Knapsack based multicast traffic grooming for optical networks

Author’s Accepted Manuscript Knapsack based Multicast Traffic Grooming for Optical Networks Ashok Kumar Pradhan, Bijoy Chand Chatterjee, Eiji Oki, Tanmay De www.elsevier.com/locate/osn

PII: DOI: Reference:

S1573-4277(17)30013-9 http://dx.doi.org/10.1016/j.osn.2017.08.002 OSN453

To appear in: Optical Switching and Networking Received date: 3 February 2017 Revised date: 7 July 2017 Accepted date: 18 August 2017 Cite this article as: Ashok Kumar Pradhan, Bijoy Chand Chatterjee, Eiji Oki and Tanmay De, Knapsack based Multicast Traffic Grooming for Optical Networks, Optical Switching and Networking, http://dx.doi.org/10.1016/j.osn.2017.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Journal Logo 00 (2017) 1–23

Knapsack based Multicast Traffic Grooming for Optical Networks Ashok Kumar Pradhana , Bijoy Chand Chatterjeeb , Eiji Okic , Tanmay Ded a Ashok

Kumar Pradhan, SRM University, Dept. of Computer Science and Engineering, Amaravati, Andhra Pradesh, India, (Email: [email protected]) b Bijoy Chand Chatterjee, Indraprastha Institute of Information Technology, Delhi (IIITD), New Delhi, India, Dept. of Computer Science and Engineering, (Email: [email protected]) c Eiji Oki, Department of Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan, (Email: [email protected]) d Tanmay De, National Institute of Technology (NIT), Department of Computer Science and Engineering, Durgapur, West Bengal, India, (Email: [email protected])

Abstract This paper proposes a light-tree based heuristic algorithm, called 0/1 knapsack based multicast traffic grooming, in order to minimize the network cost by reducing the number of higher layer electronic and optical devices, such as transmitters, receivers, and splitters, and used wavelengths in the network. The proposed algorithm constructs light-trees or sub light-trees, which satisfy sub bandwidth demands of all multicast requests. We present a light-tree based integer linear programming (ILP) formulation to minimize the network cost. We solve the ILP problem for sample four-node and six-node networks and compare the ILP results with the proposed heuristic algorithm. We observe that the performance of the proposed algorithm is comparable to the ILP in terms of cost. When the introduced ILP is not tractable for large network, the proposed algorithm still able to find the results. Furthermore, we compare the proposed heuristic algorithm to existing heuristic algorithms for different backbone networks. Numerical results indicate that the proposed heuristic algorithm outperforms the conventional algorithms in terms of cost and resource utilization. Keywords: Light-tree, multicast requests, traffic grooming, network cost

1. Introduction Wavelength division multiplexing (WDM) is an efficient technology as it supports a huge bandwidth requirement of clients. Based on the observation that the bandwidth requirement of majority of connection requests is much less than the capacity of a wavelength. Therefore, there is a huge gap between the wavelength channel capacity and bandwidth requirement of a connection request. This gap can be managed to incorporate the traffic grooming [1–5] mechanism with an routing and wavelength assignment (RWA) approach [2] where a number of low bandwidth connection requests are multiplexed onto a high capacity wavelength channel to enhance overall channel utilization. In WDM optical networks, a lightpath [2] is established to carry the information between a sourcedestination pair. In the absence of wavelength converters, a lightpath must occupy the same wavelength to its end-to-end route; this property is known as the wavelength continuity constraint [6]. In optical network, it is possible that many connection requests share the same source node, but their destination nodes 1

are different. In that case, an individual lightpath needs to be established for each source-destination pairs. This solution is costly, as it requires more number of wavelengths and higher layer devices, such transmitters and receivers. To overcome this issue, the multicast-tree concept [7, 8] is adapted, which supports one to many connections in a single logical hop; it has only a source node and different destination nodes. The multicast-tree reduces average packet hop distance and minimizes the total number of transceivers used in a network. In the logical layer, a light-tree can be represented as a set of directed links from a source node to all required set of destination nodes in the network. Since the transmission from a source node to all the destination sets takes only a single logical hop, this is called logical one hop tree (LOHT) [7]. Low bandwidth connection requests are routed by combining several LOHTs. The combinations of light-trees forms a larger tree than each light-tree. Electronic packet switching with optical-electrical-optical (OEO) conversion is required at the nodes connecting light-trees for forwarding the upstream light-tree to corresponding downstream light-tree(s). Achieving efficient and effective traffic grooming in WDM optical networks is a challenging research problem. Based on whether connection requests are known to the user a prior, the traffic grooming problem can be classified into two categories, namely static traffic grooming and dynamic traffic grooming. In the static traffic grooming scenario, information about each of traffic request, such as source, destinations and bandwidth requirement, is known to the user in advance. Studies on traffic grooming problem is useful for design and planning purpose. Whereas, in case of the dynamic traffic grooming problem, connection requests arrive dynamically based on some distribution process and both routing and wavelength assignment are dynamically decided for new connection requests. In the case of dynamic traffic request, the main objective is to accommodate the traffic requests efficiently and suppress the blocking probability; it is a ratio of the number of blocked requests to the number of offered requests in the network. In our proposed approach, we consider the static traffic scenario under wavelength continuity constraint with all nodes have finite splitting capacity. In earlier days, traffic grooming in WDM optical networks emphasized to reduce the number of used wavelengths [5, 9, 10] in order to optimize the network cost. The cost factor not only depends on the number of used wavelengths but also on the number of higher layer electronic and optical devices [7, 8, 11, 12] used in the network. The works in [5, 9, 10] do not consider the impact of higher layer electronic and optical components on network cost. To overcome this problem, R. Lin, et al. [7] designed a cost effective WDM optical networks with multicast traffic grooming considering higher layer electronic and optical devices along with used wavelengths. They introduced a light-tree based integer linear programming (ILP) formulation in order to minimize the network cost. Since the ILP is intractable for large networks, they introduced a heuristic algorithm, called sub-light-tree saturated grooming (SLTSG), in order to construct sub-light-trees. The overall time complexity of SLTSG is O(|R|3 ·|V |4 log |V |), where 2

R and V are sets of multicast requests and nodes in the networks, respectively. Our objective in this work is to develop a heuristic algorithm for multicast traffic grooming in WDM optical networks considering higher layer electronic and optical devices along with used wavelengths, which requires less computational time compared to SLTSG [7]. To achieve our objective, we propose a heuristic algorithm, called 0/1 knapsack based multicast traffic grooming, which minimizes the network cost. We present a light-tree based ILP formulation to minimize the network cost. By using an optimization solver, we solve the ILP problem for sample four-node and six-node networks and compare the ILP results with the proposed heuristic algorithm. We observe that the performance of the proposed algorithm is comparable to the ILP in terms of cost. When the introduced ILP is not tractable for large network, namely NSFNET [13], the proposed algorithm still able to find the results. Furthermore, we compare the proposed heuristic algorithm to existing heuristic algorithms for different backbone networks. Numerical results indicate that the proposed heuristic algorithm outperforms the conventional algorithms in terms of cost and resource utilization. The remainder of the paper is structured as follows. Section 2 presents the related works. Section 3 presents the ILP formulation for light-tree based multicast routing, grooming and wavelength assignment. The proposed knapsack based multicast traffic grooming approach for optical networks is presented in Section 4. Section 5 evaluates the performances of the proposed algorithms. Finally, conclusion is drawn in Section 6.

2. Related Work Multicast traffic grooming in WDM mesh networks is well known for its potential to reduce the cost related with the wavelengths requirement for a given network. Taking this direction, the authors in [6] generate multicast tree to establish the connection from the source node to all destination nodes and assign a wavelength in each and every branch of the multicast tree to construct the light-tree. Wavelength continuity constraint is maintained in their work with the objective of minimizing the wavelength usage. Three heuristic algorithms are presented in [14] for wavelength assignment in WDM optical networks with two criterion to cover the maximum number of destinations and to minimize the wavelength cost. The authors in [15] presented two algorithms, which integrate multicast routing and wavelength assignment for optimization of wavelengths used. One algorithm minimizes the number of wavelengths through reducing the maximal link load in the system while the other does it by trying to free out the least used wavelengths. The different research studies on multicast routing and wavelength assignment can be found in [16, 17]. An optimal solution for static traffic grooming can be obtained in [2, 3, 10] using ILP formulation with the objective of (i) minimizing the network resources used, such as wavelengths, add/drop ports, 3

and transceivers, and (ii) maximizing the network throughput. For static traffic scenario, the grooming is either based on lightpaths [10, 11] or light-trees [9, 12, 18–20]. The authors in [11] introduced a multi-hop static traffic grooming problem based on clique partitioning with the objective of maximizing network throughput for wavelength routed network. A light-tree based ILP formulation has been presented in [7] with the objective of minimizing the cost in terms of higher layer electronic ports and wavelengths and simultaneously introduced a sub-light-tree saturated (SLTSG) grooming algorithm to achieve the scalability. A mixed integer linear programming (MILP) has been reported in [18] in order to solve the optimal routing and wavelength assignment problem of light-trees with end-to-end delay bound, and obtain the optimal placement of power splitters and wavelength converters. The authors in [8] considered multicast traffic grooming problem in tap-and-continue networks where a node can tap a small amount of incoming optical power for the local station while forwarding the remainder to an output, and presented two heuristic algorithms, called multicast trail grooming (MTG) and multiple destination trail based grooming (MTDG), to minimize the network cost, which is associated with transmitters and receivers. The work in [9] constructs the multicast routing trees using first-fit algorithm for traffic grooming; this work also considers wavelength conversion capability of the network nodes. By intelligently grooming of several multicast requests with several sub-wavelengths into a single wavelength channel, their presented approach can significantly reduce the number of used wavelengths. A heuristic algorithm, called saturated-light-tree based multicast traffic grooming (SLTMTG), that solves grooming, routing and wavelength assignment problems with the objective of minimizing the network resources is presented in [12]. A heuristic algorithm, called dividable light-tree grooming (DLTG)[19], is introduced for optimal assignments of hop constrained light-trees for multicast connections in order to maximize the network throughput. Several studies [4, 21–30] have been performed considering dynamic traffic model in which traffic requests arrive in the network randomly over a period of time and they are served without waiting for future traffic requests. These models are more suitable for the activity mode of WDM optical networks, and hence the network performances in terms of resource utilization and blocking probability are optimized. A generic graph model has been presented in [21] for traffic grooming in heterogeneous WDM mesh networks. In their model, the edges of the auxiliary graph are manipulated to achieve various objectives using different grooming policies, while considering various constraints, such as transceivers, wavelengths, wavelength-conversion capabilities and grooming capabilities. A priority based routing and wavelength assignment scheme with incorporation of a traffic grooming mechanism (PRWATG) [22] is presented to reduce the call blocking. A light-tree based routing scheme for efficient grooming of low bandwidth requirement connection requests on a logical topology of WDM mesh network is introduced in [23]. Two link blocking models, an exact model based on stochastic knapsack problem, and an 4

approximation model based on an approximate continuous time Markov chain (CTMC) [4] have been reported with the objective of improvising the performance of a network by reducing the call blocking probability. Light-tree division adjacent node component based grooming scheme (LTD-ANCG) and light-tree division branch destination node based grooming scheme (LTD-DBNG) are suggested in [24] in order to improve the efficiency of resource utilization and reduction in the optical-electrical-optical (OEO) conversion overhead in WDM mesh networks. A new multicast multi-granular grooming approach is suggested [25] in order to accomplish the hierarchical sequential grooming for improvising the joint performances of integrated auxiliary grooming graph (IAGG), which approves both multicast traffic grooming and multicast waveband grooming. The concept of fragmentation and grouping is employed in [26], which can be applied for the wavelength assignment in multicast capable WDM optical networks. An optimized dynamic traffic grooming algorithm is also developed in their work to address the traffic grooming problem for mesh networks in a multicast scenario for maximizing the resource utilization and minimizing the blocking probability. A multicast dynamic light-tree grooming algorithm (MDTGA) is presented in [27] to support the multi-hop traffic grooming by taking advantage of light-trees. An efficient traffic grooming algorithm has been presented in [28], which grooms traffic requests efficiently, in order to suppress blocking probability and enhance network throughput. A dynamic multicast traffic grooming (DMTG) has been introduced in [30] with the objective of minimizing the blocking probability. In traditional WDM optical network, grooming of connection requests becomes complex as the traffic behavior is changing rapidly with the increasing mobility of traffic sources. Therefore, elastic optical networks (EONs) [31–34] have been considered to be a promising candidate for future high-speed optical communication as it supports bandwidth segmentation, bandwidth aggregation, efficient accommodation of multiple data rates, elastic variation of allocated resources. EONs allocates spectral resources with just enough bandwidth to satisfy the traffic demands. Traffic grooming in EONs [35, 36] is still essential for the following reasons, (i) bandwidth variable transponder is normally designed so as to maximize the traffic rate in the network, and it does not support slicing at a very early stage [37]. Electrical traffic grooming is applied in order to use transponder capacity efficiently. (ii) Generally speaking, a filter guard band between two adjacent channels should be assigned to resolve optical filter issues. Traffic grooming can minimize filter guard band usage by aggregating traffic electrically. The electrical switching fabric is still needed for traffic grooming in the elastic optical network, similar to WDM networks. Several multicast provisioning schemes [38–42] have been considered in EONs to enhance the resource utilization. Taking this direction, Z. Zhu et al. [38] designed an impairment and splitting-aware multicast provisioning schemes for EONs in order to study the procedure of adaptive modulation selection for a light-tree. They formulated the problem of impairment and splitting-aware routing, modulation and spectrum assignment (ISa-RMSA) for all-optical multicast in EONs. The work in [38] suggests that for 5

ISa-RMSA, the light-forest-based approach can use less bandwidth than the light-tree-based one, while still satisfying the quality of transmission requirement. The authors in [39] investigated multicast-capable routing, modulation, and spectrum assignment (MC-RMSA) schemes, which consider the physical-layer impairments from both the transmission and light-splitting in EONs, and introduced a scheme to serve each multicast request with a light-forest that consists of one or more light-trees. The authors in [40] presented different approaches for optimizing multicast traffic requests in EONs based on distance adaptive transmission. X. Liu et al. [41] investigated an overlay multicasting in multicast-incapable EONs and presented a spectrum-efficient multicasting scheme in order to improve the spectrum efficiency in EONs. An efficient heuristic algorithm based on genetic algorithm is suggested in [42] for EONs with the objective of better resource utilization compared to existing RMSA algorithm. Note that the multicast schemes in EONs are fundamentally different from that in the fixed-grid WDM optical networks [38]. The multicast schemes consider light-forest concept that uses several lighttrees to cover all the destinations of a multicast request. The multicasting scheme considering light-forest concept in WDM optical networks has less advantages in terms of spectrum utilization compared to the multicasting schemes considering light-forest concept in EONs. This is because WDM networks do not support elastic bandwidth allocation and distance-adaptive modulation, and hence the used bandwidth for each link for a multicast request is fixed. The minimum light-forest problem in WDM networks is reduced to the Steiner tree problem [43], where the light-forest structure is not beneficial in terms of spectrum utilization. However, multicast schemes considering light-forest concept in EONs is beneficial due to the properties of flexibility and distance-adaptive modulation.

3. ILP formulation In this section, we present an ILP based on [7] in order to minimize the network cost in terms of used transceivers, splitters, and wavelengths in the network. We model the optical network as a directed connected graph G(V, E). V is a set of nodes in the network, where v ∈ V . E is a set of links in the network, where (m, n) ∈ E is a fiber link from node m ∈ V to node n ∈ V . We assume that each link has a set of wavelengths W , where λ ∈ W . The capacity of each wavelength is denoted by C. R is a set of multicast requests. Multicast request t ∈ R is specified by one source st and a set of destination nodes Dt . The bandwidth requirement of multicast request t ∈ R is denoted by ft . Let K be a set of indices from 1 to 2|V |−1 − 1. Let l(i, k, d), where i ∈ V , k ∈ K, d ∈ V \{i}, be a flag that indicates that l(i, k, d) is set to 1 if node d ∈ V \{i} is a leaf of a logical one hop tree (LOHT) rooted node i, and 0 otherwise. k ∈ K implies a set of leaves of an LOHT. Note that a light-tree in the physical layer is represented as an LOHT in the logical layer, where the source and destinations of an LOHT are connected, within one 6

logical hop, with no interruption to the optical signal. For (m, n) ∈ E, we consider that one fiber exits from m ∈ V to n ∈ V . We assume that α, γ, δ, and β are the cost of each transmitter, receiver, splitter, and wavelength, respectively. Let Tn , n , and n be numbers of transmitters, receivers, and splitters at node n ∈ V , respectively. ψ represents the highest indexed used wavelength for any fiber link in the network. Li,k indicates the number of light-trees rooted from node i ∈ V to k ∈ K, which is a non-negative integer. Lλi,k is the number of light-trees using wavelength λ ∈ W from node i ∈ V to k ∈ K. Lλi,k > 1 indicates that there exists multiple link disjoint light-trees from node i ∈ V to k ∈ K using wavelength λ ∈ W . Hnt is the lower bound of logical hops of multicast request t ∈ R from source node st ∈ V to node n ∈ V . i,k,λ i,k,λ , Mmn , Λti,k , and Qti,k,n are used as binary decision variables in this formulation. Yλ is set Yλ , Pd,mn i,k,λ , is set to 1 if the path traverses to 1 if wavelength λ ∈ W is used in the network, and 0 otherwise. Pd,mn

from root node i ∈ V to k ∈ K for node d ∈ V \{i} through link (m, n) ∈ E using wavelength λ ∈ W , i,k,λ is set to 1 if the light-tree rooted from node i ∈ V to k ∈ K using wavelength and 0 otherwise. Mmn

λ ∈ W through link (m, n) ∈ E, and 0 otherwise. Λti,k is set to 1 if multicast request t ∈ R traverses from node i ∈ V to k ∈ K, and 0 otherwise. Qti,k,n is set to 1 if multicast request t ∈ R traverses from node i ∈ V to k ∈ K through the intermediate node n ∈ V , and 0 otherwise. We formulate an ILP problem to minimize the network cost in terms of used transceivers, splitters, and wavelengths in the following. Note that a splitter is counted if the light tree has more than one leaf node. When a splitting function is used at the transmitter, more power at the transmitter is required as the power is divided into the multiple leave nodes in the light tree. Therefore, the objective function considers the number of splitters. Eq. (1) expresses the objective function that minimizes the network cost. In this formulation, the given parameters are α, γ, δ, β, C, R, and l(i, k, d).

Minimize



(α · Tn + γ · n + δ · n ) + β · ψ

(1)

n∈V

The number of light-trees is constrained by the number of transmitters and receivers in the network, which are captured by Eqs. (2a)-(2e). Eq (2a) indicates that the number of light-trees rooted at node i ∈ V is less than or equal to the number of transmitters at node i ∈ V . Eq (2b) indicates that the number of light-trees, each of which has more than one leaf node, rooted at node i ∈ V to node k ∈ V is less than or equal to the number of splitters at node i ∈ V . Eq (2c) indicates that the number of light-trees terminated at node d ∈ V \{i} is less than or equal to the number of receivers at node d ∈ V \{i}. Eq (2d) ensures that the number of light-trees from node i ∈ V to k ∈ K equals the sum of the light-trees using wavelength λ ∈ W from node i ∈ V to k ∈ K. Eq. (2e) indicates the highest index used wavelengths in the network. Eq (2f) indicates that Yλ is set to 1 if wavelength λ ∈ W is used in 7

the network; in this equation, we consider A as a sufficiently large integer number.



Li,k ≤ Ti

k∈K

k∈K:

∀i ∈ V

 



d∈V \{i}

(2a) Li,k ≤ i

∀i ∈ V

(2b)

l(i,k,d)≥2



Li,k ≤ d

∀d ∈ V

(2c)

i∈V \{d} k∈K:l(i,k,d)=1



Lλi,k = Li,k

∀i ∈ V, ∀k ∈ K

(2d)

λ∈W

ψ ≥ λ · Yλ ∀λ ∈ W  Yλ ≥ Lλi,k /A

(2e) ∀λ ∈ W

(2f)

i∈V k∈K

Eqs. (3a)-(3e) are used to find a light-tree, which is rooted from node i ∈ V to k ∈ K. Eq. (3a) indicates that root node i ∈ V and destination node d ∈ V \{i} of a light-tree have no incoming and outgoing path, respectively. Eq. (3b) ensures that, for root node i ∈ V and destination node d ∈ V \{i}, the number of outgoing and incoming paths is equal to the number of light-trees using wavelength λ ∈ W from node i ∈ V to k ∈ K. Eq. (3c) indicates that, for an intermediate node of a path from root node i ∈ V to destination node d ∈ V \{i}, the number of incoming paths is equal to the number of outgoing paths. Eq. (3d) indicates that the physical route of light-tree traverses link (m, n) ∈ E if the light-tree rooted from node i ∈ V to k ∈ K using wavelength λ ∈ W traverses link (m, n) ∈ E. Eq. (3e) indicates that wavelength λ ∈ W is used by at most one light-tree rooted from node i ∈ V to k ∈ K using link (m, n) ∈ E.



i,k,λ Pd,mi =

m∈V :(m,i)∈E



i,k,λ Pd,dn =0

n∈V :(d,n)∈E

∀i ∈ V, ∀k ∈ K, ∀d ∈ V \{i}, ∀λ ∈ W, if l(i, k, d) = 1   i,k,λ i,k,λ Pd,in = Pd,md = Lλi,k n∈V :(i,n)∈E

m∈V :(m,d)∈E

∀i ∈ V, ∀k ∈ K, ∀d ∈ V \{i}, ∀λ ∈ W, if l(i, k, d) = 1   i,k,λ i,k,λ Pd,mq = Pd,qn m∈V :(m,q)∈E

(3a)

(3b)

n∈V :(q,n)∈E

∀i ∈ V, ∀k ∈ K, ∀d ∈ V \{i}, ∀λ ∈ W, ∀q ∈ V \{i, d}, if l(i, k, d) = 1  i,k,λ i,k,λ Pd,mn = Mmn ∀i ∈ V, ∀(m, n) ∈ E, ∀λ ∈ W, ∀k ∈ K

(3c) (3d)

d∈V \{i}:l(i,k,d)=1



i,k,λ Mmn ≤ 1 ∀(m, n) ∈ E, ∀λ ∈ W

i∈V k∈K

8

(3e)

Eqs. (4a)-(4g) are used for constructing multicast light-trees. Eqs. (4a) indicates that each multicast request is to be supported by multiple sub-light-trees. Eq. (4b) ensures that, at least one sub-light-tree starts from source node st ∈ V of each multicast request t ∈ R. Eq. (4c) ensures that for each multicast request, the sub-light-trees that are used to support that multicast request are not terminated at the source node of that request. Eq. (4d) indicates that each destination of a multicast request has only one incoming link. Eq. (4e) indicates that each intermediate node of a multicast request has at most one incoming edge. Eq. (4f) indicates that for each multicast request, except the source node, the root of a sub-light-tree must be a destination of another sub-light-tree; sub-light-tree are the part of the multicast request. Eq. (4g) indicates that for each multicast request, except the source and destinations, each intermediate node must be the root of some sub-light-trees.

Qti,k,n = Λti,k ∀t ∈ R, ∀i ∈ V, ∀k ∈ K, ∀n ∈ V \{i}, if l(i, k, n) = 1  Λtst ,k ≥ 1 ∀t ∈ R

(4a) (4b)

k∈K



i∈V k∈K



Qti,k,st = 0 ∀t ∈ R 

(4c)

Qti,k,n = 1 ∀t ∈ R, ∀n ∈ Dt

(4d)

Qti,k,n ≤ 1

(4e)

i∈V k∈K:l(i,k,n)=1





i∈V k∈K:l(i,k,n)=1



Λtq,k ≤ |K| ·

k∈K



k∈K

Λtq,k ≥





∀t ∈ R, ∀n ∈ V \{Dt , st }



Qti,k,q

∀t ∈ R, ∀q ∈ V \{st }

(4f)

i∈V k∈K:l(i,k,q)=1



Qti,k,q

∀t ∈ R, ∀q ∈ V \{Dt , st }

(4g)

i∈V k∈K:l(i,k,q)=1

Eq. (5) indicates the capacity constraint; the bandwidth used by all multicast requests must be equal to or less than the total capacity of all light-trees.



ft · Λti,k ≤ Li,k · C

∀i ∈ V, ∀k ∈ K

(5)

t∈R

Eqs. (6a)-(6b) are indicated that the logical route of each multicast request, derived by Eqs. (4a)-(4g), is loop-free. Eq. (6a) indicates that, for each multicast request, all the nodes that are not traversed by the multicast request have a zero value, and the source node has a zero value as well. Eq. (6b) indicates that, for a multicast request, if node m ∈ V and node n ∈ V \{m} are in the same light-tree that supports the multicast request, the hop number from the source of the request to node m ∈ V is larger than that from the source to node n ∈ V , where m ∈ V is the root of that light-tree; if node m ∈ V and node 9

n ∈ V \{m} are not in the same light-tree that supports the multicast request, this equation is always satisfied.

Hnt ≤ |V | ·





Qti,k,n

∀t ∈ R, ∀n ∈ V

(6a)

i∈V \{n} k∈K:l(i,k,n)=1



t + 1 − (1 − Hnt ≥ Hm

Qtm,k,n ) × |V |

∀t ∈ R, ∀m ∈ V, ∀n ∈ V \{m}

(6b)

k∈K:l(m,k,n)=1

Eqs. (7a)-(7e) express the binary variables.

Yλ ∈ {0, 1}

∀λ ∈ W

i,k,λ Pd,mn ∈ {0, 1} i,k,λ Mmn ∈ {0, 1}

Λti,k ∈ {0, 1} Qti,k,n ∈ {0, 1}

(7a)

∀i ∈ V, ∀k ∈ K, ∀d ∈ V \{i}, ∀(m, n) ∈ E

(7b)

∀i ∈ V, ∀k ∈ K, ∀λ ∈ W, ∀(m, n) ∈ E

(7c)

∀i ∈ V, ∀k ∈ K, ∀t ∈ R ∀i ∈ V, ∀k ∈ K, ∀n ∈ V \{i}, ∀t ∈ R

(7d) (7e)

4. Proposed Knapsack based Multicast Traffic Grooming Algorithm This section presents a light-tree based multicast traffic grooming algorithm, named 0/1 Knapsack based Multicast Traffic Grooming (0/1 KMTG). The main objective of this algorithm is to improve the resource utilization of different multicast requests with proper sharing of the light-trees or sub-lighttrees. In this approach, we maintain a balance between the bandwidth capacity and the number of used resources, which are wavelengths, transceivers and splitters in the network. Note that the decision version of the 0/1 Knapsack problem is an NP-complete [44]. Several heuristic approaches based on dynamic programming, branch and bound, greedy, and genetic algorithms are used to handle 0/1 Knapsack problem within a polynomial time [45]. We propose a 0/1 Knapsack-based heuristic greedy approach, similar to [45], for solving grooming problem. The detail steps of the proposed algorithm are presented in Algorithm 1. The proposed 0/1 KMTG is designed to work in three phases, namely (i) multicast tree generation, (ii) multicast traffic grooming, and (iii) wavelength assignment. These phases are discussed in the following. (i) Multicast tree generation phase: an individual path is found from the source node to a destination node using the Dijkstra’s shortest path algorithm [44]. For constructing the multicast tree, this process is repeated for the complete destination set of the request, and then union of all individual paths are considered to form a multicast tree. 10

(ii) Multicast traffic grooming phase: multicast requests are sorted in a descending order of their number of destination nodes. Let R be an order set; R = {t1 , t2 , · · · , t|R| }, where |Dt1 | ≥ |Dt2 | ≥ · · · ≥ |Dt|R| |. It then tries to find as many common destinations as possible using the longest common subsequences (LCS) algorithm [44]. If the common destination nodes among |R | multicast requests can not be found, then it tries to find the common destination nodes among |R | − 1 requests. This process is continued until all the possible multicast requests are considered. We find the number of matched destinations for all requests in R with respect t1 . Let Mi be the number of matched destination nodes between t1 and ti+1 , where i = 1, 2, · · · , |R | − 1. Now we perform Δi = |Dt1 | − Mi , where i = 1, 2, · · · , |R |. We estimate the grooming ratio Gi , which is defined as a ratio between Δi and fti . We sort the multicast requests in an increasing order of their grooming ratio. Let R be an order set; R = {t1 , t2 , · · · , t|R| }, where |gt1 | ≤ |gt2 | ≤ · · · ≤ |gt|R| |. Finally, we groom all the requests in R according to bandwidth capacity. (iii) Wavelength assignment phase: the first fit wavelength assignment policy [13] is used for establishing light-trees under wavelength continuity and bandwidth capacity constraints. The proposed algorithm is explained with an example in the following. Algorithm 1: 0/1 Knapsack based Multicast Traffic Grooming (0/1 KMTG) Input : Network, channel capacity, and set of multicast requests Output: Multicast trees, grooming of multicast trees, and wavelength assignment Step 1: Multicast tree generation: (a) Shortest paths between a source node and destination set are found of a multicast request. (b) A multicast tree is formed by taking the union of all individual shortest paths. (c) Multicast trees are constructed for all requests Step 2: Multicast traffic grooming: (a) Arrange all multicast requests in a decreasing order of their destination size. (b) Find the maximum destination size for all multicast requests. (c) Estimate the number of matched destinations using LCS method between maximum destination size and other possible destination sizes for all requests. (d) Estimate the difference between maximum destination size and matched destination size for all requests. (e) Estimate grooming ratios for all requests and arrange them in an increasing order. (f) Groom the multicast requests and update link capacity. Step 3: Wavelength Assignment: (a) The first fit wavelength selection policy is used for establishing light-trees.

11

Table 1: Ten multicast requests for the six-node network using proposed algorithm.

Request index (i)

Source (sti )

Destination OC set (Dti ) (fti )

No. of destination (|Dti |)

No. of maximum destinations (max(|(Dti |))

1 2 3 4 5 6 7 8 9 10

0 2 3 0 0 0 4 1 1 1

1, 3, 4, 1, 1, 1, 5 4, 2, 2,

4 3 2 3 4 3 1 2 3 3

4 4 4 4 4 4 4 4 4 4

2, 4, 5 3, 2, 2,

4, 5 5

12 3 1 12 12 1 3 12 12 12

5 4, 5 5

5 4, 5 4, 5

No. of matched destinations (Mi ) = max(|(Dti |) ∩ Dti 4 2 2 2 4 2 1 2 3 3

Δi fti

Δi = max(|(Dti |) -Mi

Gi =

4-4=0 4-2=2 4-2=2 4-2=2 4-4=0 4-2=2 4-1=3 4-2=2 4-3=1 4-3=1

0/12=0 2/3=0.67 2/2=1 2/12=0.167 0/12=0 2/1=2 3/3=1 2/12=0.167 1/12=0.08 1/12=0.08

Table 2: Light-tree/lightpath based logical routes during traffic grooming

Requests (ti ) 1 5 9 10

1

Light-tree (0→ 1), (1→ 2), (2→ 4, 5) (0→ 1), (1→ 2), (2→ 4, 5) (1→ 2), (2→ 4, 5) (1→ 2), (2→ 4, 5)

2

0

(0→ 1) (0→ 1)

Fiber link Request t1, t5 OC-12 Request t9, t10 OC-12 5

3

Lightpath

4

Lightpath Receiver Transmitter Splitter Grooming

Figure 1: Six-node network.

4.1. Example This subsection explains the proposed algorithm with an example. For this purpose, we consider a sample network consisting of six nodes and eight edges, as shown in Fig. 1. We consider ten randomly generated multicast requests, which are shown in Table 1. We assume that the wavelength capacity of each channel is OC-48 and the bandwidth requirement of multicast requests can be one of OC-1, OC-3, OC-12 or OC-48. Let |Dti | be the number of destinations for multicast request ti . We calculate the maximum destination nodes, denoted by max(|(Dti |), among all requests. We estimate the number of matched destinations, denoted by Mi , by performing the intersection operation between max(|(Dti |) and |Dti |. We estimate the difference, denoted by Δi , between max(|Dti |)) and Mi . For each multicast request i, we take the grooming ratio, denoted by Gi between Δi and fti . We groom the multicast 12

requests in an increasing order of grooming ratios based on the wavelength channel capacity. Multicast requests t1 and t5 have the common source (node 0) and destination set (nodes 1, 2, 4, and 5); their grooming ratio is zero. We groom these two requests and update the bandwidth of each link (0 → 1, 1 → 2, 2 → 4, 5) with OC-24, as shown in Fig 1. Furthermore, multicast requests t9 and t10 are considered for grooming according to their increasing grooming ratio, which is 0.08. The bandwidth capacity of the multicast-tree (1 → 2) and (2 → 4, 5) is satisfied with OC-48. In the similar way, other multicast requests are groomed within channel capacity. 4.2. Complexity Analysis In the first phase, the time to find a path between a source-destination pair is O(|V |2 ), where V represents the set of nodes in the network. The time to generate a multicast tree is O(|V |3 ). The time to generate multicast trees for |R| multicast requests is O(|R| · |V |3 ). In the second phase, the time to find longest common subsequence match for |R| multicast requests, each of which consists of a set of destinations, is O(|R| · |V |2 ). The time to groom of |R| multicast requests, each of which consists of a set of destinations, is O(|R| · |V |). In the third phase, the time to assign |R| multicast requests using the first fit allocation policy is O(|R| · |W |), where W represents the set of wavelengths in each link. Therefore, the overall time complexity of the proposed algorithm is O(|R| · |V |3 ). 9

7 1

10

8 3

6 5

0

11

4

2 12 13

Figure 2: 14-node NSFNET.

9

5

8

10 4 7

2 0 3 1 12

6

11 13 14

16

Figure 3: 17-node German network.

13

15

5. Performance evaluation This section presents the ILP and simulation results of the proposed knapsack based multicast traffic grooming approach. We used a 64-bit Windows-based computer with Intel (R) Core (TM) i7-5500U CPU @ 2.40 GHz and 8 GB memory for the purpose of evaluation. The presented ILP was solved by the IBM(R) ILOG(R) CPLEX(R) Interactive Optimizer 12.7.0.0. In our simulation, multicast requests are randomly generated between several source and destinations set of a given network. In the following subsections, we focus our discussion on numerical results obtained by ILP for sample networks followed by numerical results for backbone networks obtained by simulation. 5.1. ILP results for sample networks For the evaluation of ILP, we consider six-node and four-node sample networks, which are shown in Figs. 4 and 5, respectively. Five and ten randomly generated multicast requests are considered as input to the presented ILP problem for six-node and four-node networks, which are shown in Table 3. When we evaluate for five requests, we consider first five-requests from Table 3 for both networks. We assume that the capacity of a wavelength is OC-12, and the required bandwidth of multicast requests are given in Table 3. We assume that the costs of a transmitter, receiver, splitter, and a wavelength are three, three, one, and one [unit], respectively.

3

2 1

6 4

5

Figure 4: Six-node sample network.

1

2

3

4

Figure 5: Four-node sample network.

Table 4 compares the ILP results with the proposed heuristic in terms of network cost, number of wavelengths, number of transmitters, number of receivers, number of splitters used in the network, and solvable time. We observe that the solvable time of ILP increases with increase in network size. When the number of multicast requests increases in a network, the solvable time also increases. As expected, 14

Table 3: Input data for both four-node and six-node networks

Request index

Four-node network

Six-node network

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Bandwidth requirement OC-3 OC-3 OC-3 OC-3 OC-3 OC-1 OC-3 OC-1 OC-12 OC-12 OC-3 OC-3 OC-12 OC-3 OC-3 OC-3 OC-1 OC-3 OC-3 OC-3

Source

Destinations

1 4 1 2 3 1 4 1 2 3 1 4 1 6 6 1 6 5 6 2

2, 3 3 4 3 1, 4 2, 3 3 3, 4 1, 3, 4 1, 4 2, 3, 4, 6 1, 2, 3, 6 2 1, 3, 4, 5 1, 4, 5 2, 4, 5, 6 2, 3 2, 3, 6 1, 2, 4 1, 4, 5, 6

the optimal network cost using the presented ILP is less than that of the proposed heuristic algorithm. However, the proposed heuristic algorithm takes relative less time to find the near optimum solution when ten multicast requests are considered for six-node network; ILP and the proposed algorithm need 87583.89 and 0.33 seconds for obtaining the results. Table 4: Comparison results of ILP and proposed heuristic algorithm for six-node and four-node networks.

Network

Four-node

Six-node

Approach

No. of requests

Network cost

ILP Heuristic ILP Heuristic ILP Heuristic ILP Heuristic

5 5 10 10 5 5 10 10

25 32 62 69 41 47 66 74

Used wavelength 1 1 2 2 1 1 1 2

Required transmitters 4 5 10 11 6 7 9 11

Required receivers

Required splitters

Solvable time [sec]

4 5 10 11 7 8 12 12

0 1 0 1 1 1 2 3

0.14 0.29 0.75 0.30 14.25 0.31 87583.89 0.33

5.2. Results for backbone networks This subsection presents the simulation results using the proposed algorithm for two backbone networks, namely 14-node NSFNET [13] and 17-node German network [13] as shown in Fig. 2 and Fig. 3, respectively. Note that the ILP is intractable for both backbone networks. In this simulation, multicast requests are randomly generated. We assume that the wavelength channel capacity is OC-48, the bandwidth requirement of multicast requests can be one of OC-1, OC-3, OC-12 or OC-48. All multicast 15

requests are independent of each other. We ran the simulation using 100 different seeds. In this simulation, we consider three different algorithms, namely (i) 0/1 KMTG, (ii) SLTSG [7], and (iii) logical-first sequential routing with multi-hop grooming (LFSEQMH) [23]. Figs. 6 and 7 show the number of required wavelengths versus the number of multicast requests, obtained by using 0/1 KMTG, SLTSG, and LFSEQMH, for NSFNET and the German network, respectively. In this simulation, we consider that the number of destinations for each multicast request is eight. We observe that the proposed 0/1 KMTG achieves better wavelength utilization than that of existing SLTSG and LFSEQMH. This is because, in 0/1 KMTG, larger set of multicast requests that are assigned with larger bandwidths have higher precedence of grooming compared to smaller request size with less bandwidth capacity. Therefore, in 0/1 KMTG, due to better grooming effect a small number of wavelengths are required compared to SLTSG and LFSEQMH. 40 No. of destinations = 8

Number of wavelengths

35 30 25 20 15 10

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

5 0

0

20

40 60 80 Number of multicast requests

100

Figure 6: Relationship between required wavelengths and number of multicast requests for NSFNET.

40 Number of destinations = 8

Number of wavelengths

35 30 25 20 15 10

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

5 0

0

20

40 60 Number of requests

80

100

Figure 7: Relationship between required wavelengths and number of multicast requests for German network.

The comparison of 0/1 KMTG with the existing algorithms regarding the number of required transceivers 16

for each request is shown in Figs. 8 and 9 for NSFNET and the German network, respectively. In this simulation, we consider that the number of multicast requests is 100. 0/1 KMTG achieves better result than that of existing SLTSG and LFSEQMH when the number of destinations increases. In 0/1 KMTG, a large number of destinations are covered with respect to the bandwidth assigned for each multicast request, and hence it provides better grooming than existing methods. Again, SLTSG produces better result than that of LFSEQMH. This is because SLTSG uses saturated grooming that covers large number of destinations irrespective of the bandwidth assigned to the requests. In case of LFSEQMH, grooming always takes place when requests have common set of destinations with different sources, which incurs maximum number of wavelengths in order to satisfy the multicast requests.

Number of transceivers per request

9 8

No. of requests = 100

7 6 5 4

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

3 2

2

4

6 8 Number of destinations

10

12

Figure 8: Comparison of number of required transceivers for each request in NSFNET.

Number of transceivers per request

12

No. of requests = 100

10 8

6 4

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

2 2

4

6

8 10 Number of destinations

12

14

Figure 9: Comparison of number of required transceivers for each request in German network.

Figs. 10 and 11 show the number of required wavelengths versus the number of destinations, obtained by using 0/1 KMTG, SLTSG, and LFSEQMH, for NSFNET and the German network, respectively. In this simulation, we consider that the number of multicast requests is 100. When the number of destina17

tions increases, 0/1 KMTG provides better wavelength utilization compared to SLTSG and LFSEQMH. This is because, in 0/1 KMTG, traffic requests are groomed better than that of the existing approaches when the number of destinations increases in the network. As a result, 0/1 KMTG requires lesser number of wavelengths compared to existing SLTSG and LFSEQMH. In SLTSG, requests with large number of destinations have higher priority for grooming compared to the requests with small number of destinations; these request are independent of the bandwidth capacity. In 0/1 KMTG, grooming ratio of requests depends on the number of destinations and allocated bandwidth capacity.

No. of requests = 100

45

Number of wavelengths

40 35 30 25 20

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

15 2

4

6 8 Number of destinations

10

12

Figure 10: Number of required wavelengths versus number of destinations for NSFNET.

50

Number of wavelengths

45

No. of requests = 100

40 35 30 25 20

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

15 10

2

4

6

8 10 Number of destinations

12

14

Figure 11: Number of required wavelengths versus number of destinations for German network.

The comparisons of network cost for each request using three algorithms with respect to number of destinations are demonstrated in Figs. 12 and 13 for NSFNET and the German network, respectively. In this simulation, we consider that the number of multicast requests is 100. The results indicate that the network cost for each request using 0/1 KMTG is lower than that of SLTSG and LFSEQMH. This is because in 0/1 KMTG, the destinations matching among multicast requests is better than that of existing 18

algorithms. Cost factor depends on the number of used wavelengths, electronic equipments (transmitters and receivers) and optical splitters in the network, which indicates that the network resources are efficiently utilized by 0/1 KMTG compared to existing algorithms. 30

Network cost per request

No. of requests = 100 25

20

15

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

10

2

4

6 8 Number of destinations

10

12

Figure 12: Network cost versus number of destinations for each request using different algorithms in NSFNET.

40 No. of requests = 100

Network cost per request

35 30 25 20 15

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

10 5

2

4

6

8 10 Number of destinations

12

14

Figure 13: Network cost versus number of destinations for each request using different algorithms in German network.

Figs. 14 and 15 compare the splitter requirement for each multicast request using three algorithms. In this simulation, we consider that the number of multicast requests is 100. We observe that the number of required splitters using 0/1 KMTG is lower than that of existing SLTSG and LFSEQMH. This is because 0/1 KMTG shares light-trees or sub-light-trees more efficiently compared to existing methods.

6. Conclusion This paper proposed a light-tree based heuristic algorithm, called 0/1 knapsack based multicast traffic grooming, in order to minimize the network cost by reducing the number of higher layer electronic and optical devices, such as transmitters, receivers, and splitters, and used wavelengths in the network. The 19

No. of requests = 100

Number of splitters per request

3

2.5

2

1.5

1

0.5

0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional) 2

4

6 8 Number of destinations

10

12

Figure 14: Number of splitters versus number of destinations for each request in NSFNET.

3.5 Number of splitters per request

No. of requests = 100 3 2.5 2 1.5 0/1 KMTG (Proposed) SLTSG (Conventional) LFSEQMH (Conventional)

1 0.5

2

4

6

8 10 Number of destinations

12

14

Figure 15: Number of splitters versus number of destinations for each request in German network.

proposed algorithm constructs light-trees or sub light-trees, which satisfy sub bandwidth demands of all multicast requests. We presented a light-tree based integer linear programming (ILP) formulation to minimize the network cost. We solved the ILP problem for sample four-node and six-node networks and compared the ILP results with the proposed heuristic algorithm. We observed that the performance of the proposed algorithm is comparable to the ILP in terms of cost. When the introduced ILP is not tractable for large network, the proposed algorithm still able to find the results. Furthermore, we compared the proposed heuristic algorithm to the existing heuristic algorithms for different backbone networks. We observed that the proposed heuristic algorithm outperforms the conventional algorithms in terms of resource utilization, such as wavelengths, transceivers, splitters. We further observed that the network cost using the proposed heuristic algorithm significantly decreases when the number of requests increases in the network. We observed that the network costs of each request using 0/1 KMTG are 24% and 31% lower compared to existing SLTSG and LFSEQMH, respectively.

20

Acknowledgment This work was supported in part by the Inspire Faculty Scheme (No. DST/INSPIRE/04/2016/001316), Department of Science & Technology, New Delhi, India and JSPS KAKENHI Grant Number 15K00116, Japan.

References [1] A.L. Chiu and E.H. Modiano. Traffic grooming algorithms for reducing electronic multiplexing costs in WDM ring networks. J. Lightw. Technol., 18(1):2 –12, 2000. [2] K. Zhu and B. Mukherjee. Traffic grooming in an optical WDM mesh network. IEEE J. Sel. Areas Commun., 20(1):122–133, 2002. [3] A.E. Kamal and R.U. Mustafa. Multicast traffic grooming in wdm networks. in Proc. of OptiComm, pages 25–36, 2003. [4] C. Xin, C. Qiao, and S. Dixit. Traffic grooming in mesh WDM optical networks-performance analysis. IEEE J. Sel. Areas Commun., 22(9):1658–1669, 2004. [5] Y.R. Yoon, T.J. Lee, M.Y. Chung, and H. Choo. Traffic groomed multicasting in sparse-splitting wdm backbone networks. in Proc. of ICCSA, pages 534–544, 2006. [6] S. Barat, A.K. Pradhan, and T. De. A cost efficient multicast routing and wavelength assignment in WDM mesh network. in Proc. of CISIM, pages 65–73, 2011. [7] R. Lin, W.D Zhong, S.K. Bose, and M. Zukerman. Design of WDM networks with multicast traffic grooming. J. Lightw. Technol., 29(16):2337–2349, 2011. [8] R. Lin, W.D. Zhong, S.K. Bose, and M. Zukerman. Multicast traffic grooming in tap-and-continue WDM mesh networks. IEEE/OSA J. Opt. Commun. Netw., 4(11):918–935, 2012. [9] A.R.B. Billah, B. Wang, and A.A.S. Awwal. Multicast traffic grooming in WDM optical mesh networks. in Proc. IEEE GLOBECOM, pages 2755 – 2760, 2003. [10] R.U. Mustafa and A.E. Kamal. Design and provisioning of WDM networks with multicast traffic grooming. IEEE J. Sel. Areas Commun., 24(4):37–53, 2006. [11] T. De, A. Pal, and I. Sengupta. Traffic grooming, routing, and wavelength assignment in an optical WDM mesh networks based on clique partitioning. Photon. Netw. Commun., 20(2):101–112, 2010. [12] A.K. Pradhan and T. De. Multicast traffic grooming based light-tree in WDM mesh networks. Procedia Technology, 10(0):900 – 909, 2013. [13] B. Mukherjee. Optical WDM networks. Springer Science & Business Media, 2006. [14] J. Wang, X. Qi, and B. Chen. Wavelength assignment for multicast in all-optical WDM networks with splitting constraints. IEEE/ACM Trans. Netw., 14(1):169–182, 2006. [15] X.H. Jia, D.Z. Du, X.D. Hu, M.K. Lee, and J. Gu. Optimization of wavelength assignment for qos multicast in WDM networks. IEEE Trans. on Commun., 49(2):341–350, 2001. [16] A.E. Ozdaglar and D.P. Bertsekas. Routing and wavelength assignment in optical networks. IEEE/ACM Trans. Netw., 11(2):259–272, 2003. [17] G.S. Poo and Y. Zhou. A new multicast wavelength assignment algorithm in wavelength-routed WDM networks. IEEE J. Sel. Areas Commun., 24(4):2–12, 2006. [18] D.N. Yang and W. Liao. Design of light-tree based logical topologies for multicast streams in wavelength routed optical networks. in Proc. of INFOCOM 2003, pages 32–41, 2003.

21

[19] R. Lin, W.D. Zhong, S.K. Bose, and M. Zukerman. Constrained light-tree design for WDM mesh networks with multicast traffic grooming. Opt. Switching Netw., 10(3):233 – 245, 2013. [20] A.K Pradhan, S. Araiyer, and T. De. Design of light-tree based multicast traffic grooming in WDM mesh networks. J. of Opt., 43(4):330–340, 2014. [21] H. Zhu, H. Zang, K. Zhu, and B. Mukherjee. A novel generic graph model for traffic grooming in heterogeneous WDM mesh networks. IEEE/ACM Trans. Netw., 11(2):285–299, 2003. [22] B.C. Chatterjee, N. Sarma, and P.P. Sahu. Priority based routing and wavelength assignment with traffic grooming for optical networks. IEEE/OSA J. Opt. Commun. Netw., 4(6):480–489, 2012. [23] A. Khalil, C. Assi, A. Hadjiantonis, G. Ellinas, and M. A. Ali. Dynamic provisioning of low-speed unicast/multicast traffic demands in mesh-based WDM optical networks. J. Lightw. Technol., 24(2):681–693, 2006. [24] R. Lin, W.D. Zhong, S.K. Bose, and M. Zukerman. Dynamic sub-light-tree based traffic grooming for multicast in WDM networks. in Proc. of GLOBECOM, pages 1–5, 2010. [25] L. Guo, W. Hou, J. Wu, and Y. Li. Multicast multi-granular grooming based on integrated auxiliary grooming graph in optical networks. Photon. Netw. Commun., 24(2):103–117, 2012. [26] N. Kaliammal and G. Gurusamy. Performance analysis of multicast routing and wavelength assignment protocol with dynamic traffic grooming in WDM networks. Int. J. of Commun. Sys., 26(2):198–211, 2013. [27] X. Huang, F. Farahm, and J.P. Jue. Multicast traffic grooming in wavelength-routed WDM mesh networks using dynamically changing light-trees. J. Lightw. Technol., 23(10):3178–3187, 2005. [28] H.L. Liu, X. Xue, Y. Chen, Q. Fang, and S. Huang. An efficient dynamic multicast traffic-grooming algorithm for WDM networks. Photon. Netw. Commun., 26(2-3):95–102, 2013. [29] H.C. Lin, Y.X. Zhuang, and M.Y. Lin. Bandwidth-ratio-based light-tree selection in dynamic multicast traffic grooming for optical WDM mesh networks. Photon. Netw. Commun., 29(2):164–182, 2015. [30] A.K. Pradhan, S. Keshri, K. Das, and T. De. A heuristic approach based on dynamic multicast traffic grooming in WDM mesh networks. J. of Opt., pages 1–11, 2016. [31] O. Gerstel, M. Jinno, A. Lord, and S.B. Yoo. Elastic optical networking: A new dawn for the optical layer? IEEE Commun. Mag., 50(2), 2012. [32] B.C. Chatterjee, N. Sarma, and E. Oki. Routing and spectrum allocation in elastic optical networks: A tutorial. IEEE Commun. Surveys Tuts., 17(3):1776–1800, 2015. [33] P. Lu, L. Zhang, X. Liu, J. Yao, and Z. Zhu. Highly efficient data migration and backup for big data applications in elastic optical inter-data-center networks. IEEE Netw., 29(5):36–42, 2015. [34] G. Zhang, M.D. Leenheer, A. Morea, and B. Mukherjee. A survey on OFDM-based elastic core optical networking. IEEE Commun. Surveys Tuts., 15(1):65–87, 2013. [35] G. Zhang, M.D. Leenheer, and B. Mukherjee. Optical traffic grooming in OFDM-based elastic optical networks. IEEE/OSA J. Opt. Commun. Netw., 4(11):B17–B25, 2012. [36] S. Zhang, C. Martel, and B. Mukherjee. Dynamic traffic grooming in elastic optical networks. IEEE J. Sel. Areas Commun., 31(1):4–12, 2013. [37] B. Kozicki, H. Takara, and M. Jinno. Enabling technologies for adaptive resource allocation in elastic optical path network (slice). in Proc. of IEEE ACP, pages 23–24, 2010. [38] Z. Zhu, X. Liu, Y. Wang, W. Lu, L. Gong, S. Yu, and N. Ansari. Impairment- and splitting-aware cloud-ready multicast provisioning in elastic optical networks. IEEE/ACM Trans. Netw., 25(2):1220–1234, 2017. [39] L. Yang, L. Gong, F. Zhou, B. Cousin, M. Molnár, and Z. Zhu. Leveraging light forest with rateless network coding to design efficient all-optical multicast schemes for elastic optical networks. J. Lightw. Technol., 33(18):3945–3955, 2015.

22

[40] K. Walkowiak, R. Goścień, M. Klinkowski, and M. Woźniak. Optimization of multicast traffic in elastic optical networks with distance-adaptive transmission. IEEE Commun. Lett., 18(12):2117–2120, 2014. [41] X. Liu, L. Gong, and Z. Zhu. On the spectrum-efficient overlay multicast in elastic optical networks built with multicast-incapable switches. IEEE Commun. Lett., 17(9):1860–1863, 2013. [42] L. Gong, X. Zhou, X. Liu, W. Zhao, W. Lu, and Z. Zhu. Efficient resource allocation for all-optical multicasting over spectrum-sliced elastic optical networks. IEEE/OSA J. Opt. Commun. Netw., 5(8):836–847, 2013. [43] F.K. Hwang, D.S. Richards, and P. Winter. The Steiner tree problem, vol. 53. Elsevier, 1992. [44] J. Kleinberg and E. Tardos. Algorithm design. Pearson Education India, 2006. [45] M. Hristakeva and D. Shrestha. Different approaches to solve the 0/1 knapsack problem. in Proc. of the Midwest Instruction and Computing Symposium, pages 1–15, 2005.

23