Abstract Let $$S \subseteq \mathbb {R}^2$$ S ⊆ R 2 be a set of nsites in the plane, so that every site $$s \in S$$ s ∈ S has an associated radius $$r_s > 0$$ r s > 0 . Let $$\mathcal {D}(S)$$ D ( S ) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites $$s, t \in S$$ s , t ∈ S if and only if the disks with centers s, t and radii $$r_s$$ r s , $$r_t$$ r t intersect. Our goal is to design data structures that maintain the connectivity structure of $$\mathcal {D}(S)$$ D ( S ) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix $$r_s = 1$$ r s = 1 , for all sites $$s \in S$$ s ∈ S . For this case, we describe a data structure that has $$O(\log ^2 n)$$ O ( log 2 n ) amortized update time and $$O(\log n/\log \log n)$$ O ( log n / log log n ) query time. Second, we look at disk graphs with bounded radius ratio $$\Psi $$ Ψ , i.e., for all $$s \in S$$ s ∈ S , we have $$1 \le r_s \le \Psi $$ 1 ≤ r s ≤ Ψ , for a parameter $$\Psi $$ Ψ that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time $$O(\Psi \log ^{4} n)$$ O ( Ψ log 4 n ) and query time $$O(\log n/\log \log n)$$ O ( log n / log log n ) . This improves the currently best update time by a factor of $$\Psi $$ Ψ . In the incremental case, we achieve logarithmic dependency on $$\Psi $$

Abstract We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $$L_p$$ L p -norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $$O(n \log ^3 n)$$ O ( n log 3 n ) storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat, and Sharir which required $$O(n^{\varepsilon })$$ O ( n ε ) time for an update and $$O(\log n)$$ O ( log n ) time for a query [SICOMP 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $$O(n^{\varepsilon })$$ O ( n ε ) factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques from the literature. Along the way, we obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for computing the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest-neighbor context, under both types of norm). To analyze this algorithm, we also improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we combine our vertical shallow cutting construction with Chan’s algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in $${{\mathbb {R}}}^3$$ R 3 . Along the way, we also revisit Chan’s technique and present a variant that uses a single binary counter, with a simpler analysis and improved amortized deletion time (by a logarithmic factor; the insertion and query costs remain asymptotically the same).

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites, where each $s \in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t \in S$ if and only if $|st| \leq r_s + r_t$, i.e., if the disks with centers $s$ and $t$ and respective radii $r_s$ and $r_t$ intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph $T(S)$ is the directed graph with vertex set $S$ and a directed edge from a site $s$ to a site $t$ if and only if $|st| \leq r_s$, i.e., if $t$ lies in the disk with center $s$ and radius $r_s$. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in $D(S)$ and in $T(S)$. These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in $D(S)$ and in $T(S)$ can be found in $O(n \log n)$ expected time, and that the (weighted) girth of $D(S)$ can be found in $O(n \log n)$ expected time. For this, we develop new tools for batched range searching that may be of independent interest.

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.

Let $P \subset \mathbb{R}^2$ be a planar $n$-point set such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) \leq r_p$. Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H \subseteq G$ with vertex set $P$ so that for any two vertices $p,q \in P$, we have $d_H(p, q) \leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths). We show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (\log n + \log \Psi))$ time, where $\Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures, we obtain a construction that runs in $O(n \log^5 n)$ time, independent of $\Psi$. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in $G$ from any given start vertex in $O(n \log n)$ time (in addition to the time it takes to build the spanner). Second, we show how to use our spanner to extend a reachability oracle to answer geometric reachability queries. In a geometric reachability query we ask whether a vertex $p$ in $G$ can “reach” a target $q$ which is an arbitrary point in the plane (rather than restricted to be another vertex $q$ of $G$ in a standard reachability query). Our spanner allows the reachability oracle to answer geometric reachability queries with an additive overhead of $O(\log n\log \Psi)$ to the query time and $O(n \log \Psi)$ to the space.

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.

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions that includes Lp-norms and additively weighted Euclidean distances, and for general (convex, pair- wise disjoint) sites that have constant description complexity (line segments, disks, etc.). Our data structure has a polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required O(n∊) time for an update and O(log n) time for a query [1]. Our data structure has numerous applications, and in all of them it gives faster algorithms, typically reducing an O(n∊) factor in the bounds to polylogarithmic. To further demonstrate its effectiveness, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph. To obtain this data structure, we combine and extend various techniques and obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of “vertical” shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for finding the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope has linear complexity, as is the case in the dynamic nearest- neighbor context). To analyze this algorithm, we improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we plug our vertical shallow cutting construction into Chan's algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in ℝ3. While doing this, we also revisit Chan's technique and present a variant that uses a single binary counter, with a simpler analysis and an improved amortized deletion time.

Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time. After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi < sqrt(3), we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Psi >= sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2/3)log n) and S(n) = O(n^(5/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3/2)) and O(n^(5/3) log^2 n) in case (ii) and (iii), respectively.

Let k ≥ 0 be an integer. In the approximate k-flat nearest neighbor (k-ANN) problem, we are given a set P ⊂ Rd of n points in d-dimensional space and a fixed approximation factor c > 1. Our goal is to preprocess P so that we can efficiently answer approximate k-flat nearest neighbor queries: given a k-flat F, find a point in P whose distance to F is within a factor c of the distance between F and the closest point in P. The case k = 0 corresponds to the well-studied approximate nearest neighbor problem, for which a plethora of results are known, both in low and high dimensions. The case k = 1 is called approximate line nearest neighbor. In this case, we are aware of only one provably efficient data structure, due to Andoni, Indyk, Krauthgamer, and Nguyen (AIKN) [2]. For k ≥ 2, we know of no previous results.