Constructions of Optical Queues With a Limited Number of Recirculations--Part II: Optimal Constructions

One of the main problems in all-optical packet-switched networks is the lack of optical buffers, and one feasible technology for the constructions of optical buffers is to use optical crossbar Switches and fiber Delay Lines (SDL). In this two-part paper, we consider SDL constructions of optical queues with a limited number of recirculations through the optical switches and the fiber delay lines. Such a problem arises from practical feasibility considerations. In Part I, we have proposed a class of greedy constructions for certain types of optical queues, including linear compressors, linear decompressors, and 2-to-1 FIFO multiplexers, and have shown that every optimal construction among our previous constructions of these types of optical queues under the constraint of a limited number of recirculations must be a greedy construction. Specifically, given M ≥ 2 and 1 ≤ k ≤ M − 1, we have shown that to find an optimal construction, it suffices to find an optimal sequence d1 ∈ GM,k such that B(d1 ; k) = maxdM1 ∈GM,k B(d M 1 ; k), where B(d M 1 ; k) is the maximum representable integer with respect to d1 and k (defined in Part I) and GM,k is the set of sequences d1 = (d1, d2, . . . , dM ) given by dsi+j = B(d si+j−1 1 ; i + 1) + 1 for i = 0, 1, . . . , k − 1 and j = 1, 2, . . . , ni+1 for some sequence n1 = (n1, n2, . . . , nk) with n1 ≥ 2, n2, n3, . . . , nk ≥ 1, and ∑k i=1 ni = M , in which s0 = 0 and si = ∑i `=1 n` for i = 1, 2, . . . , k. In Part II, the present paper, we further show that there are at most two optimal constructions and give a simple algorithm to obtain the optimal construction(s). The main idea in Part II is to use pairwise comparison to remove a sequence d1 ∈ GM,k such that B(d1 ; k) < B(d′ M 1 ; k) for some d′ M 1 ∈ GM,k. To our surprise, the simple algorithm for obtaining the optimal construction(s) is related to the well-known Euclid’s algorithm for finding the greatest common divisor (gcd) of two integers. In particular, we show that if gcd(M,k) = 1, then there is only one optimal construction; if gcd(M,k) = 2, then there are two optimal constructions; and if gcd(M,k) ≥ 3, then there are at most two optimal constructions. Index Terms Euclid’s algorithm, FIFO multiplexers, integer representation, linear compressors, linear decompressors, maximum representable integer, optical buffers, optical queues, packet switching. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Contract NSC 96-2221-E-007-076, Contract NSC 97-2221-E-007-105-MY3, and the Program for Promoting Academic Excellence of Universities NSC 94-2752E-007-002-PAE. The authors are with the Department of Electrical Engineering and the Institute of Communications Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C. (e-mails: [email protected]; [email protected]). ar X iv :1 00 4. 42 77 v1 [ cs .I T ] 2 4 A pr 2 01 0