Hee-Kap Ahn

Competitive facility location: the Voronoi game

Abstract. We consider a competitive facility location problem with two players.Pla yers alternate placing points, one at a time, into the playing arena, until each of them has placed n points.The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins.W e present a winning strategy for the second player, where the arena is a circle or a line segment.

A fast nearest neighbor search algorithm by nonlinear embedding

We propose an efficient algorithm to find the exact nearest neighbor based on the Euclidean distance for large-scale computer vision problems. We embed data points nonlinearly onto a low-dimensional space by simple computations and prove that the distance between two points in the embedded space is bounded by the distance in the original space. Instead of computing the distances in the high-dimensional original space to find the nearest neighbor, a lot of candidates are to be rejected based on the distances in the low-dimensional embedded space; due to this property, our algorithm is well-suited for high-dimensional and large-scale problems. We also show that our algorithm is improved further by partitioning input vectors recursively. Contrary to most of existing fast nearest neighbor search algorithms, our technique reports the exact nearest neighbor - not an approximate one - and requires a very simple preprocessing with no sophisticated data structures. We provide the theoretical analysis of our algorithm and evaluate its performance in synthetic and real data.

Spatial Skyline Queries: An Efficient Geometric Algorithm

As more data-intensive applications emerge, advanced retrieval semantics, such as ranking and skylines, have attracted attention. Geographic information systems are such an application with massive spatial data. Our goal is to efficiently support skyline queries over massive spatial data. To achieve this goal, we first observe that the best known algorithm VS 2, despite its claim, may fail to deliver correct results. In contrast, we present a simple and efficient algorithm that computes the correct results. To validate the effectiveness and efficiency of our algorithm, we provide an extensive empirical comparison of our algorithm and VS 2 in several aspects.

Competitive Facility Location along a Highway

We consider a competitive facility location problem with two players.Pla yers alternate placing points, one at a time, into the playing arena, until each of them has placed n points.The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins.W e present a winning strategy for the second player, where the arena is a circle or a line segment.

Constructing optimal highways

For two points p and q in the plane, a (unbounded) line h, called a highway, and a real v > 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a high-way speed v, we consider the problem of finding an axis-parallel line, the highway, that minimizes the maximum travel time over all pairs of points in S. We achieve a linear-time algorithm both for the L1- and the Euclidean metric as the underlying metric. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

We consider the problem of computing the discrete Frechet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time \(2^{O(d^2)} m^2n^2\log^2(mn)\) the Frechet distance lower bound between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O( mn log2(mn) + (m 2 + n 2)log(mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L ∞ distance, we give an O(dmn log(dmn))-time algorithm.

Reachability by paths of bounded curvature in convex polygons

Work by Cheong was supported by Mid-career Researcher Program through NRF grant funded by the MEST (No. R01-2008-000-11607-0). Work by Ahn was supported by the National IT Industry Promotion Agency (NIPA) under the program of Software Engineering Technologies Development.

Maximum overlap of convex polytopes under translation

We study the problem of maximizing the overlap of two convex polytopes under translation in R^d for some constant d>=3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any @e>0, finds an overlap at least the optimum minus @e and reports the translation realizing it. The running time is O(n^@?^d^/^2^@?^+^1log^dn) with probability at least 1-n^-^O^(^1^), which can be improved to O(nlog^3^.^5n) in R^3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error @e, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the big-O constants in these bounds are independent of @e.

Computing minimum-area rectilinear convex hull and L-shape

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O(n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O(n^2) time and O(n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.

Covering points by disjoint boxes with outliers

For a set of n points in the plane, we consider the axis-aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+klogk) time for p=1, and O(nlogn+k^plog^pk) time for p=2,3. For rectangles we get O(n+k^3) for p=1 and O(nlogn+k^2^+^plog^p^-^1k) time for p=2,3. In all cases, our algorithms use O(n) space.