# Naoki Katoh

• Symposium on Computational Geometry
• 1994
In this paper we consider the<italic>k</italic>-clustering problem for a set <italic>S</italic> of <italic>n</italic> points <inline-equation><f><inf>i</inf>=<fen lp="par"><b>x<inf>i</inf></b><rp post="par"> </fen></f> </inline-equation> in the<italic>d</italic>-dimensional space with variance-based errors as clustering criteria, motivated from the color(More)
• J. Algorithms
• 1991
We consider the problem of selecting a specified number of points, k, from a given set S, subject to some optimization criterion. Problems of this type often arise in statistical clustering and pattern recognition (see Andrews [3] and Hartigan [7]). From an algorithmic standpoint, these problems usually can be solved in time O(nk+c), where 12 is the number(More)
• 8
• SODA
• 2000
1 I n t r o d u c t i o n . The problem of optimizing the sum of linear fractional functions (SOLF) is defined as follows: max f(Zl,...,Xd)= ~ [ : 7 : ~ d ) (Xl ..... z~t)ES i=1 ~ a r t m e n t of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, USA, {chen,odaescu,xwu,jxu}@cse.nd.edu. This research was supported in part by(More)
• STOC
• 1997
Let P be a set of n points in the plane. A k-tour through P is a tour in the plane that starts and ends at the fixed origin and visits at most k points of P. Our goal is to cover all the points of P by k-tours so as to minimize the total length of the tours. We give a polynomial time approximation scheme (PTAS) for this problem whose running time is(More)
Despite its simplicity and its linear time, a serial K-means algorithm's time complexity remains expensive when it is applied to a problem of large size of multidimensional vectors. In this paper we show an improvement by a factor of O(K/2), where K is the number of desired clusters, by applying theories of parallel computing to the algorithm. In addition(More)
• SODA
• 1996
Separating an object in an image from its background is a central problem (called segmentation) in pattern recognition and computer vision. In this paper, we study the complexity of the segmentation problem, assuming that the object forms a connected region in an intensity image. We show that the optimization problemof separating a connected region in an(More)