In this paper, a new class of low-hit-zone (LHZ) frequency-hopping sequence sets (LHZ FHS sets) is constructed based upon the Cartesian product, and the periodic partial Hamming correlation within its LHZ are studied. Studies have shown that the new LHZ FHS sets are optimal according to the periodic partial Hamming correlation bounds of FHS set, and some known FHS sets are the special cases of this new construction.

A queue layout of a graph G consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The queuenumber qn(G) is the minimum number of queues required in a queue layout of G. The Cartesian product of two graphs G1 = (V1,E1) and G2 = (V2,E2), denoted by G1 × G2, is the graph with {:v1 ∈ V1 and v2 ∈ V2} as its vertex set and an edge (,) belongs to G1×G2 if and only if either (u1,v1) ∈ E1 and u2 = v2 or (u2,v2) ∈ E2 and u1 = v1. Let Tk1,k2,...,kn denote the n-dimensional toroidal grid defined by the Cartesian product of n cycles with varied lengths, i.e., Tk1,k2,...,kn = Ck1 × Ck2 × … × Ckn, where Cki is a cycle of length ki ≥ 3. If k1 = k2 = … = kn = k, the graph is also called the k-ary n-cube and is denoted by Qnk. In this paper, we deal with queue layouts of toroidal grids and show the following bound: qn(Tk1,k2,...,kn) ≤ 2n-2 if n ≥ 2 and ki ≥ 3 for all i = 1,2,...,n. In particular, for n = 2 and k1,k2 ≥ 3, we acquire qn(Tk1,k2) = 2. Recently, Pai et al. (Inform. Process. Lett. 110 (2009) pp.50-56) showed that qn(Qnk) ≤ 2n-1 if n ≥1 and k ≥9. Thus, our result improves the bound of qn(Qnk) when n ≥2 and k ≥9.

System level diagnosis that can identify the faulty units in the system was introduced by Preparata, Metze, and Chien. In this area, the fundamental problem is to decide the diagnosability of given networks. We study the diagnosability of networks represented by the cartesian product. Our result is the optimal one with respect to the restriction of degrees of vertices of graphs that represent the networks.

The disk allocation problem examined in this paper is finding a method to distribute a Binary Cartesian Product File on multiple disks to maximize parallel disk I/O accesses for partial match retrieval. This problem is known to be NP-hard, and heuristic approaches have been applied to obtain suboptimal solutions. Recently, efficient methods such as Binary Disk Modulo (BDM) and Error Correcting Code (ECC) methods have been proposed along with the restrictions that the number of disks in which files are stored should be a power of 2. In this paper, a new Disk Allocation method based on Genetic Algorithm (DAGA) is proposed. The DAGA does not place restrictions on the number of disks to be applied and it can allocate the disks adaptively by taking into account the data access patterns. Using the schema theory, it is proven that the DAGA can realize a near-optimal solution with high probability. Comparing the quality of solution derived by the DAGA with the General Disk Modulo (GDM), BDM, and ECC methods through the simulation, shows that 1) the DAGA is superior to the GDM method in all the cases and 2) with the restrictions being placed on the number of disks, the average response time of the DAGA is always less than that of the BDM method and greater than that of the ECC method in the absence of data skew and 3) when data skew is considered, the DAGA performs better than or equal to both BDM and ECC methods, even when restrictions on the number of disks are enforced.