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Register a new userRouting on the Visibility Graph
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Added by
Matias Korman
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.
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For a fixed virtual scene (=collection of simplices) S and given observer position p, how many elements of S are weakly visible (i.e. not fully occluded by others) from p? The present work explores the trade-off between query time and preprocessing space for these quantities in 2D: exactly, in the approximate deterministic, and in the probabilistic sense. We deduce the EXISTENCE of an O(m^2/n^2) space data structure for S that, given p and time O(log n), allows to approximate the ratio of occluded segments up to arbitrary constant absolute error; here m denotes the size of the Visibility Graph--which may be quadratic, but typically is just linear in the size n of the scene S. On the other hand, we present a data structure CONSTRUCTIBLE in O(n*log(n)+m^2*polylog(n)/k) preprocessing time and space with similar approximation properties and query time O(k*polylog n), where k<n is an arbitrary parameter. We describe an implementation of this approach and demonstrate the practical benefit of the parameter k to trade memory for query time in an empirical evaluation on three classes of benchmark scenes.
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.
An important problem in terrain analysis is modeling how water flows across a terrain creating floods by forming channels and filling depressions. In this paper we study a number of emph{flow-query} related problems: Given a terrain $Sigma$, represented as a triangulated $xy$-monotone surface with $n$ vertices, a rain distribution $R$ which may vary over time, determine how much water is flowing over a given edge as a function of time. We develop internal-memory as well as I/O-efficient algorithms for flow queries. This paper contains four main results: (i) We present an internal-memory algorithm that preprocesses $Sigma$ into a linear-size data structure that for a (possibly time varying) rain distribution $R$ can return the flow-rate functions of all edges of $Sigma$ in $O(rho k+|phi| log n)$ time, where $rho$ is the number of sinks in $Sigma$, $k$ is the number of times the rain distribution changes, and $|phi|$ is the total complexity of the flow-rate functions that have non-zero values; (ii) We also present an I/O-efficient algorithm for preprocessing $Sigma$ into a linear-size data structure so that for a rain distribution $R$, it can compute the flow-rate function of all edges using $O(text{Sort}(|phi|))$ I/Os and $O(|phi| log |phi|)$ internal computation time. (iii) $Sigma$ can be preprocessed into a linear-size data structure so that for a given rain distribution $R$, the flow-rate function of an edge $(q,r) in Sigma$ under the single-flow direction (SFD) model can be computed more efficiently. (iv) We present an algorithm for computing the two-dimensional channel along which water flows using Mannings equation; a widely used empirical equation that relates the flow-rate of water in an open channel to the geometry of the channel along with the height of water in the channel.
Let $P$ be an $x$-monotone orthogonal polygon with $n$ vertices. We call $P$ a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points $p$ and $q$ in $P$ are co-visible if and only if the (axis-parallel) rectangle spanned by $p$ and $q$ completely lies in $P$. In the $r$-visibility graph $G(P)$ of $P$, we connect two vertices of $P$ with an edge if and only if they are co-visible. We consider routing with preprocessing in $G(P)$. We may preprocess $P$ to obtain a label and a routing table for each vertex of $P$. Then, we must be able to route a packet between any two vertices $s$ and $t$ of $P$, where each step may use only the label of the target node $t$, the routing table and neighborhood of the current node, and the packet header. We present a routing scheme for double histograms that sends any data packet along a path whose length is at most twice the (unweighted) shortest path distance between the endpoints. In our scheme, the labels, routing tables, and headers need $O(log n)$ bits. For the case of simple histograms, we obtain a routing scheme with optimal routing paths, $O(log n)$-bit labels, one-bit routing tables, and no headers.
We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with $m>1$ viewpoints (or guards) located on the terrain surface. A point on the terrain is considered emph{visible} if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the emph{colored} visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting. To the best of our knowledge, the other structures have not been studied before.
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