Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Fast Cell Discovery in mm-wave 5G Networks with Context Information

The exploitation of mm-wave bands is one of the key-enabler for 5G mobile radio networks. However, the introduction of mm-wave technologies in cellular networks is not straightforward due to harsh propagation conditions that limit the mm-wave access availability. Mm-wave technologies require high-gain antenna systems to compensate for high path loss and limited power. As a consequence, directional transmissions must be used for cell discovery and synchronization processes: this can lead to a non-negligible access delay caused by the exploration of the cell area with multiple transmissions along different directions. The integration of mm-wave technologies and conventional wireless access networks with the objective of speeding up the cell search process requires new 5G network architectural solutions. Such architectures introduce a functional split between C-plane and U-plane, thereby guaranteeing the availability of a reliable signaling channel through conventional wireless technologies that provides the opportunity to collect useful context information from the network edge. In this article, we leverage the context information related to user positions to improve the directional cell discovery process. We investigate fundamental trade-offs of this process and the effects of the context information accuracy on the overall system performance. We also cope with obstacle obstructions in the cell area and propose an approach based on a geo-located context database where information gathered over time is stored to guide future searches. Analytic models and numerical results are provided to validate proposed strategies.Comment: 14 pages, submitted to IEEE Transaction on Mobile Computin

Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

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

An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithm

Improved Handover Through Dual Connectivity in 5G mmWave Mobile Networks

The millimeter wave (mmWave) bands offer the possibility of orders of magnitude greater throughput for fifth generation (5G) cellular systems. However, since mmWave signals are highly susceptible to blockage, channel quality on any one mmWave link can be extremely intermittent. This paper implements a novel dual connectivity protocol that enables mobile user equipment (UE) devices to maintain physical layer connections to 4G and 5G cells simultaneously. A novel uplink control signaling system combined with a local coordinator enables rapid path switching in the event of failures on any one link. This paper provides the first comprehensive end-to-end evaluation of handover mechanisms in mmWave cellular systems. The simulation framework includes detailed measurement-based channel models to realistically capture spatial dynamics of blocking events, as well as the full details of MAC, RLC and transport protocols. Compared to conventional handover mechanisms, the study reveals significant benefits of the proposed method under several metrics.Comment: 16 pages, 13 figures, to appear on the 2017 IEEE JSAC Special Issue on Millimeter Wave Communications for Future Mobile Network

Visibility Graphs, Dismantlability, and the Cops and Robbers Game

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.Comment: 23 page