A graph G = (V, E) with N nodes is called an N-hyper-ring if V = {0, ..., N-1} and E = {(u, v)(u-v) modulo N is power of 2}. We study embeddings of the 2n-hyper-ring in the n-dimensional hypercube. We first show a greedy embedding with dilation 2 and congestion n+1. We next modify the greedy embedding, and then we obtain an embedding with dilation 4 and congestion 6.

We give an efficient shortest path algorithm on a mesh-connected processor array for nn banded matrices with bandwidth b. We use a b/2b/2 semisystolic processor array. The input data is supplied to the processor array from the host computer. The output from the processor array can be also supplied to itself through the host computer. This algorithm computes all pair shortest distances within the band in 7n4b/21 steps.

We consider a class of unknown scenes Sk(n) with rectangular obstacles aligned with the axes such that Euclidean distance between the start point and the target is n, and any side length of each obstacle is at most k. We propose a strategy called the adaptive-bias heuristic for navigating a robot in such a scene, and analyze its efficiency. We show that a ratio of the total distance walked by a robot using the strategy to the shortest path distance between the start point and the target is at most 1+(3/5) k, if k=o(n) and if the start point and the target are at the same horizontal level. This ratio is better than a ratio obtained by any strategy previously known in the class of scenes, Sk(n), such that k=o(n).

We propose two fault-tolerant broadcasting schemes in star graphs. One of the schemes can tolerate up to n2 faults of the crash type in the n-star graph. The other scheme can tolerate up to (n3d1)/2 faults of the Byzantine type in the n-star graph, where d is the smallest positive integer satisfying nd!. Each of the schemes is designed for the single-port mode, and it completes the broadcasting in O(n log n) time. These schemes are time optimal. For the former scheme we analyze the reliability in the case where faults of the crash type are randomly distributed. It can tolerate (n!)α faults randomly distributed in the n-star graph with a high probability, where α is any constant less than 1.

A directed graph G = (V,E) is called the n-rotator graph if V = {a1a2an|a1a2an is a permutation of 1,2,,n} and E = {(a1a2an,b1b2bn)| for some 2 i n, b1b2bn = a2aia1ai+1an}. We show that for any pair of distinct nodes in the n-rotator graph, we can construct n - 1 disjoint paths, each length < 2n, connecting the two nodes. We propose a nonadaptive fault-tolerant file transmission algorithm which uses these disjoint paths. Then the probabilistic analysis of its reliability is given.

We study robot navigation in unknown environment with rectangular obstacles aligned with the x and y axes. We propose a strategy called the modified-bian heuristic, and analyze its efficiency. Let n be the distance between the start point and the target of robot navigation, and let k be the maximum side length among the obstacles in a scene. We show that if k=(o(n) and if the summation of the widths of the obstacles on the line crossing the target and along the y axis is o(n), then ratio of the total distance walked by the robot to the shortest path length between the start point and the target is at most arbitrarily close to 1+k/2, as n grows. For the same restrictions as above on the sizes of the obstacles, the ratio is also at most arbitrarily close to 1+3/4n, as n grows, where is the summation of lengths of the obstacles in y axis direction.