Koji Obokata

Independent spanning trees of chordal rings

Abstract A chordal ring, denoted by CR(N, d), is a graph G = (V, E) with V = {0, 1,…, N − 1} and E = {(u, v) ¦[v − u] N = 1 or d} , where 2 ⩽ N ⩽ N 2 and [r]N denotes r modulo N. We show that for 2 ⩽ d N 2 , CR(N, d) has four independent spanning trees rooted at the same vertex.

Independent Spanning Trees of Product Graphs

A graph G is called an n-channel graph at vertex r if there are n independent spanning trees rooted at r. A graph G is called an n-channel graph if for every vertex u, G is an n-channel graph at u. Independent spanning trees of a graph play an important role in faulttolerant broadcasting in the graph. In this paper we show that if G1 is an n1-channel graph and G2 is an n2-channel graph, then G1×G2 is an (n1+n2)-channel graph. We prove this fact by a construction of n1+n2 independent spanning trees of G1 × G2 from n1 independent spanning trees of G1 and n2 independent spanning trees of G2.

Hyper-ring connection machines

A graph G=(V,E) is called a hyper-ring with N nodes (N-HR for short) if V={0,...,N-1} and E={{u,v}|v-u modulo N is a power of 2}. We study constructions and spanners of HRs, and embeddings into HRs. The stretch factors of three types of spanners given in this paper are at most [log/sub 2/ N], 2k-1 for any 1/spl les/k/spl les/[log/sub 2/ N], and 2k-1 for any 0/spl les/k/spl les/[log/sub 2/ N]-1, respectively. The numbers of edges of these types of spanners are N-1, at most N[(log/sub 2/ N)/k] and at most N([log/sub 2/ N]-k)/(2k)+Nk, respectively. Some of these spanners are superior in both stretch factors and numbers of edges to corresponding spanners for synchronizer /spl gamma/ of HRs.<<ETX>>

Independent Spanning Trees of Chordal Rings

A chordal ring, denoted by CR(N, d), is a graph G = (V, E) with V = {0,1,..., N − 1} and E = {(u, v) | [v − u]N = 1 or d}, where 2 ≤ < N ≤ N/2 and [r]N denotes r modulo N. We show that for 2 ≤ d ≤ N/2, CR(N, d) has 4 independent spanning trees rooted at the same vertex, and for 2 ≤ d = N/2, CR(N, d) has 3 independent spanning trees rooted at the same vertex. We can design a fault-tolerant broadcasting scheme for CR(N, d) using independent spanning trees.

Combinatorial and geometric problems related to digital halftoning

Digital halftoning is a technique to convert a continuoustone image into a binary image consisting of black and white dots. It is an important technique for printing machines and printers to output an image with few intensity levels or colors which looks similar to an input image. The purposes of this paper are to reveal that there are a number of problems related to combinatorial and computational geometry and to present some solutions or clues to those problems.

A probably optimal embedding of hyper-rings in hypercubes

Abstract A graph G = (V,E) with N nodes is called an N-hyper-ring if V = {0,…,N − 1} and E = {(u,v) ¦(v − u) modulo N is a power of 2} . We present an embedding of the 2n-hyper-ring in the n-dimensional hypercube with dilation 2 and congestion 4. This is probably an optimal embedding of hyper-rings in hypercubes.