Takeshi Tokuyama

Mining Optimized Association Rules for Numeric Attributes

Given a huge database, we address the problem of finding association rules for numeric attributes, such as(Balance?I)?(CardLoan=yes),which implies that bank customers whose balances fall in a rangeIare likely to use card loan with a probability greater thanp. The above rule is interesting only if the rangeIhas some special feature with respect to the interrelation betweenBalanceandCardLoan. It is required that the number of customers whose balances are contained inI(called thesupportofI) is sufficient and also that the probabilitypof the conditionCardLoan=yesbeing met (called theconfidence ratio) be much higher than the average probability of the condition over all the data. Our goal is to realize a system that finds such appropriate ranges automatically. We mainly focus on computing twooptimized ranges: one that maximizes the support on the condition that the confidence ratio is at least a given threshold value, and another that maximizes the confidence ratio on the condition that the support is at least a given threshold number. Using techniques from computational geometry, we present novel algorithms that compute the optimized ranges in linear time if the data are sorted. Since sorting data with respect to each numeric attribute is expensive in the case of huge databases that occupy much more space than the main memory, we instead apply randomized bucketing as the preprocessing method and thus obtain an efficient rule-finding system. Tests show that our implementation is fast not only in theory but also in practice. The efficiency of our algorithm enables us to compute optimized rules for all combinations of hundreds of numeric and Boolean attributes in a reasonable time.

Finding a minimum-weightk-link path in graphs with the concave Monge property and applications

AbstractLetG be a weighted, complete, directed acyclic graph (DAG) whose edge weights obey the concave Monge condition. We give an efficient algorithm for finding the minimum-weightk-link path between a given pair of vertices for any givenk. The time complexity of our algorithm is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9vqFf0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVe0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacaWGUbWaaOaaaeaacaWGRbGaciiBaiaac+gacaGGNbGaamOBaaWc% beaakiabgUcaRiaad6gaciGGSbGaai4BaiaacEgacaWGUbGaaiykaa% aa!46B3!

Minimizing interference of a wireless ad-hoc network in a plane

O(n\sqrt {k\log n} + n\log n)

Efficient algorithms for optimization-based image segmentation

. Our algorithm uses some properties of DAGs with the concave Monge property together with the parametric search technique. We apply our algorithm to get efficient solutions for the following problems, improving on previous results: (1) Finding the largestk-gon contained in a given convex polygon. (2) Finding the smallestk-gon that is the intersection ofk half-planes out ofn half-planes defining a convexn-gon. (3) Computing maximumk-cliques of an interval graph. (4) Computing length-limited Huffman codes. (5) Computing optimal discrete quantization.

WALKING IN AN ARRANGEMENT TOPOLOGICALLY

Abstract. Let Γ be a collection of unbounded x -monotone Jordan arcs intersecting at most twice each other, which we call pseudoparabolas, since two axis parallel parabolas intersect at most twice. We investigate how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once. We give an Ω( n4/3 ) lower bound and O(n5/3) upper bound on the number of cuts. We give the same bounds for an arrangement of circles. Applying the upper bound, we give an O(n23/12) bound on the complexity of a level in an arrangement of pseudoparabolas, and an O(n11/6) bound on the complexity of a combinatorially concave chain of pseudoparabolas. We also give some upper bounds on the number of transitions of the minimum weight matroid base when the weight of each element changes as a quadratic function of a single parameter.

Covering points in the plane by k -tours: towards a polynomial time approximation scheme for general k

We consider interference minimization in wireless ad-hoc networks. This is formulated as assigning a suitable transmission radius to each of the given points in the plane, so as to minimize the maximum number of transmission ranges overlapping any point. Using ideas from computational geometry and @e-net theory, we attain an O(@D) bound for the maximum interference where @D is the interference of a uniform-radius ad-hoc network. This generalizes a result given in [P. von Rickenbach, S. Schmid, R. Wattenhofer, A. Zollinger, A robust interference model for wireless ad-hoc networks, in: Proc. 5th International Workshop on Algorithms for Wireless, Mobile, Ad Hoc and Sensor Networks (WMAN), Denver, Colorado, USA, April 2005] for the special case of highway model (i.e., one-dimensional problem) to the two-dimensional case. We show how a distributed algorithm can achieve a slightly weaker bound. We also give a method based on quad-tree decomposition and bucketing that has another provable interference bound in terms of the ratio of the minimum distance to the radius of a uniform-radius ad-hoc network.

On minimum and maximum spanning trees of linearly moving points

We introduce a new string matching problem called order-preserving matching on numeric strings, where a pattern matches a text if the text contains a substring of values whose relative orders coincide with those of the pattern. Order-preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves. Solving order-preserving matching is closely related to the representation of order relations of a numeric string. We define the prefix representation and the nearest neighbor representation of the pattern, both of which lead to efficient algorithms for order-preserving matching. We present efficient algorithms for single and multiple pattern cases. For the single pattern case, we give an O(nlogm) time algorithm and optimize it further to obtain O(n+mlogm) time. For the multiple pattern case, we give an O(nlogm) time algorithm.

Greedily Finding a Dense Subgraph

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 computational complexity of the segmentation problem, assuming that the sought object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in a grid of N×N pixels is NP-hard under the interclass variance, a criterion that is often used in discriminant analysis. More importantly, we consider the basic case in which the object is bounded by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing the optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(N4) time; this algorithm is based on several techniques such as a parametric optimization formulation, a hand-probing algorithm for the convex hull of an unknown planar point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an ∊-approximate solution in O(∊-1 N2log L) time, where ∊ is any fixed constant with 1>∊>0, and L is the total sum of the absolute values of the brightness levels of the image.