Elsevier

Information Processing Letters

Volume 113, Issue 18, 15 September 2013, Pages 677-680
Information Processing Letters

Routing and wavelength assignment for 3-ary n-cube communication patterns in linear array optical networks for n communication rounds

https://doi.org/10.1016/j.ipl.2013.06.007Get rights and content

Highlights

  • In this paper, we considered Qn3,b communication patterns and Lnb optical networks.

  • All packets are transmitted one dimension after another.

  • Embedding scheme ϕ was designed and the congestion about dimensions under ϕ was obtained.

  • The optimal number of wavelengths under ϕ was achieved.

  • An optimal routing and wavelength assignment strategy was designed.

Abstract

k-ary n-cubes are a class of communication patterns that are employed by a number of typical parallel algorithms. This paper addresses the implementation of parallel algorithms with bidirectional 3-ary n-cube communication patterns on a bidirectional linear array WDM optical networks when the information is transmitted one dimension after another. By giving an embedding scheme ϕ, we prove the optimal number of wavelengths under ϕ and design a routing and wavelength assignment strategy of it.

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 x,y 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 Qn3 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 4×3n2+2.

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 Qnk (k2 and n1) has N=kn vertices, each of the form x=(xn1,xn2,,x0), where 0xik1 for every 0in1. Two vertices x=(xn1,xn2,,x0) and y=(yn1,yn2,,y0) in Qnk are adjacent if and only if there exists an integer j, 0jn1, such that xj=yj±1(modk) and xi=yi for i{0,1,2,,n1}{j}.

Fig. 1 depicts a 3-ary 3-cube.

Definition 2

(See [9].) The natural numbering of Qnk assigns to each vertex x=(xn1,xn2,,x0) the number 1+i=0n1xiki.

Denote by DIMn,i, where 0in1, the set

Embedding scheme ϕ of Qn3 in Ln

First, we give an embedding scheme ϕ of Qn3 in Ln as follows:

(1) Embed the vertex whose natural numbering is 1 into the first vertex of Ln, let i=1;

(2) Embed the vertices which are not embedded into Ln before such that all the vertices are adjacent to i, and the natural numberings of them increase one by one. Denote by Si the set of all these vertices;

(3) If i<3n1, then let i=i+1, goto (2);

It is obvious that all the vertices of Qn3 have been embedded into Ln. In the following, we consider the

The congestion about dimensions of Qn3 in Ln

Lemma 1

DiCong(Qn3;Ln;ϕ)Dn1Cong(Qn3;Ln;ϕ), where 0i<n1.

Proof

Case 1: i=n2. We can partition Qn3 along dimension n1, by deleting all the edges of DIMn,(n1) into three disjoint subcubes, denoted Q(n1),03, Q(n1),13, Q(n1),23, respectively. If 1<j+13n2, {j+1+3n2, j+1+2×3n2, j+1+3n1, j+1+2×3n1}=ASj+1, i.e., |Sj+1|4 and for any vertex kSj+1A, k is adjacent to two vertices out of Sj+1. Combining with kjn2=Dn2Cong(Qn3;Ln;ϕ;(j+3n1,j+2×3n1)), we have κj+1n2=κjn2+2+2(|Sj+1|4). Then κj+1n2>

Routing and wavelength assignment

Since bidirectional graph can be considered as undirected graph, combining Theorem 2 and [1], we have:

Lemma 3

λn,ϕ(Qn3,b;Lnb)4×3n2+2.

Theorem 3

The optimal number of wavelengths required to realize Qn3,b on Ln under ϕ is 4×3n2+2.

A routing and wavelength assignment is given below, in which the number of wavelengths is just 4×3n2+2, where ωcyclet(k,i) is the wavelength assigned to cyclet(k,i) (shown in Fig. 4), where t{1,2}.

Conclusion

In this paper, we have discussed the routing and wavelength assignment for bidirectional Qn3 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 4×3n2+2 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)

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