Nobuji Saito

Algorithms for routing around a rectangle

Abstract Simple efficient algorithms are given for three routing problems around a rectangle. The algorithms find routing in two or three layers for two-terminal nets specified on the sides of a rectangle. All algorithms run in linear time. One of the three routing problems is the minimum area routing previously considered by LaPaugh and Gonzalez and Lee. The algorithms they developed run in time O(n3) and O(n) respectively. Our simple linear time algorithm is based on a theorem of Okamura and Seymour and on a data structure developed by Suzuki, Ishiguro and Nishizeki.

A linear 5-coloring algorithm of planar graphs

A simple linear algorithm is presented for coloring planar graphs with at most five colors. The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.

On the f-coloring of multigraphs

An f-coloring of a multigraph is a coloring of edges E such that each color appears at each vertex at most f times. The minimum number of colors needed to f-color G is called the f-chromatic index of G. Various scheduling problems on networks are reduced to finding an f-coloring of a multigraph. An upper bound on the f-chromatic index is given. >

An Efficient Algorithm for Finding Multicommodity Flows in Planar Networks

This paper presents an efficient algorithm for finding multicommodity flows in planar graphs. Suppose that G is an undirected planar graph with all sources and sinks on the boundary of the outer face and that a real-valued demand is given for each source–sink pair. The algorithm decides whether G has multicommodity flows, each from a source to a sink and of a given demand, and actually finds them if G has. It spends

A Linear-Time Routing Algorithm for Convex Grids

O(kn + n^2 (\log n)^{1/2} )

Planar multicommodity flows, maximum matchings and negative cycles

time and

An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs

O(kn)

A linear algorithm for finding Hamiltonian cycles in 4-connected maximal planar graphs

space if G has n vertices and k source–sink pairs.

A linear algorithm for five-coloring a planar graph

In this paper, we consider the channel routing problem involving two-terminal nets on rectilinear grids. An efficient algorithm is described which necessarily finds a routing in a given grid whenever it exists. The algorithm is not a heuristic but an exact one, and works for a rather large class of grids, called convex grids, including the grids of rectangular, T-, L-, or X-shape boundaries. Both the running time and required space are linear in the number of vertices of a grid.

Algorithms for Multicommodity Flows in Planar Graphs

This paper shows that the multicommodity flow problem on a class of planar undirected graphs can be reduced to another famous combinatorial problem, the weighted matching problem. Assume that in a given planar graph G all the sources can be joined to the corresponding sinks without destroying the planarity. Then we show that the feasibility of multicommodity flows can be tested simply by solving, once, the weighted matching problem on a certain graph constructed from G, and that the multicommodity flows of given demands can be found by solving the matching problem