SONET ADMs Minimization with Divisible Paths

Abstract

We consider an optical routing problem. SONET add-drop multiplexers (ADMs) are the dominant cost factor in SONET/WDM rings. The number of SONET ADMs required by a set of traffic streams is determined by the routing and wavelength assignment of the traffic streams. In this paper we consider the version where a traffic stream may be divided into several parts and assigned different wavelengths. A specific division may increase or decrease the number of ADMs needed for a given input. Following previous work, we consider two versions. In the arc version the route of each traffic stream is given as input, and we need to decide on divisions of streams, and then to assign wavelengths so as to minimize the total number of used SONET ADMs. In the chord version the route is not prespecified, but is assigned by the algorithm, and only after this step are the divisions done and wavelengths assigned. The previously best known approximation algorithm for the arc version has a performance guarantee of 5 4 = 1.25 whereas the previously best known approximation algorithm for the chord version has a performance guarantee of 3 2 = 1.5. We improve both these results. We present a 36 29 ≈ 1.24138-approximation algorithm for the arc version and a 7 5 = 1.4-approximation algorithm for the chord version.

The chord version for SONET ADMs minimization

“…Therefore, OPT = 4(2j + 1) whereas PIM = 12j + 4. The ratio between the costs approaches 3 2 as j grows.…”

“…The optimal solution is composed of the following cycles. (0, 1), (1,3), (3,0), (1,2), (2,4), (4, 1) and (2, 0), (0, 2). Clearly, OPT = 8.…”

“…The paper [2] showed that the performance guarantee of PIM is in the interval [ 4 3 , 3 2 ]. We show that the upper bound of 3 2 on the performance guarantee of PIM given in [2] is tight, by giving a lower bound of 3 2 on the performance guarantee of the algorithm PIM. This means that PIM cannot be analyzed in a better way, and alternative algorithms need to be designed.…”

“…Therefore, we conclude that either the pseudo-cycle is empty, or it consists of two edges. In both cases, we charge the edges of the pseudo-cycle by 3 2 each, and each removed edge by 1.…”

“…Shalom and Zaks presented a different algorithm with approximation ratio 10 7 + ε [7]. Another version of the problem, where traffic streams can be divided into several parts, such that each part is treated as a separate stream was studied in [3,1], both for the arc version and the chord version.…”