Nonpreemptive Scheduling of Optical Switches

Abstract

Abstract-Many high-speed routers today use Input-Queued (IQ) architectures with a crossbar switching fabric based on optical technology. Packets in the input queues are divided into cells of unit length and the goal is to find a schedule of minimum makespan that forwards all packets to the output ports. The problem is complicated since in optical switches so called configuration delay, that is the time required to reconfigure the switching fabric, is non-negligible with respect to the cell transmission time. We aim to design a scheduler whose complexity does not depend on the number of packets in the input queues. Thus, we focus on the Non-Preemptive Bipartite Scheduling (NPBS) problem, where each input queue is connected to each output port in at most one configuration. We demonstrate that the NPBS problem is NP-hard for any value of the configuration delay and approximation within a ratio smaller than 7/6 is NP-hard as well. For the offline version of the NPBS problem, we show that a simple greedy algorithm achieves an approximation factor of 2 for arbitrary configuration delay. Then we consider the online version of the NPBS problem, where the switch gathers the incoming traffic periodically and then schedules the accumulated batches (batch scheduling). We propose a scheduling algorithm which guarantees strict delay for any admissible traffic provided that the switch has a moderate speedup of two.

Approximating the Max Edge-Coloring Problem

“…The most interesting and general result for the MEC problem is due to Kesselman and Kogan [18] who proposed the following greedy algorithm: Algorithm KK 1. Let E = e 1 , e 2 , .…”

“…In [18], it has been shown that Algorithm KK is a 2-approximation one and an example has been presented yielding an approximation ratio of 2 − 1 ∆ . By a slightly tighter analysis we prove here the next lemma.…”

“…Concerning the approximability of the MEC problem, a natural greedy 2-approximation algorithm for general graphs has been proposed in [18]. For bipartite graphs of maximum degree ∆ = 3, an algorithm that attains the 7/6 inapproximability bound has been presented in [8].…”

“…Although a 2-approximation algorithm is known for the MEC problem on general graphs, no algorithm of ratio 2 − δ, for any small constant δ > 0, is known for any special graph class. Apart from their theoretical interest, special graph classes, like bipartite graphs and trees, are also motivated by practical applications [18,24]. Towards this direction we present approximation algorithms for the MEC problem on general and bipartite graphs, trees and graphs with bi-valued edge weights.…”

“…The next section starts with our notation and a remark on the known greedy 2-approximation algorithm [18]. By combining this remark with a simple idea, we present a first approximation algorithm for general and bipartite graphs which already beats the best known ratios for these classes.…”