Implementing duplex crossed cube communication patterns on optical linear arrays
Introduction
Due to appealing properties such as extremely high bandwidth, extremely low power consumption and extremely low time delay, optical networks have been widely regarded as a promising alternative to the conventional electrical networks as a means of exchanging data between processors in a parallel computer. Wavelength division multiplexing (WDM) technique provides a feasible approach to the realization of optical networks; by dividing the bandwidth of an optical fiber into multiple communication channels, represented by their respective wavelengths, a multiplicity of different data streams can be transmitted simultaneously across a same optical fiber. The current WDM technology has already allowed sixty wavelengths per fiber.
To execute a parallel algorithm on a WDM network efficiently, it is crucial to solve the corresponding routing and wavelength assignment (RWA) problem, i.e., that of finding all source-destination paths in the underlying communication pattern and assigning appropriate wavelengths to these paths so that the total number of wavelengths is minimized. In recent years, the RWA problem has been studied for various combinations of communication pattern and optical network [1], [4], [6], [7], [13], [14]. In most previous work, it is assumed that no wavelength converter is available in a WDM optical network, implying that, for the purpose of effective communication between a source node and a destination node along a path, a dedicated lightpath, which uses a same wavelength on all the links along the path, has to be established. In this context, different lightpaths sharing a same link (optical fiber) must be assigned distinct wavelengths. Noting that the emerging optical network-on-chips (ONoCs) possess no wavelength converters [12], the obtained results also apply to ONoCs.
Mainly due to smaller diameter (nearly half that of hypercubes of the same size), on one hand, crossed cubes are regarded as appealing communication patterns. Indeed, quite a number of parallel algorithms have been designed based on crossed cube patterns [2], [3], [8], [9], [10], [11]. Due to ease in implementation, on the other hand, linear arrays are promising topologies of WDM networks [5], [6], [13]. To the best of our knowledge, however, the RWA problem for the combination of crossed cube and linear array remains yet to be solved.
This paper addresses the efficient implementation of a crossed cube communication pattern on a light linear array. The main contributions achieved include: (1) a simple embedding scheme is proposed, together with a proof of its congestion and (2) under the proposed embedding scheme, a wavelength assignment strategy for the half-duplex pattern and full-duplex pattern is designed, accompanied with its optimality proof.
The subsequent materials of this paper are organized as follows. Section 2 introduces the basic notations and terminologies. In Section 3, the congestion of embedding a crossed cube on a linear array is presented. In Section 4, the two wavelength assignment schemes, together with their optimality proofs, are described. Finally, Section 5 summarizes this work.
Section snippets
Crossed cube
For our purpose, the communication pattern underlying a parallel algorithm can be represented by a graph G = (V, E), where vertices and edges represent processes in the pattern and communication relationships between processes. To define the crossed cube communication pattern, the following notion is necessary.
Definition 2.1 Two length-(d + 1) binary strings, x = xdxd−1 … x0 and y = ydyd−1 … y0, are pair-related, denoted x ∼ y, if either d = 1, (x, y) ∈ R = {(00, 00), (10, 10), (01, 11), (11, 01)}, or d > 1, xd = yd when d is even,
The congestion of a crossed cube in a linear array
Let LN denote a linear array with vertex set V = {0, 1, …, N − 1} and edge set E = {(i, i + 1), 0 ≤ i ≤ N − 1}, where N = 2n. In this paper, our purpose is to solve the RWA problem for CQn communication pattern on LN optical network. From Lemma 2.2, the congestion of CQn in LN is critical for the RWA problem. Thus our primarily discuss the congestion of CQn in LN in this section. First, it is critical to present an embedding scheme. Second, it is to calculate the congestion of the embedding scheme.
Now,
Half-duplex communication of CQn
By half-duplex communication, the two adjacent vertices can communicate in both directions, but only one direction at a time (not simultaneously). So one connection in CQn can be realized by one wavelength. Thus, from Lemma 2.2 and Theorem 3.1, a lower bound on the number of wavelengths for realizing half-duplex communication of crossed cube on linear array WDM optical network by NE can be obtained as follows. Lemma 4.1 The number of wavelengths required to realize half-duplex communication of CQn on Ln
Conclusion
This paper has investigated the routing and wavelength assignment problem for half-duplex and full duplex communication patterns of crossed cube in linear array WDM optical network, respectively. Employing the congestion estimation technique, we have proved that by natural embedding scheme the minimum number of wavelengths required is ⌊2n+1/3⌋ and 2· ⌊2n+1/3 ⌋, respectively. Wavelength assignment strategies reaching this minimum number of wavelengths have been designed, respectively. In addition,
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 (Grant No. 20110191110022) and Natural Science Foundation of Chongqing (Grant No. cstc2012jjA40039).
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