A subdivision of a rectangle into rectangular faces with horizontal and vertical line segments is called a rectangular drawing or floorplan. It has been an open problem to determine whether there exist a polynomial time algorithm for computing R(n). We affirmatively solve the problem, that is, we introduce an O(n4)-time and O(n3)-space algorithm for R(n). The algorithm is based on a recurrence for R(n), which is the main result of the paper. We also implement our algorithm and computed R(n) for n 3000.

Approximate pattern matching plays an important role in various applications. In this paper we focus on (δ, γ)-matching, where a character can differ at most δ and the sum of these errors is smaller than γ. We show how to find these matches when the pattern is transformed by y=αx + β, without knowing α and β in advance.

Given a set of strings U = {T1, T2, ...,T}, the longest common repeat problem is to find the longest common substring that appears at least twice in each string, considering direct, inverted, and mirror repeats. We define the generalised longest common repeat problem and present a linear time solution.

Given a combinatorial problem on a set of weighted elements, if we change the weight using a parameter, we obtain a parametric version of the problem, which is often used as a tool for solving mathematical programming problems. One interesting question is how to describe and analyze the trajectory of the solution. If we consider the trajectory of each weight function as a curve in a plane, we have a set of curves from the problem instance. The curves induces a cell complex called an arrangement, which is a popular research target in computational geometry. Especially, for the parametric version of the problem of computing the minimum weight base of a matroid or polymatroid, the trajectory of the solution becomes a subcomplex in an arrangement. We introduce the interaction between the two research areas, combinatorial optimization and computational geometry, through this bridge.

Distributed algorithms that entail successive rounds of message exchange are called decentralized consensus protocols. Several consensus protocols use a finite projective plane as a communication structure and require 4nn messages in two rounds, where n is the number of nodes. This paper presents an efficient communication structure that uses a finite projective plane with a duality of indices. The communication structure requires 2nn messages in two rounds, and can therefore halve the number of messages. It is shown that a finite projective plane with a duality can be constructed from a difference set, and that the presented communication structure has two kinds of symmetry.