Hee-Kap Ahn

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Papers by Hee-Kap Ahn

Computing the Discrete Fréchet Distance with Imprecise Input

International Journal of Computational Geometry and Applications, Feb 1, 2012

We consider the problem of computing the discrete Fréchet distance between two polygonal curves w... more We consider the problem of computing the discrete Fréchet 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 2 n 2 log 2 (mn) the Fréchet 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 log 2 (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. We also give efficient O(dmn)-time algorithms to approximate the Fréchet distance upper bound, as well as the smallest possible Fréchet distance lower/upper bound that can be achieved between two imprecise point sequences when one is allowed to translate them. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

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Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Symposium on Computational Geometry, Jun 8, 2017

Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into ... more Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m ≤ n/ polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m ≤ n/ polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

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Dilation-Optimal Edge Deletion in Polygonal Cycles

Springer eBooks, 2007

Let C be a polygonal cycle on n vertices in the plane. A randomized algorithm is presented which ... more Let C be a polygonal cycle on n vertices in the plane. A randomized algorithm is presented which computes in O(n log 3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that for each edge e of C, a (1 −)approximation to the dilation of the path C \ {e} can be computed in O(n log n) total time.

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Farthest-point Voronoi diagrams in the presence of rectangular obstacles

arXiv (Cornell University), Mar 7, 2022

We present an algorithm to compute the geodesic L 1 farthest-point Voronoi diagram of m point sit... more We present an algorithm to compute the geodesic L 1 farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes O(nm + n log n + m log m) construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthestneighbor query in O(log(n + m)) time.

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Covering convex polygons by two congruent disks

Computational Geometry: Theory and Applications, Feb 1, 2023

We consider the planar two-center problem for a convex polygon: given a convex polygon in the pla... more We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an O(n log n)-time algorithm for the two-center problem for a convex polygon, where n is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.

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Minimum-Width Square Annulus Intersecting Polygons

Lecture Notes in Computer Science, 2018

For k (possibly overlapping) polygons of total complexity n in the plane, we present an algorithm... more For k (possibly overlapping) polygons of total complexity n in the plane, we present an algorithm for computing a minimum-width square annulus that intersects all input polygons in \(O(n^2\alpha (n)\log ^3 n)\) time, where \(\alpha (\cdot )\) is the inverse Ackermann function. When input polygons are pairwise disjoint, the running time becomes \(O(n\log ^3 n)\). We also present an algorithm for computing a minimum-width square annulus for k convex polygons of total complexity n. The running times are \(O(n\log k)\) for possibly overlapping convex polygons and \(O(n+k\log n)\) for pairwise disjoint convex polygons.

Aperture-Angle and Hausdor-Approxim ation of Convex Figures

The aperture angle α(x, Q) of a point x ∈ Q in the plane with respect to a convex polygon Q is th... more The aperture angle α(x, Q) of a point x ∈ Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q ⊂ C is the minimum aperture angle of any x ∈ C \ Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q ⊂ C with aperture angle approximation error`1 − 2 k+1´π. This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance σ, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k + 1)-gon. This implies the following result: For any k > 2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most 1 k+1 sin π k+1 .

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AS urvey on Multidimensional Methods

The extraordinary format of spatial data and the fact that there is no straightforward mapping of... more The extraordinary format of spatial data and the fact that there is no straightforward mapping of spatial objects from the multidimensional space to the 1-dimensional space, stimulated various researchers during the past two decades to develop multidimensional access methods that facilitate efficient indexing of spatial objects in large databases. This survey paper tries a classification of existing multidimensional access methods, according to the types of data they are most suitable for (points or objects with spatial extent), their structure (hierarchical or flat), and their performance over spatial queries. Most of this work is based on an excellent survey paper[Gaed97] * This work was conducted for the purposes of COMP630c, "Spatial, Image and Multimedia Databases," Oct. 1997

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A Time-Space Trade-Off for Triangulations of Points in the Plane

Springer eBooks, 2017

In this paper, we consider time-space trade-offs for reporting a triangulation of points in the p... more In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm.

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An improved data stream algorithm for clustering

Computational Geometry: Theory and Applications, Oct 1, 2015

ABSTRACT We present a single-pass, (1.8 + ε)-factor, O(1/ε)-space data stream algorithm for the E... more ABSTRACT We present a single-pass, (1.8 + ε)-factor, O(1/ε)-space data stream algorithm for the Euclidean 2-center problem for any fixed d ≥ 1. This is an improvement on the approximation factor over the (2 + ε)-factor and O(1/ε)-space algorithms of Ahn et al. [3] and Guha [8]. It can also be considered as an improvement on the space over the (1 + ε)-factor and O(1/ε d )-space algorithm of Zarrabi-Zadeh [11], while sacrificing the approximation factor a little bit. To our best knowledge, this is the first breakthrough with an approximation factor below 2 using O(1/ε) space for any fixed d. Our algorithm also extends to the k-center problem with k &gt; 2.

Casting a polyhedron with directional uncertainty

Computational Geometry: Theory and Applications, Oct 1, 2003

Casting is a manufacturing process in which molten material is poured into a cast (mould), which ... more Casting is a manufacturing process in which molten material is poured into a cast (mould), which is opened after the material has solidified. As in all applications of robotics, we have to deal with imperfect control of the casting machinery. In this paper, we consider directional uncertainty: given a 3-dimensional polyhedral object, is there a polyhedral cast such that its two parts can be removed in opposite directions with uncertainty α without inflicting damage to the object or the cast parts? We give a necessary and sufficient condition for castability, and an algorithm that verifies castability and produces two polyhedral cast parts for a polyhedral object of arbitrary genus. Its running time is O´n log nµ. The resulting cast parts have O´nµ vertices in total. We also consider the case where the removal direction is not specified in advance, and give an algorithm that finds all feasible removal directions with uncertainty α in time O´n 2 log n α 2 µ.

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Efficient planar two-center algorithms

Computational Geometry

We consider the planar Euclidean two-center problem in which given n points in the plane we are t... more We consider the planar Euclidean two-center problem in which given n points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic O(n log n)-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic O(n log n)-time algorithm for the case that the input points are in convex position. Both results improve the previous best O(n log n log log n) bound on the problems.

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Statistical Similarity of Critical Infrastructure Network Traffic Based on Nearest Neighbor Distances

Research in Attacks, Intrusions, and Defenses

Industrial control systems (ICSs) operate a variety of critical infrastructures such as waterwork... more Industrial control systems (ICSs) operate a variety of critical infrastructures such as waterworks and power plants using cyber physical systems (CPSs). Abnormal or malicious behavior in these critical infrastructures can pose a serious threat to society. ICS networks tend to be configured such that specific tasks are performed repeatedly. Further, for a specific task, the resulting pattern in the ICS network traffic does not vary significantly. As a result, most traffic patterns that are caused by tasks that are normally performed in a specific ICS have already occurred in the past, unless the ICS is performing a completely new task. In such environments, anomaly-based intrusion detection system (IDS) can be helpful in the detection of abnormal or malicious behaviors. An anomaly-based IDS learns a statistical model of the normal activities of an ICS. We use the nearest-neighbor search (NNS) to learn patterns caused by normal activities of an ICS and identify anomalies. Our method learns the normal behavior in the overall traffic pattern based on the number of network packets transmitted and received along pairs of devices over a certain time interval. The method uses a geometric noise model with lognormal distribution to model the randomness on ICS network traffic and learns solutions through cross-validation on random samples. We present a fast algorithm, along with its theoretical time complexity analysis, in order to apply our method in real-time on a large-scale ICS. We provide experimental results tested on various types of large-scale traffic data that are collected from real ICSs of critical infrastructures.

NAVER : Design and Implementation of Networked Virtual Environments Based on PC Cluster

Proceedings of the Korean Society for Emotion and Sensibility Conference, 2002

Maximum-area and maximum-perimeter rectangles in polygons

Computational Geometry, 2021

Abstract We study the problem of finding maximum-area and maximum-perimeter rectangles that are i... more Abstract We study the problem of finding maximum-area and maximum-perimeter rectangles that are inscribed in polygons in the plane. There has been a fair amount of work on this problem when the rectangles have to be axis-aligned or when the polygons are convex. We consider this problem in polygons with n vertices that are not necessarily convex, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle and a maximum-perimeter rectangle in O ( n 3 log ⁡ n ) time using O ( k n 2 + n ) space, where k is the number of reflex vertices of the polygon. Our algorithm can report all maximum-area rectangles in the same time using O ( n 3 ) space. We also present a simple algorithm that finds a maximum-area rectangle inscribed in a convex polygon with n vertices in O ( n 3 ) time using O ( n ) space.

Adaptive Computational Geometry (Computational Geometry and Discrete Mathematics)

Bundling Two Simple Polygons to Minimize Their Convex Hull

Given two simple polygons P and Q in the plane, we study the problem of finding a placement \(\va... more Given two simple polygons P and Q in the plane, we study the problem of finding a placement \(\varphi P\) of P such that \(\varphi P\) and Q are disjoint in their interiors and the convex hull of their union is minimized. We present exact algorithms for this problem that use much less space than the complexity of the Minkowski sum of P and Q. When the orientation of P is fixed, we find an optimal translation of P in \(O(n^2m^2\log n)\) time using O(nm) space, where n and m (\(n\ge m\)) denote the number of edges of P and Q, respectively. When we allow reorienting P, we find an optimal rigid motion of P in \(O(n^3m^3\log n)\) time using O(nm) space. In both cases, we find an optimal placement of P using linear space at the expense of slightly increased running time. For two polyhedra in three dimensional space, we find an optimal translation in \(O(n^3m^3 \log n)\) time using O(nm) space or in \(O(n^3m^3(m+\log n))\) time using linear space.

Maximum Overlap of Two Convex Polygons

KIISE Transactions on Computing Practices, 2021

On Romeo and Juliet problems: Minimizing distance-to-sight

Computational Geometry, 2019

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility ... more We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points s and t in a simple polygon P with no holes, we want to minimize the distance they travel in order to see each other in P. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple n-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in O(log 2 n) time for any two query positions s, t where the two persons start.

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Adaptive Algorithms for Planar Convex Hull Problems

Lecture Notes in Computer Science, 2010

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