Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\}$ time, where $\omega<2.373$ denotes the matrix multiplication exponent and $\chi\in \Omega(n)\cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in $O(n^2 \log n)$ time

β\beta-Stars or On Extending a Drawing of a Connected Subgraph

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

A Novel Approach for Extraction of Polygon Regions

This paper presents a new algorithm to find out whether a polygon exists around a reference point given within the graphical domain. The algorithm is based on creating discrete line segments and then searching them using the orientations formed at segments intersections. The computational complexity of the searching algorithm has been determined as O( n2

Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). Let P be a trapezoided rectilinear simple polygon with n vertices. In O(log n) time using O(n/log n) processors we can optimally compute: 1. Minimum réctilinear link paths, or shortest paths in the L1 metric from any point in P to all vertices of P. 2. Minimum rectilinear link paths from any segment inside P to all vertices of P. 3. The rectilinear window (histogram) partition of P. 4. Both covering radii and vertex intervals for any diagonal of P. 5. A data structure to support rectilinear link-distance queries between any two points in P (queries can be answered optimally in O(log n) time by uniprocessor). Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which used O(n log n) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon

Shortest path queries in rectilinear worlds

Abstract In this paper, a data structure is given for two and higher dimensional shortest path queries. For a set of n axis-parallel rectangles in the plane, or boxes in d-space, and a fixed target, it is possible with this structure to find a shortest rectilinear path avoiding all rectangles or boxes from any point to this target. Alternatively, it is possible to find the length of the path. The metric considered is a generalization of the Ll-metric and the link metric, where the length of a path is its L1-Iength plus some (fixed) constant times the number of turns on the path. The data structure has size 0« n log n )d-l), and a query takes O(logd-l n) time (plus the output size if the path must be reported). As a byproduct, a relatively simple solution to the single shot problem is obtained; the shortest path between two given points can be computed in time O(ndlogn) for d ~ 3, and in time 0(n 2 ) in the plane

Multi-Dimensional Medial Geometry: Formulation, Computation, and Applications

Medial axis is a classical shape descriptor. It is a piece of geometry that lies in the middle of the original shape. Compared to the original shape representation, the medial axis is always one dimension lower and it carries many intrinsic shape properties explicitly. Therefore, it is widely used in a large amount of applications in various fields. However, medial axis is unstable to the boundary noise, often referred to as its instability. A small amount of change on the object boundary can cause a dramatic change in the medial axis. To tackle this problem, a significance measure is often associated with the medial axis, so that medial points with small significance are removed and only the stable part remains. In addition to this problem, many applications prefer even lower dimensional medial forms, e.g., shape centers of 2D shapes, and medial curves of 3D shapes. Unfortunately, good significance measures and good definitions of lower dimensional medial forms are still lacking. In this dissertation, we extended Blum\u27s grassfire burning to the medial axis in both 2D and 3D to define a significance measure as a distance function on the medial axis. We show that this distance function is well behaved and it has nice properties. In 2D, we also define a shape center based on this distance function. We then devise an iterative algorithm to compute the distance function and the shape center. We demonstrate usefulness of this distance function and shape center in various applications. Finally we point out the direction for future research based on this dissertation