Kazuo Iwano

On minimum and maximum spanning trees of linearly moving points

In this paper we investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Here, a transition means a change on the combinatorial structure of the spanning trees. Suppose that we are given a set ofn points ind-dimensional space,S={p1,p2, ...pn}, and that all points move along different straight lines at different but fixed speeds, i.e., the position ofpi is a linear function of a real parametert. We investigate the numbers of transitions of MinST and MaxST whent increases from-∞ to +∞. We assume that the dimensiond is a fixed constant. Since there areO(n2) distances amongn points, there are naivelyO(n4) transitions of MinST and MaxST. We improve these trivial upper bounds forL1 andL∞ distance metrics.Letkp(n) (resp.) be the number of maximum possible transitions of MinST (resp. MaxST) inLp metric forn linearly moving points. We give the following results in this paper: κ1(n)=O(n5/2α(n)),κ∞(n)=O(n5/2α(n)),, and where α(n) is the inverse Ackermanns function. We also investigate two restricted cases, i.e., thec-oriented case in which there are onlyc distinct velocity vectors for movingn points, and the case in which onlyk points move.

Efficient algorithms for finding the most vital edge of a minimum spanning tree

Let G(V,E) be an undirected graph with m edges and n vertices such that each edge e has a real valued weight w(e). Let MST(G) be a minimum spanning tree in G. Let ƒ(G) be the weight of a minimum spanning tree of G if G is connected; otherwise ƒ(G)∞. We define a most vital edge with respect to a minimum spanning tree in a connected undirected graph G as an edge e such that ƒ(G−e)⩾ƒ(G−e′) for every edge e′ in G. In this paper, we give O(m+n log n) and O(mα(m,n)) time algorithms, which improve O(m log m) and O(n2) time bounds by Hsu et al. in Inform. Process. Lett. 39 (1991) 277–281.

Self-Adaptable Autonomic Computing Systems: An Industry View

This paper reflects different industry views on self-adaptable autonomic computing, as perceived by representatives from Enigmatec, IBM, Intel, MCNC, and Oracle. The statements in this paper serve as starting points for the panel discussion on methodologies, engineering requirements, and technical challenges, during the SAACS Conference

A new scaling algorithm for the maximum mean cut problem

We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A networkN=(G, lower, upper) with lower and upper arc capacities is said to be δ-feasible ifN has a feasible circulation when we relax the capacity bounds by δ; that is, we use (lower(a)- δ, upper(a)+δ) bounds instead of (lower(a), upper(a)) bounds for each arca εA. During an approximate binary search we maintain two bounds,LB andUB, such thatN is LB-infeasible andUB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph withn vertices, m arcs, and integer capacities bounded byU, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under thesimilarity condition (that is,U=O(no(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

The Traveling Cameraman Problem, with Applications to Automatic Optical Inspection

We are given a finite set of disjoint regions in the plane. We wish to cover all the regions by unit squares, and compute a path that visits the centers of all the unit squares in the cover. Our objective is to minimize the length of this path. The problem arises in the automatic optical inspection of printed circuit boards and other assemblies.

Efficient algorithms for minimum range cut problems

Let G = (V, E) be an undirected graph with n vertices and m edges such that a real-valued weight, denoted by w(e), is associated with each edge e. This paper studies what we call the minimum range cut problem that asks to find a cut in G such that the range of all edge weights in the cut is minimum. Here, the range of a cut C is defined to be the maximum difference among weights of edges in the cut, i.e., maxeϵcw(e) - mineϵcw(e). This paper proposes an O(m + n log n) algorithm for the minimum range cut problem. It is also shown that this running time is optimal. We also study two variants of this problem. One is the minimum range target cut problem. Given a prespecified value called a target, this problem asks to find a cut with minimum range among all cuts such that the target value is between the minimum and maximum of weights of edges in the cut. The second is the minimum range s – t cut problem that asks to find an s – t cut with minimum range. This paper proposes O(m + n log n) algorithms for these problems. For the second problem, we show that an ancestor tree of O(n) space recently developed by Cheng and Hu effectively represents all pairs minimum range cuts, which can be constructed in O(n2) time, and enables us to answer any minimum range s – t cut query in O(1) time [resp., in O(n) time] if we want to obtain only the range value of the cut (resp., the bipartition of vertices induced by the cut).

FINDING k FARTHEST PAIRS AND k CLOSEST/FARTHEST BICHROMATIC PAIRS FOR POINTS IN THE PLANE

We study the problem of enumerating k farthest pairs for n points in the plane and the problem of enumerating k closest/farthest bichromatic pairs of n red and n blue points in the plane. We propose a new technique for geometric enumeration problems which iteratively reduces the search space by a half and provides efficient algorithms. As applications of this technique, we develop algorithms, using higher order Voronoi diagrams, for the above problems, which run in O(min{n2, n log n+k4/3log n/log1/3 k}) time and O(n+k4/3/log1/3 k+k log n) space for general Lp metric with p≠2, and O(min{n2, n log n+k4/3}) time and O(n+k4/3+k log n) space for L2 metric. For the problem of enumerating k closest/farthest bichromatic pairs, we shall also discuss the case where we have different numbers of red and blue points. To the authors’ knowledge, no nontrivial algorithms have been known for these problems and our algorithms are faster than trivial ones when k=o(n3/2).

New TSP Construction Heuristics and Their Relationshipsto the 2-Opt

Correction heuristics for the traveling salesman problem (TSP), with the 2-Opt applied as a postprocess, are studied with respect to their tour lengths and computation times. This study analyzes the “2-Opt dependency,” which indicates how the performance of the 2-Opt depends on the initial tours built by the construction heuristics. In accordance with the analysis, we devise a new construction heuristic, the recursive-selection with long-edge preference (RSL) method, which runs faster than the multiple-fragment method and produces a comparable tour when they are combined with the 2-Opt.

Efficient sequential and parallel algorithms for planar minimum cost flow

This paper presents efficient sequential and parallel algorithms for the minimum cost flow problem on planar networks. Our algorithms are based on the interior point method for linear programming, and make full use of the planarity of networks in solving a system of linear equations in sequential and parallel ways. For the planar minimum cost flow problem with n vertices and integer costs and capacities on edges whose absolute values are bounded by γ, we give a sequential algorithm with O(n1.594\(\sqrt {\log n}\)log(nγ)) time and O(nlogn) space and a parallel algorithm with O(\(\sqrt n\)log3n log(nγ)) parallel time using O(n1.094) processors. These algorithms are currently best for γ=poly(n). These results can be generalized to the minimum cost flow problem on s(n)-separable networks such as three-dimensional grid networks.

Finding Subsets Maximizing Minimum Structures

We consider the problem of finding a set of k vertices in a graph that are in some sense remote. Stated more formally, given a graph G and an integer k, find a set P of k vertices for which the total weight of a minimum structure on P is maximized. In particular, we are interested in three problems of this type, where the structure to be minimized is a spanning tree (Remote-MST), Steiner tree, or traveling salesperson tour. We study a natural greedy algorithm that simultaneously approximates all three problems on metric graphs. For instance, its performance ratio for Remote-MST is exactly 4, while this problem is NP -hard to approximate within a factor of less than 2. We also give a better approximation for graphs induced by Euclidean points in the plane, present an exact algorithm for graphs whose distances correspond to shortest-path distances in a tree, and prove hardness and approximability results for general graphs.