OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS

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Abstract- Design/planning of WMNs is the key phase before any deployment. Few proposals can be found in the open literature that deal with the design ...
OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS A. Beljadid, A. S. Hafid, and M. Gendreau Network Research Lab, University of Montreal {abeljadid, ahafid}@iro.umontreal.ca, [email protected] Abstract- Design/planning of WMNs is the key phase before any deployment. Few proposals can be found in the open literature that deal with the design problem; moreover, they do not take into account all the parameters that have an impact on the outcome of the design and they assume the existence of a physical topology where the location and the characteristics of nodes (e.g., number of channels, number of radios) are fixed. In this paper, we define a generalized model for the WMNs design problem that takes into account all the parameters that have a significant impact on the network (interference, multi-channel, transmission power, etc.), the requirements of providers (expected amount of traffic/users), the constraints of the physical environment (potential locations of wireless routers, e.g., poles, and gateways, e.g., data centers), etc. The objective is to minimize the cost of the network and its operations while satisfying the requirements. The proposed model is shown to outperform considerably existing solutions. Keywords- Design, Infrastructure/Backbone WMNs, Gateways placement, General network model.

I. INTRODUCTION Wireless mesh networks (WMNs) consist of stationary wireless routers interconnected by wireless links with a small fraction acting as gateways to the Internet via wired (e.g., Ethernet) or wireless (e.g., WIMAX) links. In this paper, mesh clients (e.g., mobile hosts) do not act as routers for mesh networking; we are concerned with Infrastructure/backbone WMNs [7]. The design of WMNs is a challenging issue that did not receive much attention in the literature. We believe that the planning phase of WMNs is one of the most important optimization tasks before any deployment. Most of proposals that deal with WMNs assume the existence of a physical topology where the location and characteristics of the nodes are fixed/predefined. Indeed, most research efforts for WMNs have been focused on developing efficient strategies for routing, channel assignment and scheduling in order to maximize throughput [4], [5], [6]. Only a few proposals can be found in the open literature that deal with the design problem; however, they do not take into account all parameters involved in the design. By fixing the topology and capacity assignments, the design problem is reduced to a routing problem. To date and to the best of our knowledge, no research has dealt with an unfixed topology. The authors in [9] calculate the per-node throughput for a given WMN topology and gateways locations. The authors in [1], [2], and [8] propose techniques to place and minimize the number of gateways while supporting a specific amount of traffic to and from the Internet for fixed topology. It is worth noting that most of existing models are based on the concept of connectivity graph and the conflict graph. The connectivity graph model determines which two nodes have wireless connectivity using some type of measurements while the conflict graph indicates which groups of links mutually interfere and hence cannot be active simultaneously. The

connectivity graph model is formulated as an LP and the conflict graph is used to define a set of constraints. The major problem with these models is that (a) the use of the conflict graph is highly complex: for even a moderately-sized network the number of interference constraints can be hundreds of thousands; (b) the objective of the LP is partial (i.e., not global): it does not really optimize WMNs. The deployment and management cost of gateways in WMNs is significant. However, by optimizing the number of gateways [1, 2, 8] we just reduce the total cost and do not minimize it because, for example, (a) a “good” assignment of channels, (b) changing the characteristics of one or more nodes (e.g., adding a radio, using radios with an optimum number of channels [3]), or (c) adding one or more nodes can be far more profitable. In this paper, we propose a unified/generalized model for the design of WMNs. The goal is to determine a topology and configuration of WMNs that satisfy the requirements, in terms of throughput and delay, with a least cost. The proposed solution requires a set of inputs that consists of the traffic demands (throughput and delay), a set of potential locations for nodes (routers and gateways) and cost information. The resulting optimal configuration that satisfies the requirements is described in terms of the total cost, number and locations of routers and gateways, characteristics of the nodes (number of channels and radios per node, power level/range, channel assignment), and routing information. Our contributions can be summarized as follows: (1) The uniqueness of our proposal lies in the fact that rather than treating each of the components (e.g., multi-channel, power control, placement of gateways, cost, etc.) separately, as has been done so far in the literature, we address the above issues using a unified model by exploiting the intimate relationships among them. The proposed model can be used both for a new WMNs solution and for any expansion of the network; (2) The proposed model is formulated as a linear program; its complexity has been considerably reduced by using/defining a relaxation of the interference constraints and an aggregation of the coverage constraints; and (3) Even though we propose, in this paper, an objective function that minimizes the cost of the network while satisfying the requirements, the proposed model can be used with any other objective function, e.g., maximizing the throughput while satisfying the requirements. The paper is organized as follows. Section 2 presents the proposed formulation of the WMNs design problem. Section 3 presents an evaluation of the proposed model. Section 4 concludes the paper and presents future research directions. II. FORMULATION A. Problem description In this section, we present a formulation of the WMNs design problem that takes into account all the parameters that

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OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS have an impact on the network. The objective is to minimize the cost of the network subject to technical and geographical/physical constraints. We consider a fixed Multi-Channel Multiple Radios wireless network (MC-MR wireless networks). Stationary wireless routers are assumed to have multiple Network Interface Cards (NICs). A router can establish a link with a neighboring router when each router has one of its interfaces using the same channel and the distance between the two routers is less than the power transmission (i.e., range) of the transmitter. Since the number of channels is limited, some links in the WMNs may be allocated to the same channel. In this case, interference will occur if these links are close to each other. To take into account interferences, we use the protocol interference model reported in [10]. In this model, a transmission on channel k is successful when all potential interferers in the neighborhood of the sender and the receiver are silent on channel k for the duration of the transmission (this is similar to the model used in IEEE 802.11). The main characteristics of the WMNs considered in this study can be summarized as follows: 1. There are multiple potentials locations for routers/gateways and there is no obstacle between routers; we assume line-of-sight links. 2. Routers are not mobile and have multiple radio transceivers, which allow them to communicate, interference-free, simultaneously with more than one neighbor at the same time using different/orthogonal channels. A router can be receiving from or transmitting to a neighbor Ri on channel A, while transmitting to or receiving from neighbor R j on channel B, where A ≠ B. 3. There are multiple wireless channels and these channels are orthogonal to each other. The number of orthogonal wireless channels is limited; this means that more than one node in a given region could contend for the same channel at the same time, thereby resulting in interference and collisions. 4. Transmission power (i.e., range) of routers can be selected from a discrete set of possible ranges. B. Network model We consider MC-MR wireless networks. The network is modeled as a directed graph G = (V, A) where V represents the set of nodes in the network and A the set of arcs (directed links) that can carry data (data links). If node R i can transmit directly to node R j (and vice versa), then we represent this by directed arc < R i , R j > between node R i and node R j with the arc belonging to the set A . In the following, we present definitions and notations that will be used in the remainder of the paper. • R denotes the number of radios per node and C denotes the number of channels per radio. We assume that all the nodes have R radios and C channels per radio. Two nodes are connected if the distance between them is smaller than the maximum communication range (in the

set of possible ranges); in the worst case, we will have a complete graph with O(n 2 ) arcs.

Θ ijk denotes the capacity of arc < R i , R j > over channel



k.

G i denotes a virtual node; G i is introduced if the router R i is a potential gateway. < R i , G i > denotes a virtual arc between each R i and

• •

G i ; this arc is introduced only in the case R i is a potential gateway. Generally, only a subset of the routers are potential gateways; in the worst case scenario, we have O(n) arcs (this corresponds to the model of wireless hot



spots). The capacity of arc < R i , G i > is Θ i the capacity of the gateway. S and T denote two virtual nodes.



< S , R i > denotes a virtual arc between S and router R i and D i denotes the traffic demand on R i , which is equal to the capacity of < S , R i > ; < S , R i > is a saturated arc.

< G i , T > denotes a virtual arc between gateway G i and T ; it is used to describe a wired link in the Internet. The capacity of this arc is unlimited. Let us consider the design of a WMN that consists of up to three routers (R1, R2 and R3), two of which (R1 and R3) are potential gateways. Furthermore, the distance between R1 and R3 is larger than the maximum range (in the set of potential ranges); thus, no connection is possible between R1 and R3. Figure 1 shows a representation of this network using the proposed network model. 7 virtual links are introduced: , , , , , , and . •

R1 G1 R2

S

T G3

R3

Figure 1: Sample of the network model

G

Graph

V

Set of vertices

A

Set of arcs

n

Number of nodes (potential routers)

g

Number of potential gateways

C

Maximum number of orthogonal channels

R

Maximum numbers of radios (NICs)

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OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS

P

Number of possible communication ranges

Ri

Wireless router (Access Point)

Gi

Gateway if R i is wired

i R1i K RR

Pi

We assume that the transmission power for all radios of R i (1 ≤ i ≤ n) is the same. The proposed model can be easily extended to support different transmission powers for different radios, by adding an extra index for radios. Thus, instead of P i (Table 1), we would have P ir , where r would be used to

Radios of R i

indicate which radio is used. Let Aij ∈ A denote the arc < R i , R j > if it exists (i.e., if

Actual communication range of router R i

p th element

Ppi

in

the

set/list

of

potential

P i = Ppi , where 1 ≤ p ≤ P ; this binary variable indicates that

communication ranges of router R i i

i

i Pmax

Maximum communication range of R , i.e., P P .

i Pmin

Minimum communication range of R , i.e., P1i .

i

D

Set of demands Traffic demand at node R i . We will have D i = 0 ,

Di

< Ri , R j >

Θ ijk

transmission power Ppi will be used. Let us denote by d ij (= Ppi ij ) the smallest communication range that enables connectivity between R i and R j (i.e., such that d ( R i , R j ) ≤ Ppi ij and d ( R i , R j ) > Ppi ij −1 .); in this case,

if R i can be used only as relay.

d ij = d ji . Now let B ip be the cost associated with using the

Arc that represents the fact that R i can com-

range Ppi .

municate with R j if R i is within the actual

To minimize the cost associated with using ranges, we have to

communication range of R j and uses the same channel.

minimize the objective function

i

Capacity of the gateway G

P

∑B Q

i

i p

i p

subject to

p =1

j

Capacity of the arc < R , R > over channel k

Θi

i d ( R i , R j ) ≤ Pmax ), and Q ip be a binary variable set to 1 if

P

∑Q

i p

≤ 1, as well as communication range constraints

p =1

d (Ri , R j ) B ip Ci

distance between R i and R j

needed to transmit on selected arcs < R i , R j > (see below).

Cost of R i (router cost) if communication range p (transmission power) is used

The solution obtained will specify the range p (i.e., Q ip = 1

i

Cost to wire R (gateway cost)

H

Maximum number of hops; it is used to limit the delay.

Ν

Set of integers Table 1. Formulation: list of symbols/parameters

C. Formulation We formulate the design problem as a mixed integer linear program (MIP). We present the formulation in four steps; in each step we define a set of variables and constraints that will be used in defining the unified/generalized model. In the first step, we model the communication range of the nodes; in the second step, we concentrate on channel assignment; in the third step, we model the QoS requirements; in the fourth step, we present an objective function that minimizes the cost of the network. The analysis of WMN scalability is based on the following scaling relationships: traffic increases with the number of nodes, and traffic also increases with the distance over which each node wishes to communicate. Table 1 shows the list parameters/symbols used in the formulation. Step 1: Communication range

P

and Q ip ' = 0 ∀ p ≠ p' ) to be used. The constraint

∑Q

i p

≤ 1,

p =1

will always be satisfied since it cannot be optimal to use more than one range; thus, we can relax it. Step 2: Channel assignment Let X ikr js be a binary variable that indicates whether channel

k of radio Rri and radio Rsj is used to transmit over the arc < R i , R j > . It is obvious that if X ikr js = 1 then one must have Aij ∈ A . For any given node, it is not useful to have two radios tuned to the same channel since local interferences will ensure that at most one of them is active at any time. To avoid this situation, we define the following constraints: R

R

∑∑ X

k ir j s

≤ 1, ∀ 1 ≤ k ≤ C and ∀ 1 ≤ i, j ≤ n.

r =1 s =1 R

Let us denote X ijk =

R

∑∑ X

k ir js . (13).

Thus, if X ijk = 1, then

r =1 s =1

Aij ∈ A , which means that we must have

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OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS

Then the flow on < R i , R j > for all demand using the

P

d ij X ijk ≤



Ppi Q ip ; ∀ 1 ≤ k ≤ C , ∀ 1 ≤ i, j ≤ n. (9a)

channel k can be defined as

p =1

With this formulation, we have C n 2 constraints; however, only a subset is applicable/active. We reduce the number of constraints to C n constraints ( C constraints per node ) by adding, for each node, the n constraints defined for neighborring nodes. This induces a set of relaxed constraints, which we call communication range constraints: P

n



d ij X ijk ≤ n

∑P Q

j =1

i p

i p ,∀1 ≤

(9)

k ≤ C, ∀ 1 ≤ i ≤ n

p =1

We define the interference constraints over data links using the model reported in [10]. These constraints are

∑X

k ij

+

∑X

k li

≤1, ∀1≤ k ≤ C , ∀1 ≤ i ≤ n

l Ali ∈A

j Aij ∈ A

Since X ijk = 0 whenever node R j is not in the communication range of R i , the condition Aij ∈ A can be relaxed in these constraints, thus yielding for the interference constraints: n

∑X j =1

n k ji

+ ∑ X ≤ 1; ∀1≤ k ≤ C ; ∀1≤ i ≤ n k li

(8)

l =1

When taking into account constraints (8), we have (9) equivalent to (9a). Step 3: QoS requirements In this step, we take into account the quality-of-service (QoS) requirements such as delay and bandwidth. We assume that the capacity of a wireless link over a channel k is the same for all radios. The proposed model can be easily extended to support different capacities by adding an extra index for radios. Let Z i a binary variable that indicates whether the gateway G i is set and Y i a binary variable that indicates whether the router R i is set. Y i = 1 means that R i is not a gateway ( Z i = 0 ), i.e., the flow that crosses R i is directed to another router/gateway and not to the Internet. Thus, Z i +Y i ≤1; ∀1≤ i ≤ g (2) To prevent gateways from transmitting traffic to routers (gateways send traffic, coming from WMN routers, directly to the Internet), we must have X ijk = 0 if Y i = 0 for 1 ≤ k ≤ C and 1 ≤ i, j ≤ n which can be written as: n

C

∑∑ X

k ij

≤ C Y i , ∀ 1 ≤ i ≤ n.

(3)

j =1 k =1

Let ω ijkd be the flow of the demand d ∈ D on < R i , R j > when using the channel k and ω i the flow on < R i , G i > ( ωi

= 0 when the link < R i , G i > doesn’t exist).

ω ijk =



d ∈D

ω ijkd .

(7)

The conservation of flows requires that n

ωi +

C

∑∑

n

ω ijk = D i +

C

∑∑ ω

j =1 k =1

k li ,

∀1 ≤ i ≤ n.

(4)

l =1 k =1

A flow can cross over < R i , R j > using channel k if

X ijk = 1 (otherwise ω ijk = 0 ); and a flow can cross over < R i , G i > only if Z i = 1 . The flow activation constraints are (where M is a big/infinite value): ω ijk ≤ M X ijk , ∀ 1 ≤ k ≤ C and ∀ 1 ≤ i, j ≤ n, (5a)

ω i ≤ M Z i , ∀1 ≤ i ≤ g.

(6a) Bandwidth requirements are of two forms. First, the traffic is bounded by the capacity of the gateway (based on its connectivity to the Internet and its processing speed) and second the throughput for individual flows is constrained by link capacity. Let Θ ijk denotes the capacity of the link < R i , R j > over the channel k . The value of Θ ijk can be computed using P i and the distance d ( R i , R j ) (more parameters can be used, such as the type of NIC associated to channel, etc.). If Θ i denotes the capacity of the gateway, the capacity constraints can be defined as follows: ω ijk ≤ Θ ijk , ∀ 1 ≤ k ≤ C and ∀ 1 ≤ i, j ≤ n, (5b)

ω i ≤ Θ i , ∀1 ≤ i ≤ g.

(6b)

i

Remark: D ≤ Θ i , ∀ 1 ≤ i ≤ g , otherwise the solution is infeasible. The delay is a function of the number of communication hops between the source and the gateway. We define the hop variable hid . When transmitting over < R i , R j > using channel k , the number of hops is increased by 1, i.e., we have

ω ijkd > 0 ⇒ hid ≥ h dj + 1. We define Ω ijkd a binary variable such that Ω ijkd = 1 when

ω ijkd > 0 . We have ω ijkd ≤ M Ω ijkd . Let H be the maximum number of hops; the delay constraint is translated into: ω ijkd ≤ M Ω ijkd , (10)

hid + 1 − h dj ≤ M (1 − Ωijkd ), ∀i, j , k ∀d ∈ D,

(11)

hid ≤ H , ∀1 ≤ i ≤ n, ∀d ∈ D.

(12)

The integrality constraints on

hid

can be relaxed.

Step 4: Objective function Let C i be the cost of the gateway G i . The objective function is then written as follows:

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OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS g

n



min

CiZ i +

∑∑

i =1

III. ANALYSIS AND SIMULATIONS

P

B ip Q ip

i =1 p =1

When minimizing the cost without taking into account throughput maximization, we can aggregate (5a)-(5b) into constraint (5) and (6a)-(6b) into constraint (6) as follow: ω ijk ≤ Θ ijk X ijk , ∀ 1 ≤ k ≤ C and ∀ 1 ≤ i, j ≤ n, (5)

ω i ≤ Θ i Z i , ∀1 ≤ i ≤ g.

(6)

The (MIP) can then be formulated as: g

n

P

min ∑ C i Z i + ∑∑ B ipQ ip i =1

(1)

i =1 p =1

Subject to:

Z i + Y i ≤ 1, ∀ 1 ≤ i ≤ n n

(2)

C

∑∑ X

k ij

≤ CYi, ∀1≤ i ≤ n

(3)

j =1 k =1 C

n

C

n

ωi + ∑∑ ωijk = D i + ∑∑ ωlik , ∀1 ≤ i ≤ n j =1 k =1 k ij

In this section, we present very preliminary results; we consider one possible utilization of our model. Indeed, our model is very general to be used to optimize different requirements (e.g., throughput, QoS, number of channels, etc.). Analysis If the topology is not fixed, a low cost solution for WMNs design can be produced. As in the example of Figure 2, we suppose demands are D1 = D 3 = 1 and gateway capacities are Θ1 = 1 and Θ 3 = 3 for the two potentials routers. If the topology is fixed, as in existing solutions, we will have both routers wired (i.e., gateways) as a solution when performing any optimization for this problem. But if we can modify the topology (as we suggest), we can reduce the number of gateways (only 1 gateway is necessary). We can either increase the power transmission of routers to add a link between R1 and R3 (if it is possible) or add a router R2 with demand D 2 = 0 as illustrated in Figure 3.

(4)

R1 D1

l =1 k =1 k ij

G1

k ij

ω ≤ Θ X , ∀ 1 ≤ k ≤ C and ∀ 1 ≤ i, j ≤ n (5)

Cap1

S

T Cap3

D3

ωi ≤ Θi Z i , ∀1 ≤ i ≤ g (6) k kd ωij = ∑ ωij , 1 ≤ i, j ≤ n and 1 ≤ k ≤ C (7)

G3 R3

d ∈D

Figure 2: two routers and two gateways

n

∑X

k ji

≤1; ∀1 ≤ k ≤ C ; ∀1≤ i ≤ n

(8)

R1

j =1 P

n

∑ d ij X ≤ n∑ P Q , ∀ 1 ≤ k ≤ C , ∀ 1 ≤ i ≤ n (9) k ij

j =1

Cap12

D1

i p

i p

S

D2

R2 D3

p =1

Cap23

T Cap3 G3

R3

ωijkd ≤ M Ωijkd

(10)

hid + 1 − h dj ≤ M (1 − Ωijkd ), ∀i, j , k ∀d ∈ D,

(11)

hid ≤ H , ∀1 ≤ i ≤ n, ∀d ∈ D R

X ijk =

(12)

R

∑∑ X

k ir js .

…….(13)

r =1 s =1

X ijk , Y i , Z i , Qki ∈ {0,1}, ∀1 ≤ i, j ≤ n, ∀1 ≤ k ≤ C (14) Ω ijkd ∈ {0,1}, ωijkd ∈ Ν, hid ≥ 0, ∀i, j , k , ∀d ∈ D

(15)

Constraints (2)-(13) have been already introduced before. Constraints (14)-(15) simply indicate that X , Y , Z , Q and

Ω are binary variables, ω are integer and h are real. The problem formulated above is NP-hard. LP solvers, such as CPLEX and CLP, can only handle small sized networks; we therefore plan to use heuristics to solve this program for realistically-sized instances (to be published in a future paper).

Figure 3: two routers 1 gateway

Simulations To study the benefits of being able to change the topology, we compute solutions for small networks. We used the CLP solver [11] that can be invoked by CBC (Branch and Cut) as a Mixed Integer Program (MIP) solver [12], and for modeling purposes, we used the ZIMPL modeling language [13]. For simulations, we consider a 8 x 8 grid topology with 64 nodes, of which only 40 nodes have a demand and 20 can be gateways. We suppose (a) C i is the same for all gateways, (b)

D i = 1 , Θi = 10 ,

Θijk = 2 and (c) B ip = P i . When

optimizing the cost using parameters P=1, R=2, C=4 and H=4; we found 12 gateways as an optimal number to satisfay the demand over the 40 nodes (same result as existing solutions, e.g., [8]). Now, let us study the effects of changing the topology by adding potential routers/gateways, changing communication range and varying number of channels per radio. We present the results in Figures 4, 5 and 6.

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OPTIMAL DESIGN OF BROADBAND WIRELESS MESH NETWORKS

is reduced by 50% when using up to 4 radios and 5 channels. Channel affectation gives a better solution than adding routers/gateway and than varying range.

Figure 4: adding potentials nodes/gateways where R=2, C=4, P=1 and H=4. 16

optimal number of gateways

14 12 10 g fixed

8

g vary

6 4 2 0 35

40

45

50

55

60

65

number of potentials nodes

Figure 4 shows that the optimal number of gateways decreases as the number of potential nodes/gateways increases. We present two case: (1) the number of nodes that can be wired is fixed (20 potentials gateways) we just add routers and (2) the number of gateways vary, we add potentials routers/gateways. For example, the number of gateways is reduced by 33% by adding less than 10 routers (less than 25% increase in the number of nodes) in case (2). Figure 5: varying the number of possible communication range where n=40, R=2, C=4, and H=4.

REFERENCES

16

optimal number of gateways

14 12 10 8

range

6 4 2 0 0

1

2

3

4

5

6

number of possible communication range

When the number of possible communication ranges is large, most nodes are reachable from one another within few hops, thus the optimal number of gateways decreases when the connectivity increases as illustrated in Figure 5. The number of gateways is reduced by 33% when varying the range of 14 nodes (35 %). Figure 6: varying the number of channel and radio where n=40, P=1 and H=4. 18

optimal number of gateways

16 14 12 2 Radio

10

3 Radio 8

4 Radio

6 4 2 0 1

2

3

4

5

6

IV. CONCLUSION In this paper, we proposed a new unified/generalized model for the design of WMNs. The uniqueness of our approach lies in the fact that rather than treating each of the components (e.g., multi-channel, power control, cost, QoS, etc.) separately, as has been done so far in the literature, we developed a new model that integrates all the components by exploiting the intimate relationships among them. In this paper, we used this model to determine a design that minimizes the number of gateways while satisfying all technical and topological constraints. The proposed model is more general and can be used to resolve designs that optimize other parameters of interest, such maximizing the throughput. We are currently working to develop heuristics to resolve the model for larger networks; furthermore, we are planning to investigate the stability of solutions in cases of failures, as well as network expansion in terms of geography and traffic.

7

number of channel/radio

An optimal number of channel/radio allows the maximization of throughput, thus the demand is satisfied by a small number of gateways (Figure 6). The number of gateways

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