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, pairwise 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 ε ) 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 ε ) 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 * A full version is available on the arXiv [20] has linear complexity, as is the case in the dynamic nearestneighbor 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 R 3 . 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.

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 P ⊂ R d be a set of n points in the d dimensions such that each point p ∈ P has an associated radius r p > 0. The transmission graph G for P is the directed graph with vertex set P such that there is an edge from p to q if and only if d(p, q) ≤ r p , for any p, q ∈ P .A reachability oracle is a data structure that decides for any two vertices p, q ∈ G whether G has a path from p to q. The quality of the oracle is measured by the space requirement S(n), the query time Q(n), and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in O(n log n) time an oracle with Q(n) = O(1) and S(n) = O(n). For planar point sets, the ratio Ψ between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on Ψ: the first works only for Ψ < √ 3 and achieves Q(n) = O(1) with S(n) = O(n) and preprocessing time O(n log n); the second data structure gives Q(n) = O(Ψ 3

Let P ⊂ R 2 be a planar n-point set such that each point p ∈ 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 ∈ P , there is an edge from p to q if and only if d(p, q) ≤ r p .Let t > 1 be a constant. A t-spanner for G is a subgraph H ⊆ G with vertex set P so that for any two vertices p, q ∈ P , we have d H (p, q) ≤ td 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 Ψ)) time, where Ψ 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 Ψ.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 Ψ) to the query time and O(n log Ψ) to the space.

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that all points in a cell have the same nearest neighbor in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1, . . . , n}, an s-workspace algorithm has readonly access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing an arbitrary triangulation of S in time O(n 2 /s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n 2 /s) log s + n log s log * s). * WS and PS were supported in part by DFG Grants MU 3501/1 and MU 3501/2. YS was supported by the DFG within the research training group "Methods for Discrete Structures" (GRK 1408).The exact setting may vary, but there is a common theme: the input resides in read-only memory, the output must be written to a write-only structure, and we can use O(s) additional variables to find the solution (for a parameter s). The goal is to design algorithms whose running time decreases as s grows, giving a time-space trade-off [23]. One of the first problems considered in this model is sorting [19,20]. Here, the time-space product is known to be Ω(n 2 ) [8], and matching upper bounds for the case b ∈ Ω(log n) ∩ O(n/ log n) were obtained by Pagter and Rauhe [21] (b denotes the available workspace in bits).Our current notion of memory constrained algorithms was introduced to computational geometry by Asano et al. [4], who showed how to compute many classic geometric structures with O(1) workspace (related models were studied before [9]). Later, time-space trade-offs were given for problems on simple polygons, e.g., shortest paths [1], visibility [6], or the convex hull of the vertices [5].We consider a model in which the set S of n points is in an array such that random access to each input point is possible, but we may not change or even reorder the input. Additionally, we have O(s) variables (for a parameter s ∈ {1, . . . , n}). We assume that each variable or pointer contains a data word of Θ(log n) bits. Other than this, the model allows the usual word RAM operations. In this setting we study two problems: computing an arbitrary triangulation for S and computing the Voronoi diagram VD(S) for S. Since the output cannot be stored explicitly, the goal is to report the edges of the triangulation or the vertices of VD(S) successively, in no particular order. Dually, the latter goal may be phrased in terms of Delaunay triangulations. We focus on Voronoi diagrams, as they lead to a more natural presentation.Both problems can be solved in O(n 2 ) time with O(1) workspace [4] or in O(n log n) ...