In the topologies for interconnected nodes, it is desirable to have a low degree and a small diameter. For the same number of nodes, a dual-cube topology has almost half the degree compared to a hypercube while increasing the diameter by just one. Hence, it is a promising topology for interconnection networks of massively parallel systems. We propose here a stochastic fault-tolerant routing algorithm to find a non-faulty path from a source node to a destination node in a dual-cube.

An algorithm is described for solving the node-to-set disjoint paths problem in bi-rotator graphs, which are obtained by making each edge of a rotator graph bi-directional. The algorithm is of polynomial order of n for an n-bi-rotator graph. It is based on recursion and divided into three cases according to the distribution of destination nodes in the classes into which the nodes in a bi-rotator graph are categorized. We estimated that it obtains 2n-3 disjoint paths with a time complexity of O(n5), that the sum of the path lengths is O(n3), and that the length of the maximum path is O(n2). Computer experiment showed that the average execution time was O(n3.9) and, the average sum of the path lengths was O(n3.0).

Many researchers have used hypercube interconnection networks for their good properties to construct many parallel processing systems. However, as the number of processors increases, the probability of occurrences of faulty nodes also increases. Hence, for hypercube interconnection networks which have faulty nodes, several efficient dynamic routing algorithms have been proposed which allow each node to hold status information of its neighbor nodes. In this paper, we propose an improved version of the algorithm proposed by Chiu and Wu by introducing the notion of full reachability. A fully reachable node is a node that can reach all nonfaulty nodes which have Hamming distance l from the node via paths of length l. In addition, we further improve the algorithm by classifying the possibilities of detours with respect to each Hamming distance between current and target nodes. We propose an initialization procedure which makes use of an equivalent condition to perform this classification efficiently. Moreover, we conduct a simulation to measure the improvement ratio and to compare our algorithms with others. The simulation results show that the algorithms are effective when they are applied to low-dimensional hypercube interconnection networks.

In this paper, we give an algorithm for the node-to-set disjoint paths problem in pancake graphs with its evaluation results. The algorithm is of polynomial order of n for an n-pancake graph. It is based on recursion and divided into two cases according to the distribution of destination nodes in classes into which all the nodes in a pancake graph are categorized. The sum of lengths of paths obtained and the time complexity of the algorithm are estimated and the average performance is evaluated based on computer simulation.

We developed a system called DELTA that supports the students' use of backward chaining (BC) to prove the congruence of two triangles. DELTA is designed as an interactive learning environment and supports the use of BC by providing hints and a function to automatically check the proofs inputted by the students. DELTA also has coloring, marking, and highlighting functions to support students' attempts to prove the congruence of two triangles. We evaluated the efficacy of DELTA with 36 students in the second grade of a junior high school in Japan. We found that (1) the mean number of problems, which the experimental group (EG) completely solved, was statistically higher than that of the control group on the post-test; (2) the EG effectively used the BC strategy to solve problems; and (3) the students' attempt to use both the forward chaining strategy and the BC strategy led to solving the problems completely.

Extending the very popular tori interconnection networks[1]-[3], Torus-Connected Cycles (TCC) have been proposed as a novel network topology for massively parallel systems [5]. Here, the set-to-set disjoint paths routing problem in a TCC is solved. In a TCC(k,n), it is proved that paths of lengths at most kn2+2n can be selected in O(kn2) time.