Let V ⊂ ℝ² be a set of n sites in the plane. The unit disk graph DG(V) of V is the graph with vertex set V where two sites v and w are adjacent if and only if their Euclidean distance is at most 1. We develop a compact routing scheme ℛ for DG(V). The routing scheme ℛ preprocesses DG(V) by assigning a label 𝓁(v) to every site v in V. After that, for any two sites s and t, the scheme ℛ must be able to route a packet from s to t as follows: given the label of a current vertex r (initially, r = s), the label of the target vertex t, and additional information in the header of the packet, the scheme determines a neighbor r' of r. Then, the packet is forwarded to r', and the process continues until the packet reaches its desired target t. The resulting path between the source s and the target t is called the routing path of s and t. The stretch of the routing scheme is the maximum ratio of the total Euclidean length of the routing path and of the shortest path in DG(V), between any two sites s, t ∈ V. We show that for any given ε > 0, we can construct a routing scheme for DG(V) with diameter D that achieves stretch 1+ε, has label size (1/ε)^{O(ε^(-2))} log Dlog³n/log log n, and the header has at most O(log²n/log log n) bits. In the past, several routing schemes for unit disk graphs have been proposed. Our scheme achieves poly-logarithmic label and header size, small stretch and does not use any neighborhood oracles.

Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

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 P , where 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.

The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n / logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.