Routing in Polygonal Domains

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex. Then, we must be able to route a data packet between any two vertices p and q of Pwhere each step must use only the label of the target node q and the routing table of the current node. For any fixed eps > 0, we pre ent a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps. The labels have O(log n) bits, and the routing tables are of size O((eps^{-1} + h) log n). The preprocessing time is O(n^2 log n + hn^2 + eps^{-1}hn). It can be improved to O(n 2 + eps^{-1}n) for simple polygons

Practical Distance Functions for Path-Planning in Planar Domains

Path planning is an important problem in robotics. One way to plan a path between two points $x,y$ within a (not necessarily simply-connected) planar domain $\Omega$, is to define a non-negative distance function $d(x,y)$ on $\Omega\times\Omega$ such that following the (descending) gradient of this distance function traces such a path. This presents two equally important challenges: A mathematical challenge -- to define $d$ such that $d(x,y)$ has a single minimum for any fixed $y$ (and this is when $x=y$), since a local minimum is in effect a "dead end", A computational challenge -- to define $d$ such that it may be computed efficiently. In this paper, given a description of $\Omega$, we show how to assign coordinates to each point of $\Omega$ and define a family of distance functions between points using these coordinates, such that both the mathematical and the computational challenges are met. This is done using the concepts of \emph{harmonic measure} and \emph{$f$-divergences}. In practice, path planning is done on a discrete network defined on a finite set of \emph{sites} sampled from $\Omega$, so any method that works well on the continuous domain must be adapted so that it still works well on the discrete domain. Given a set of sites sampled from $\Omega$, we show how to define a network connecting these sites such that a \emph{greedy routing} algorithm (which is the discrete equivalent of continuous gradient descent) based on the distance function mentioned above is guaranteed to generate a path in the network between any two such sites. In many cases, this network is close to a (desirable) planar graph, especially if the set of sites is dense

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$\theta_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Maximum thick paths in static and dynamic environments

AbstractWe consider the problem of finding a large number of disjoint paths for unit disks moving amidst static or dynamic obstacles. The problem is motivated by the capacity estimation problem in air traffic management, in which one must determine how many aircraft can safely move through a domain while avoiding each other and avoiding “no-fly zones” and predicted weather hazards. For the static case we give efficient exact algorithms, based on adapting the “continuous uppermost path” paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem.Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. Deciding the existence of just one path, even for a 0-radius disk, moving with bounded speed is NP-hard, as shown by Canny and Reif [J. Canny, J.H. Reif, New lower bound techniques for robot motion planning problems, in: Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987, pp. 49–60]. Moreover, we observe that determining the existence of a given number of paths is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying “dual” approximations, compromising on the radius of the disks and on the maximum speed of motion.Our main result is a pseudopolynomial-time dual-approximation algorithm. If K unit disks, each moving with speed at most 1, can be routed through an environment, our algorithm finds (at least) K paths for disks of radius somewhat smaller than 1 moving with speed somewhat larger than 1

Chasing Puppies: Mobile Beacon Routing on Closed Curves

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure