Chan-Su Shin

Area-efficient algorithms for straight-line tree drawings

Abstract We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment. Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O (n log n) (Crescenzi et al., 1992) to O (n loglog n) . This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type. Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O (n loglog n) . If T is binary, we can obtain an O (n loglog n) -area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O (n/ log n) bends in total. Secondly, we present O (n loglog n) -area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.

Labeling Points with Weights

Abstract Annotating maps, graphs, and diagrams with pieces of text is an important step in information visualization that is usually referred to as label placement. We define nine label-placement models for labeling points with axis-parallel rectangles given a weight for each point. There are two groups: fixed-position models and slider models. We aim to maximize the weight sum of those points that receive a label. We first compare our models by giving bounds for the ratios between the weights of maximum-weight labelings in different models. Then we present algorithms for labeling n points with unit-height rectangles. We show how an O(n\log n)-time factor-2 approximation algorithm and a PTAS for fixed-position models can be extended to handle the weighted case. Our main contribution is the first algorithm for weighted sliding labels. Its approximation factor is (2+\varepsilon), it runs in O(n2/\varepsilon) time and uses O(n/\varepsilon) space. We show that other than for fixed-position models even the projection to one dimension remains NP-hard. For slider models we also investigate some special cases, namely (a) the number of different point weights is bounded, (b) all labels are unit squares, and (c) the ratio between maximum and minimum label height is bounded.

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 farthest neighbors on a convex polytope

Let N be a set of n points in convex position in R3. The farthest point Voronoi diagram of N partitions R3 into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log2 n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log2 n), and to perform farthest-neighbor queries on N in O(log2 n) time with high probability.

Computing the Optimal Bridge between Two Polygons

Abstract. Let P and Q be disjoint polygons in the plane. We consider the problem of finding an optimal bridge (p,q) , p∈ \partial P and q∈ \partial Q , such that the length of the longest path from a point in P , passing through the bridge (p,q) , to a point Q is minimized. We propose efficient algorithms for three cases according to whether P and Q are convex or not. These problems are motivated from the bridge construction between two islands (or the canal construction between two lakes).

Efficient Algorithms for Two-Center Problems for a Convex Polygon

Let P be a convex polygon with n vertices. We want to find two congruent disks whose union covers P and whose radius is minimized. We also consider its discrete version with centers restricted to be at vertices of P. Standard and discrete two-center problems are respectively solved in O(n log3 n log log n) and O(n log2 n) time. Furthermore, we can solve both of the standard and discrete two-center problems for a set of points in convex positions in O(n log2 n) time.

Constructing the city Voronoi diagram faster

A rug cleaning attachment for a vacuum cleaner is disclosed. The attachment has a suds confining chamber and a scrubbing brush roll mounted thereon for rotation about a horizontal axis within the chamber. The brush roll has a multiplicity of tufts which are adapted to scrub suds into the rug and which define a cylindrical envelope upon rotation of the brush roll. A shield closely conforms to the cylindrical element and guards against suds being centrifugally thrown into the drive and/or suction components of the vacuum cleaner. A scalloped doctor blade is provided for the brush roll which projects into the cylindrical envelope defined by the tufts to scrape spent suds and lint from the brush roll.

Area-efficient algorithms for upward straight-line tree drawings

In this paper, we investigate planar upward straight-line grid drawing problems for bounded-degree rooted trees so that a drawing takes up as little area as possible. A planar upward straight-line grid tree drawing satisfies the following four constraints: (1) all vertices are placed at distinct grid points (grid), (2) all edges are drawn as straight lines (straight-line), (3) no two edges in the drawing intersect (planar), and (4) no parents are placed below their children (upward). Our results are summarized as follows. First, we show that a bounded-degree tree T with n vertices admits an upward straight-line drawing with area O(n log log n). If T is binary, we can obtain an O(n log log n)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/log n) bends in total. Second, we show that bounded-degree trees in some classes of balanced trees, frequently used as search trees, admit strictly upward straight-line drawings with area O(n loglog n). They include k-balanced trees, red-black trees, BB [α]-trees, and (a, b)-trees. In addition, trees in the same classes admit O(n(loglog n)2)-area strictly upward straight-line drawings that preserve the left-to-right ordering of the children of each vertex. Finally, we discuss an extension of our drawing algorithms to non-upward straight-line drawing algorithms in 2- and 3-dimensions.

Labeling a rectilinear map with sliding labels

A rectilinear map consists of a set of mutually non-intersecting rectilinear (i.e., horizontal or vertical) line segments, and each segment is allowed to use a rectangular label of height B and length the same as the segment. Sliding labels are not restricted to any finite number of predefined positions but can slide and be placed at any position as long as it intersects the segment. This paper considers three versions of the problem of labeling a rectilinear map with sliding labels and presents efficient exact and approximation algorithms for them.

Computing Weighted Rectilinear Median and Center Set in the Presence of Obstacles

Given a set B of obstacles and a set S of source points in the plane, the problem of finding a set of points subject to a certain objective function with respect to B and S is a basic problem in applications such as facility location problem 5.