Routing in Histograms

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.Comment: 18 pages, 11 figure

Approximability of guarding weak visibility polygons

The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of O(log n), which was improved up to O (log log OPT) by King and Kirkpatrick (2011). It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in O(n(2)) time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than ((1 - is an element of)/12) Inn for any is an element of > 0, unless NP = P. We also show that, for the special class of polygons without holes that are orthogonal as well as weakly visible from an edge, the approximation ratio can be improved to 3. Finally, we consider the point guard problem and show that it is NP-hard in the case of polygons weakly visible from an edge. (C) 2016 Elsevier B.V. All rights reserved