizontal S strip congruent to i mod(k + 1) 5: Gi = 1≤j≤r Gi,j 6: for j = 0 to r do 7: Compute the MIS in Gi,j 8: end for S 9: M IS(Gi ) = 1≤j≤r M IS(Gi,j ) 10: end for 11: Independent Set IS(G) = max0≤i≤k M IS(Gi )

Overall execution time complexity of the algorithm will be O(nO(k) ) where n is the total number of ODDs in the two dimensional plane A. B. Performance Guarantee of Approximate Algorithm

Fig. 6.

Grouping 1 of the strips

In this section, we prove that the size of the independent set computed by the approximate algorithm is close to the size of the MIS. Formally, we will show that |IS(G)| ≥ (1 − ǫ)|OP T (G)|. The proof of this claim follows along the same lines as the proof for the unit disk graphs given in [7]. From the description of the algorithm, it is clear that during each iteration (for different values of i), the algorithm excludes some ODDs from independent set computation. First, we establish the fact that the number of ODDs excluded from independent set computation is a small fraction of the MIS, at least during one iteration of the algorithm. During each iteration of i, we exclude the ODDs (and hence the vertices in the ODD graph) in the strips j1 , j2 , . . . , jp where jl = i mod(k + 1), 1 ≤ l ≤ p. Suppose that Si denotes the set of ODDs excluded during the ith iteration and ISopt (Si ) denotes the vertices (corresponding to the ODDs) in the set Si that are part of the optimal (maximum) independent set OP T (G). Lemma 2: max |OP T (Gi )| ≥

Fig. 7.

Grouping 2 of the strips

A. Computation of MIS(Gi,j ) Consider the ODDs whose centers lie in a rectangular slice RS of height 4k and width 4 as shown in figure 8. Since Gi,j is an ODD graph, RS can contain no more than 6k+3 mutually non-intersecting ODDs. This gives a bound on the size of the MIS of ODDs whose centers lie in RS. Furthermore, removal of ODDs in RS divides the set of ODDs into disjoint sets L and R. Suppose that the ρ is the number of ODDs in RS. By examining each combination of at most 6k+3 ODDs (or nodes in Gi,j ), we get all the independent sets in RS. This can be done in O(ρ6k+3 ) L

RS

R

0≤i≤k

k |OP T (G)| k+1

Proof: Noting that different strips are considered during different iterations, we make the S following observations: 0 ≤ t=k i, j ≤ k, i 6= j, Si ∩ Sj = ∅ and t=0 Si = V (G) From the above equation, |ISopt (S0 )| + |ISopt (S1 )| + . . . + |ISopt (Sk )| = |OP T (G)| This leads to: min0≤t≤k |ISopt (St )| ≤ =⇒ max |OP T (Gi )| ≥ 0≤i≤k

|OP T (G)| . k+1

k |OP T (G)| k+1

Theorem 3: |IS(G)| ≥ (1 − ǫ)|OP T (G)| Proof: We note that |OP T (Gi )| =

j=r X

|OP T (Gi,j )|

j=1

Fig. 8.

Rectangular slice RS divides the area into L and R

Suppose that the disks associated with the independent sets in RS are {ISRS1 , ISRS2 , . . . , ISRSt }. The following principle of optimality holds for L, R and RS. M IS(L ∪ RS ∪ R) = max1≤i≤t (M IS(L − L ∩ ISRSi )+ |ISRSi | + M IS(R − R ∩ ISRSi )) The computational time for the procedure will be given by T (L ∪ RS ∪ R) ≤ T (L) + O(ρ6k+3 ) + T (R)

From the above equation and lemma 2, we make the following derivation |IS(G)| ≥ max |IS(Gi )| 0≤i≤k j=r X

|IS(G)| ≥ max

0≤i≤k

|IS(G)| ≥ max

0≤i≤k

j=1 j=r X

|IS(Gi,j )|

|OP T (Gi,j )|

j=1

|IS(G)| ≥ max |OP T (Gi )| 0≤i≤k

|IS(G)| ≥

k |OP T (G)| k+1

|IS(G)| ≥ (1 − ǫ)|OP T (G)|

VI. C HANNEL A SSIGNMENT IN M ULTICHANNEL M ULTIRADIO W IRELESS M ESH N ETWORK In this section, we present a technique of solving the channel assignment problem optimally using Integer Linear Programming (ILP). We then present a heuristic that applies the PTAS described in the previous section repeatedly to compute the channel assignment efficiently. We evaluate the channel assignment produced by the heuristic with that of the optimal. A. Optimal solution using ILP Let G(V, E) and G′ (V ′ , E ′ ) be PCG and LIG respectively. Let N be number of channels in the network. Let k be number of radios per node. The binary variables are defined as follows. ∀v ∈ V, 1 ≤ i ≤ N,  1, if channel i is used on node v v Xi = 0, otherwise ∀e ∈ V ′ , 1 ≤ i ≤ N,  1, if channel i is used on both end points of e Yie = 0, otherwise

The objective is to maximize the number links that can be activated simultaneously. X X Yie M aximize ∀e∈V ′ 1≤i≤N

The set of constraints are as follows. • Radio constraints: The number of channels assigned to a node should not exceed the number of radios on the node. That is, ∀v ∈ V,

N X

Xiv ≤ k

i=1

•

•

Link-Channel Usage Constraints: Channel i is used on link e if and only if channel i is set on both the end points of e.  Xiu + Xiv ≥ 2Yie ∀e = (u, v) ∈ E, Xiu + Xiv − 1 ≤ Yie

Capacity Constraints: The set of links in G corresponding to a maximal clique in G′ defines the set of mutually interfering links. In this set, at most one link can be active. For every maximal clique C in G′ , X Yie ≤ 1 ∀1 ≤ i ≤ N, e∈C

•

Connectivity Constraints: To ensure connectivity, we try to route unit flows from an arbitrary source node s to all

other nodes in the graph. Define for convenience, a t-flow as the flow destined towards node t. t Let F(u,v) = 1 if there is a t-flow on link (u, v), 0 otherwise. The flow conservation states that the total incoming flow should be equal to the total outgoing flow at the intermediate nodes. That is, ∀t ∈ V \ s, ∀v ∈ V \ {s, t}, X X t t F(v,u) F(u,v) = (u,v)∈E

(u,v)∈E

There is a unit flow going out of source and no flow into the source. ( P Ft =1 P(s,v)∈E (s,v) ∀t ∈ V, t F (v,s)∈E (v,s) = 0

Note that there can be a t-flow through link (u, v) only when u and v have at least one channel in common. This can be represented by the following constraints. ( PN t F(u,v) ≤ i=1 Yie PN ∀t ∈ V \ s, ∀e = (u, v), t F(v,u) ≤ i=1 Yie

B. Heuristic

The heuristic for the channel assignment takes as input the location of the routers, transmission radius, interference radius, number of channels, number of radios on each node and outputs the channels assigned to the radios of each router. The heuristic starts by reserving one radio on each node for ensuring connectivity later. The channel assignment heuristic invokes two functions namely, MIS channel assignment and ensure connectivity. Algorithm 2 heuristic channel assignment(L, RT , RI , N, K) Input: Location of the routers L, Transmission range RT , Interference Range RI , Number of channels N, Number of radios K Output: Channels assigned to each router 1: Reserve one radio on each node for ensuring connectivity 2: G ← MIS channel assignment(location of the routers, RT , RI , N) 3: Free the reserved radio on each node 4: ensure connectivity(G, N) The objective of MIS channel assignment is to assign channels such that the number of links that can be activated simultaneously is maximized. The algorithm computes largest independent sets of the nodes of LIG repeatedly and assigns a channel to the nodes in each independent set. Recall that the nodes of the LIG are the links in the PCG. Thus, this algorithm in each iteration tries assigning the same channel to as many non-interfering links (more specifically to the end routers of the link) as possible in the PCG. The algorithm applies the PTAS MIS of ODDGraphs to compute the (largest) independent set in the LIG. The nodes in this set are

Algorithm 3 MIS channel assignment(location of the routers, RT , RI , N) 1: construct the potential communication graph G from location of the routers and RT . 2: c ← 1 3: while |E(G)| > 0 do 4: construct an isomorphic copy H of graph G with V (H) = V (G) and E(H) = E(G). 5: construct conflict graph H ′ of graph H. 6: while |E(H)| > 0 do 7: M IS ← MIS ODD graph(H ′ ). {Note that M IS ⊆ ′ E(H) since S V (H ) = E(H)} 8: let VH = u, v such that (u, v) = e ∈ M IS. Let VG be the corresponding set of vertices in graph G. 9: if channel c is already assigned to all nodes in VG then 10: find the next channel c that is not assigned to at least one node in VG . 11: end if 12: ∀v ∈ VG , assign channel c to a free radio in node v. 13: for all v ∈ VG such that v has zero radios left do 14: remove vertex v and all edges incident on v from graph G and graph H. Also, remove the corresponding vertices and the edges incident on these vertices from H ′ . 15: end for 16: E(H) ← E(H) \ M IS. Remove the corresponding vertices and the incident edges from the conflict graph H ′ . 17: c ← (c + 1) mod N 18: end while 19: end while 20: return G

temporarily removed from the LIG and another independent set is computed. This process is repeated until all nodes in PCG have been assigned some channel. At this stage, the LIG is constructed again and the process is repeated. Throughout, the number of free radios on each node in PCG is kept track of and whenever all radios in a node are used up, the node and all its edges are removed from the PCG and LIG. At the end of the MIS channel assignment algorithm, the topology resulting from the channel assignment may have several connected components. The ensure connectivity algorithm uses the single radio that was reserved earlier to connect all the components. The algorithm maintains a set S consisting of a single connected component. At each iteration, all paths that connect S with some component not in S are examined. The interference degree of a path is the largest number of edges interfered by an edge in the path. The path and the channel to be assigned on all nodes of this path that lead to the least interference is computed. Channel assignments are done on this path and the component Ci connected by this path is included into S. This procedure is repeated until all components are merged into S.

Algorithm 4 ensure connectivity(G, N) Input: Potential Communication Graph G with partial channel assignments, number of channels N Output: Potential Communication Graph with all the channel assignments 1: Compute all the connected components C1 , C2 , ..., Cp in G. 2: S ← C1 {S is the connected subgraph of G} 3: while ∃Ci ∈ / S do 4: next path ← φ, next channel ← 0 5: smallest int degree ← ∞ 6: for all path P such that P connects S with a component not in S do 7: channel(P ) ← channel assignment that results in least interference on path P . 8: interf erence degree(P ) ← interference degree on P by assigning channel(P ) on P . 9: if (interf erence degree(P ) < smallest int degree) then 10: smallest int degree ← interf erence degree(P ) 11: next path ← P . 12: next channel ← channel(P ). 13: end if 14: end for 15: Assign channel next channel on all links of path next path. 16: Let next S path connect S with component Ci . S ← S Ci . 17: end while C. Simulation Environment, Results and Discussion We conducted experiments to evaluate the efficiency of our heuristic. In all experiments, there were 30 routers each having a transmission radius of 150m and an interference radius of 300m. The locations for the routers were randomly generated in an area of width 1500m and height 1500m.

Fig. 9. Maximum number links that can be active simultaneously for different number of radios and for 8 channels

In the first set of experiments (figures 9,10), we studied the effect of increasing the number of radios in each router on the total number of links that can be activated simultaneously. We

demonstrate that our channel assignment algorithm produces near optimal solution in almost all instances of the problem. VII. C ONCLUSION In this paper, we have provided a heuristic for the channel assignment problem in Wireless Mesh Networks. In the process, we characterize the LIG as ODD graphs and provide a PTAS to compute MIS for ODD graphs. Our results demonstrate the effectiveness of the heuristic. R EFERENCES Fig. 10. Maximum number links that can be active simultaneously for different number of radios and for 10 channels

varied the number of radios from 2 to 8. The experiment was conducted for 8 and 10 channels.

Fig. 11. Maximum number links that can be active simultaneously for different number of channels and each router having 4 radios

Fig. 12. Maximum number links that can be active simultaneously for different number of channels and each router having 5 radios

In the second set of experiments (figures 11,12), we studied the effect of increasing the number of channels on the number of links that can be active simultaneously. We varied the number of channels from 5 to 12. The experiment was conducted for 4 and 5 radios on each router. For the PTAS for computing MIS, we chose ǫ = 0.1. Additional experiments were conducted to measure the effect of transmission radius and the interference radius on the number of links that can be activated simultaneously. These results are not included here due to the paucity of space. The experimental results

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