The Geodesic Edge Center of a Simple Polygon

The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete & Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete & Computational Geometry, 2016]. The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary

The Visibility Center of a Simple Polygon

We introduce the visibility center of a set of points inside a polygon - a point c_V such that the maximum geodesic distance from c_V to see any point in the set is minimized. For a simple polygon of n vertices and a set of m points inside it, we give an O((n+m) log (n+m)) time algorithm to find the visibility center. We find the visibility center of all points in a simple polygon in O(n log n) time. Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an O((n+k) log (n+k)) time algorithm for this problem, where k is the number of half-polygons

Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line

We study three geometric facility location problems in this thesis. First, we consider the dispersion problem in one dimension. We are given an ordered list of (possibly overlapping) intervals on a line. We wish to choose exactly one point from each interval such that their left to right ordering on the line matches the input order. The aim is to choose the points so that the distance between the closest pair of points is maximized, i.e., they must be socially distanced while respecting the order. We give a new linear-time algorithm for this problem that produces a lexicographically optimal solution. We also consider some generalizations of this problem. For the next two problems, the domain of interest is a simple polygon with n vertices. The second problem concerns the visibility center. The convention is to think of a polygon as the top view of a building (or art gallery) where the polygon boundary represents opaque walls. Two points in the domain are visible to each other if the line segment joining them does not intersect the polygon exterior. The distance to visibility from a source point to a target point is the minimum geodesic distance from the source to a point in the polygon visible to the target. The question is: Where should a single guard be located within the polygon to minimize the maximum distance to visibility? For m point sites in the polygon, we give an O((m + n) log (m + n)) time algorithm to determine their visibility center. Finally, we address the problem of locating the geodesic edge center of a simple polygon—a point in the polygon that minimizes the maximum geodesic distance to any edge. For a triangle, this point coincides with its incenter. The geodesic edge center is a generalization of the well-studied geodesic center (a point that minimizes the maximum distance to any vertex). Center problems are closely related to farthest Voronoi diagrams, which are well- studied for point sites in the plane, and less well-studied for line segment sites in the plane. When the domain is a polygon rather than the whole plane, only the case of point sites has been addressed—surprisingly, more general sites (with line segments being the simplest example) have been largely ignored. En route to our solution, we revisit, correct, and generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored to work specifically for point sites. We give an optimal linear-time algorithm for finding the geodesic edge center of a simple polygon

Distant Representatives for Rectangles in the Plane

The input to the distant representatives problem is a set of n objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the L?, L?, and L_? distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP)

Shortest Beer Path Queries in Interval Graphs

Our interest is in paths between pairs of vertices that go through at least one of a subset of the vertices known as beer vertices. Such a path is called a beer path, and the beer distance between two vertices is the length of the shortest beer path. We show that we can represent unweighted interval graphs using 2n log n + O(n) + O(|B|log n) bits where |B| is the number of beer vertices. This data structure answers beer distance queries in O(log^? n) time for any constant ? > 0 and shortest beer path queries in O(log^? n + d) time, where d is the beer distance between the two nodes. We also show that proper interval graphs may be represented using 3n + o(n) bits to support beer distance queries in O(f(n)log n) time for any f(n) ? ?(1) and shortest beer path queries in O(d) time. All of these results also have time-space trade-offs. Lastly we show that the information theoretic lower bound for beer proper interval graphs is very close to the space of our structure, namely log(4+2?3)n - o(n) (or about 2.9 n) bits

Token Swapping on Trees

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of \emph{swaps}, where a swap exchanges the tokens on the endpoints of an edge. Token swapping on a tree, also known as "sorting with a transposition tree", is not known to be in P nor NP-complete. We present some partial results: 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves---as all known approximation algorithms for the problem do---has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem---weighted coloured token swapping---is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved