This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k 3 as far as k is a fixed constant. For the case k=2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.

Triangulations have been one of main research topics in computational geometry and have many applications in computer graphics, finite element methods, mesh generation, etc. This paper surveys properties of triangulations in the two- or higher-dimensional spaces. For triangulations of the planar point set, we have a good triangulation, called the Delaunay triangulation, which satisfies several optimality criteria. Based on Delaunay triangulations, many properties of planar triangulations can be shown, and a graph structure can be constructed for all planar triangulations. On the other hand, triangulations in higher dimensions are much more complicated than in planar cases. However, there does exist a subclass of triangulations, called regular triangulations, with nice structure, which is also touched upon.

This paper surveys two methods for designing numerically robust geometric algorithms. The first method is the exact-arithmetic method, in which numerical computations are done in sufficiently high precision so that all the topological judgements can be done correctly. This method is usually accompanied with lazy evaluation and symbolic perturbation in order to reduce the computational cost and the implementation cost. The second method is the topology-oriented method, in which the consistency of the topological structure is considered as higher-priority information than numerical computation, and thus inconsistency is avoided. Both of the methods are described with the implementation examples.

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.

In this paper, a puzzle called Cell-Maze is analyzed. In this puzzle, cells are arranged in checker board squares. Each cell is rotated when a player arrives at the cell. Cell-Maze asks whether or not a player started from a start cell can reach a goal cell. The reachability problem for ordinary graphs can be easily solved in linear time, however a reachability problem for the network such as Cell-Maze may be extremely difficult. In this paper, NP-hardness of this puzzle is proved. It is proved by reducing Hamiltonian Circuit Problem of directed planar graph G such that each vertex involved in just three arcs. Furthermore, we consider subproblems, which can be solved in polynomial time.

Consider a set of parties who do not trust each other but want to compute some agreed function of their inputs in a secure way. This problem is known as multi-party computation. It has various interesting applications including election over the internet, electric contracts, private and secret database, joint signatures, and others. A number of techniques for the problem have been proposed. Secure protocols for multi-paty computation known so far are mainly based on threshold secret sharing, verifiable secret sharing, zero-knowledge proofs, and error-correcting codes. We survey important and interesting results on secure multi-party computation under the existence of various types of adversaries.

In this paper, we propose a new secure server-aided RSA secret computation protocol which guards against not only the attacks in [1],[2],[15],[18] but also the new powerful active attacks in [3],[4]. The new protocol is also efficient to support high security level.

We sketch two algorithms that solve the undirected st-connectivity problem in a small amount of space. One is due to Nisan, Szemeredy and Wigderson, and takes space O(log3/2n), where n denotes the number of nodes in a give undirected graph. This is the first algorithm that overcame the Savitch barrier on the space complexity of the problem. The other is due to Tarui and this author, and takes space O(sw(G)2 log2 n), where sw(G) denotes the separation-width of a given graph G. Their result implies that the st-connectivity problem can be solved in logarithmic space for any class of graphs with separation-width bounded above by a predetermined constant. This class is one of few nontrivial classes for which the st-connectivity problem can be solved in logarithmic space.

The two dimensional mesh is widely considered to be a promising parallel architecture in its scalability. In this architecture, processors are naturally placed at intersections of horizontal and vertical grids, while there can be three different types of communication links: (i) The first type is the most popular model, called a mesh-connected computer: Each processor is connected to its four neighbours by local connections. (ii) Each processor of the second type is connected to a couple of (row and column) buses. The system is then called a mesh of buses. (iii) The third model is equipped with both buses and local connections, which is called a mesh-connected computer with buses. Mesh routing has received considerable attention for the last two decades, and a variety of algorithms have been proposed. This paper provides an overview of lower and upper bounds for algorithms, with pointers to the literature, and suggests further research directions for mesh routing.

A convex hull is one of the most fundamental and interesting geometric constructs in computational geometry. Considerable research effort has focused on developing algorithms, both in serial and in parallel, for computing convex hulls. In particular, there are few problems whose parallel algorithms are so thoroughly studied as convex hull problems. In this paper, we review the convex hull parallel algorithms and their paradigm. We provide a summary of results and introduce several interesting topics including typical techniques, output-size sensitive methods, randomized approaches, and robust algorithms for convex hull problems, with which we may see the highlights of the whole research for parallel algorithms. Most of our discussion uses the PRAM (Parallel Random Access Machine) computational model, but still we give a glance at the results of the other parallel computational models such as mesh, mesh-of-trees, hypercube, recofigurable array, and models of coarse grained multicomputers like BSP and LogP.

Two different examples have been respectively given by Aggarwal and Viswanathan to establish the necessity of (n + 2)/5 edge guards for spiral polygons. However, the former example is incorrect. To show why it is wrong, we give an alternate proof of sufficiency of (n + 2)/5 edge guards for spiral polygons. Our proof is simpler than the sufficiency proof given by Viswanathan.

This paper investigates some fundamental properties of 2-way alternating multi-counter automata (2amca's) with only existential (universal) states which have sublinear space and 1 inkdot. It is shown that for any function s(n) log n such that log s(n)=o(log n), s(n) space-bounded 1-inkdot 2amca's with only existential states are incomparable with the ones with only universal states, and the ones with only existential (universal) states are not closed under complementation.

This paper surveys recent progress in the investigation of the underlying discrete proximity structures of geometric clustering with respect to the divergence in information geometry. Geometric clustering with respect to the divergence provides powerful unsupervised learning algorithms, and can be applied to classifying and obtaining generalizations of complex objects represented in the feature space. The proximity relation, defined by the Voronoi diagram by the divergence, plays an important role in the design and analysis of such algorithms.

In pattern recognition using neural networks, it is very difficult for researchers or users to design optimal neural network architecture for a specific task. It is possible for any kinds of neural network architectures to obtain a certain measure of recognition ratio. It is, however, difficult to get an optimal neural network architecture for a specific task analytically in the recognition ratio and effectiveness of training. In this paper, an evolutional method of training and designing feedforward neural networks is proposed. In the proposed method, a neural network is defined as one individual and neural networks whose architectures are same as one species. These networks are evaluated by normalized M. S. E. (Mean Square Error) which presents a performance of a network for training patterns. Then, their architectures evolve according to an evolution rule proposed here. Architectures of neural networks, in other words, species, are evaluated by another measurement of criteria compared with the criteria of individuals. The criteria assess the most superior individual in the species and the speed of evolution of the species. The species are increased or decreased in population size according to the criteria. The evolution rule generates a little bit different architectures of neural network from superior species. The proposed method, therefore, can generate variety of architectures of neural networks. The designing and training neural networks which performs simple 3 3 and 4 4 pixels which include vertical, horizontal and oblique lines classifications and Handwritten KATAKANA recognitions are presented. The efficiency of proposed method is also discussed.

Recent developments in computer technology allow us to analyze all the data in a huge database. Data mining is to analyze all the data in such a database and to obtain useful information for database users. One of the well-studied problems in data mining is the search for meaningful association rules in a market basket database which contains massive amounts of transactions. One way to find meaningful association rules is to find all the large itemsets first, and then to find meaningful association rules from the large itemsets. Although a number of algorithms for computing all the large itemsets have been proposed, the computational complexity of them is scarcely disscussed. In this paper, we show that it is NP-complete to decide whether there exists a large itemset that has a given cardinality. Also, we propose subclasses of databases in which all the meaningful association rules can be computed in time polynomial of the size of a database.

A novel analog-computation system using quantum-dot spin glass is proposed. Analog computation is a processing method that solves a mathematical problem by applying an analogy of a physical system to the problem. A 2D array of quantum dots is constructed by mixing two-dot (antiferromagnetic interaction) and three-dot (ferromagnetic interaction) systems. The simulation results show that the array shows spin-glass-like behavior. We then mapped two combinatorial optimization problems onto the quantum-dot spin glasses, and found their optimal solutions. The results demonstrate that quantum-dot spin glass can perform analog computation and solve a complex mathematical problem.

Equipment simulation can provide valuable support in reactor design and process optimization. This article describes the physical and chemical models used in this technique and the current state of the art of the available software tools is reviewed. Moreover, the potential of equipment simulation will be highlighted by means of three recent examples from advanced quarter micron silicon process development. These include a vertical batch reactor for LPCVD of arsenic doped silicon oxide, a multi station tungsten CVD reactor, and a plasma reactor for silicon etching.

A new concept of "an imperfect singular solution" is defined as an approximate solution which becomes a singular solution by adding a suitable small perturbation to the original equations. A numerical method is presented for proving the existence of imperfect singular solutions of nonlinear equations with guaranteed accuracy. A few numerical examples are also presented for illustration.

This paper presents a method of calculating an interval including a bifurcation point. Turning points, simple bifurcation points, symmetry breaking bifurcation points and hysteresis points are calculated with guaranteed accuracy by the extended systems for them and by the Krawczyk-based interval validation method. Taking several examples, the results of validation are also presented.

This paper presents a new approach to designing arithmetic circuits by using a graph-based evolutionary optimization technique called Evolutionary Graph Generation (EGG). The key idea of the proposed method is to introduce a higher level of abstraction for arithmetic algorithms, in which arithmetic circuit structures are modeled as data-flow graphs associated with specific number representation systems. The EGG system employs evolutionary operations to transform the structure of graphs directly, which makes it possible to generate the desired circuit structure efficiently. The potential capability of EGG is demonstrated through an experiment of generating constant-coefficient multipliers.