Routing and wavelength assignment for 3-ary n-cube communication patterns in linear array optical networks for n communication rounds
Introduction
In a typical WDM optical network, every fiber link can support a certain number of wavelengths, and each wavelength can carry a separate stream of data. To efficiently utilize the bandwidth resources and to eliminate the high cost and bottleneck caused by opoelectrical conversion and processing at intermediate nodes, end-to-end lightpaths are usually set up between each pair of source–destination nodes. A connection or lightpath in a WDM network is an ordered pair of nodes corresponding to transmission of a packet from source x to destination y.
The central task for WDM optical networks is to select a suitable path and wavelength for each connection of a given communication pattern so that the number of wavelengths used is minimized, with the following constrains: (1) Wavelength-continuity constraint. (2) Distinct wavelength constraint. There are some results about routing and wavelength assignments in optical networks when only one communication round is used in [2], [3], [9], [10]. But in reality, the number of wavelengths feasible is limited. So when the connections are large enough, the number of wavelengths required will outnumber that of the network can afford. In [4], [5], [6], [8] the authors considered wavelength assignment for parallel FFT communication pattern on a class of regular optical networks by giving some embedding schemes.
This paper addresses the routing and wavelength assignment for bidirectional communication patterns in linear array WDM optical networks when n communication rounds are used. This paper confines that the information is transmitted one dimension after another. By giving an embedding scheme ϕ and a routing and wavelength assignment strategy, we prove that the number of wavelengths under ϕ is .
The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, we design embedding scheme ϕ and provide some properties. In Section 4, the congestion about dimensions under ϕ is obtained. In Section 5, the optimal number of wavelengths under ϕ is achieved when the packets are transmitted one dimension after another and a routing and wavelength assignment strategy is designed. Finally, we conclude this paper in Section 6.
Section snippets
Preliminary
Definition 1 (See [7].) The k-ary n-cube ( and ) has vertices, each of the form , where for every . Two vertices and in are adjacent if and only if there exists an integer j, , such that and for .
Fig. 1 depicts a 3-ary 3-cube.
Definition 2 (See [9].) The natural numbering of assigns to each vertex the number . Denote by , where , the set
Embedding scheme ϕ of in
First, we give an embedding scheme ϕ of in as follows:
(1) Embed the vertex whose natural numbering is 1 into the first vertex of , let ;
(2) Embed the vertices which are not embedded into before such that all the vertices are adjacent to i, and the natural numberings of them increase one by one. Denote by the set of all these vertices;
(3) If , then let , goto (2);
It is obvious that all the vertices of have been embedded into . In the following, we consider the
The congestion about dimensions of in
Lemma 1 , where .
Proof Case 1: . We can partition along dimension , by deleting all the edges of into three disjoint subcubes, denoted , , , respectively. If , , , , , i.e., and for any vertex , k is adjacent to two vertices out of . Combining with , we have . Then
Routing and wavelength assignment
Since bidirectional graph can be considered as undirected graph, combining Theorem 2 and [1], we have:
Lemma 3 .
Theorem 3 The optimal number of wavelengths required to realize on under ϕ is .
A routing and wavelength assignment is given below, in which the number of wavelengths is just , where is the wavelength assigned to (shown in Fig. 4), where .
Conclusion
In this paper, we have discussed the routing and wavelength assignment for bidirectional communications in bidirectional linear array WDM optical networks under a given embedding scheme ϕ when the packets are transmitted one dimension after another. We have proved that the optimal number of wavelengths required is and have designed a routing and wavelength assignment strategy which reaches the optimal number of wavelengths. Furthermore, we will try to find the optimal number of
Acknowledgements
The authors would like to express their gratitude to the editor and the anonymous referees for their valuable suggestions about this paper. This work was supported by Doctorate Foundation of Educational Ministry of China (No. 20110191110022); the Natural Science Foundation of Chongqing (No. cstc2012jjA40039); and Research Project of Chongqing Education Committee (No. KJ120508).
References (10)
- et al.
Routing and wavelength assignment for hypercube in array-based WDM optical networks
Journal of Parallel and Distributed Computing
(2010) - et al.
Routing and wavelength assignment for 3-ary n-cube in array-based optical network
Information Processing Letters
(2012) - et al.
Graph problems arising from wavelength routing in all optical networks
- et al.
Wavelength assignment for directional hypercube communications on a class of WDM optical networks
- et al.
Wavelength assignment for realizing parallel FFT on regular optical networks
Journal of Supercomputing
(2006)
Cited by (12)
Routing and wavelength assignment for exchanged hypercubes in linear array optical networks
2015, Information Processing LettersCitation Excerpt :The primary issue for WDM optical networks is to select a proper path and wavelength satisfying the wavelength-continuity constraint and the distinct wavelength constraint for each connection of a given communication pattern so that the number of used wavelengths is minimized [13–16].
Wavelength assignment for locally twisted cube communication pattern on optical bus network-on-chip
2014, Optical Fiber TechnologyCitation Excerpt :The routing and wavelength assignment (RWA) problem is very important in optical NoC. In recent years, the RWA problem has been studied for various combinations of communication pattern and optical network [3–5,20,21]. This paper proposes a static routing and wavelength assignment scheme for locally twisted cube (LTQ) communication pattern on global bus optical NoC.
Optimal wavelength assignment in the implementation of parallel algorithms with ternary n-cube communication pattern on mesh optical network
2014, Theoretical Computer ScienceCitation Excerpt :In this paper, a lightpath must use a same wavelength on all the links along its path from the source to the destination, and all lightpaths sharing a same link (fiber) must be assigned distinct wavelengths. In recent years, the study of the RWA problem for various communication patterns implemented on a variety of WDM networks has received considerable interest [3–5,14–16]. Due to excellent properties such as symmetry, structural recursiveness and high connectivity, on one hand, the k-ary n-cube has been regarded as appealing communication pattern; indeed, quite a number of parallel algorithms have been designed based on k-ary n-cube patterns [6,8,9,12,13].
Realizing the ternary n-cube communication pattern on a ring-connected WDM optical network
2014, Optical Fiber TechnologyCitation Excerpt :In order to efficiently execute a parallel algorithm with static communication pattern on a WDM optical network, routes and wavelengths must be planned carefully so that the total number of wavelengths used is minimized, and the corresponding problem is referred to as the routing and wavelength assignment (RWA) problem. In recent years, the RWA problem has been resolved for different combinations of communication pattern and WDM network [6–9,25–29]. Due to ease in operation, on one hand, administration and maintenance, WDM networks with ring topology have been considered as promising WDM optical networks, and have been widely deployed in today’s metropolitan area network infrastructures.
Routing and Wavelength Allotment for Exchanged Folded Hypercube Communications Embedded in Bus-Topology WDM Optical Networks
2022, Mobile Networks and ApplicationsAll-to-All Broadcast in WDM Linear Array with 3-length extension
2022, WSEAS Transactions on Circuits and Systems