A 3D-dissection (A rectangular solid dissection) is a dissection of a rectangular solid into smaller rectangular solids by planes. In this paper, we propose an O-sequence, a string of representing any 3D-dissection which is dissected by only non-crossing rectangular planes. We also present a necessary and sufficient condition for a given string to be an O-sequence.

We consider the routing for a multicast in a WDM all-optical network. We prove a min-max theorem on the number of wavelengths necessary for routing a multicast. Based on the min-max theorem, we propose an efficient on-line algorithm for routing a multicast.

This paper proves that for every positive integers n,k and any positive number ε, we can explicitly construct a DAG G with n+O(k1+ε) vertices and a constant degree such that even after removing any k vertices from G, the remaining digraph still contains an n-vertex dipath. This paper also proves that for every positive integers n,k and any positive number ε, we can explicitly construct a graph H with n+O(k2+ε) vertices and a constant degree such that even after removing any k vertices from H, the remaining graph still contains an n-vertex 2-dimensional square mesh.

For a given N-vertex graph H, a graph G obtained from H by adding t vertices and some edges is called a t-FT (t-fault-tolerant) graph for H if even after deleting any t vertices from G, the remaining graph contains H as a subgraph. For the n-dimensional cube Q(n) with N vertices, a t-FT graph with an optimal number O(tN+t2) of added edges and maximum degree of O(N+t), and a t-FT graph with O(tNlog N) added edges and maximum degree of O(tlog N) have been known. In this paper, we introduce some t-FT graphs for Q(n) with an optimal number O(tN+t2) of added edges and small maximum degree. In particular, we show a t-FT graph for Q(n) with 2ctN+ct2((logN)/C)C added edges and maximum degree of O(N/(logC/2N))+4ct.

It is a fundamental problem to construct a virtual path layout minimizing the hop number as a function of the congestion for a communication network. It is known that we can construct a virtual path layout with asymptotically optimal hop number for a mesh of trees network, butterfly network, cube-connected-cycles network, de Bruijn network, shuffle-exchange network, and complete binary tree network. The paper shows a virtual path layout with minimum hop number for a complete binary tree network. A generalization to complete k-ary tree networks is also mentioned.

Motivated by the design of fault-tolerant multiprocessor interconnection networks, this paper considers the following problem: Given a positive integer t and a graph H, construct a graph G from H by adding a minimum number Δ(t, H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate Δ(t, H) for the hypercube and torus, which are well-known as important interconnection networks for multiprocessor systems. If we denote the hypercube and the square torus on N vertices by QN and DN respectively, we show, among others, that Δ(t, QN) = O(tN log(log N/t + log 2e)) for any t and N (t 2), and Δ(1, DN) = N/2 for N even.

This paper presents a practical fault-tolerant architecture for mesh parallel machines that has t spare processors and has 2(t+2) communication links per processor while tolerating at most t+1 processor and link faults. We also show that the architecture presented here can be laid out efficiently in a linear area with wire length at most O(t).

This note considers a problem of minimum length scheduling for a set of messages subject to precedence constraints for switching and communication networks, and shows some improvements upon previous results on the problem.

The finite reconfigurability and local reconfigurability of graphs were proposed by Sha and Steiglitz [1], [2] in connection with a problem of on-line reconfiguraion of WSI networks for run-time faults. It is shown in [2] that a t-locally-reconfigurable graph for a 2-dimensional N-vertex array AN can be constructed from AN by adding O(N) vertices and edges. We show that Ω(N) vertices must be added to an N-vertex graph GN in order to construct a t-locally-reconfigurable graph for GN, which means that the number of added vertices for the above mentioned t-locally-reconfigurable graph for AN is optimal to within a constant factor. We also show that a t-finitely-reconfigurable graph for an N-vertex graph GN can be constructed from GN by adding t vertices and tN + t (t+1)/2 edges.

This note shows an efficient implementation of de Bruijn networks by the Optical Transpose Interconnection System (OTIS) extending previous results by Coudert, Ferreira, and Perennes [2].