A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures. This revision contains a few improvements in the expositio

Abstract. We investigate ways in which an algorithm can improve its expected performance by fine-tuning itself automatically with respect to an unknown input distribution D. We assume here that D is of product type. More precisely, suppose that we need to process a sequence I 1 , I 2 , . . . of inputs I = (x 1 , x 2 , . . . , xn) of some fixed length n, where each x i is drawn independently from some arbitrary, unknown distribution D i . The goal is to design an algorithm for these inputs so that eventually the expected running time will be optimal for the input distribution D = i D i .We give such self-improving algorithms for two problems: (i) sorting a sequence of numbers and (ii) computing the Delaunay triangulation of a planar point set. Both algorithms achieve optimal expected limiting complexity. The algorithms begin with a training phase during which they collect information about the input distribution, followed by a stationary regime in which the algorithms settle to their optimized incarnations.

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.

A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n 2 ) time and constant workspace. We also consider the problem of preprocessing a simple polygon P for shortest path queries, where P is given by the ordered sequence of its n vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n 2 /s) time.

We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffle-operation in constant time; (ii) if we know the ordering of a planar point set in x-and in y-direction, its DT can be found by a randomized algebraic computation tree of expected linear depth; (iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any P ⊆ U , D can find the DT of P in time O(|P | log log |U |); (iv) given a universe U of points in 3-space in general convex position, there is a data structure D for convex hull queries: for any P ⊆ U , D can find the convex hull of P in time O(|P |(log log |U |) 2 ); (v) given a convex polytope in 3-space with n vertices which are colored with χ > 2 colors, we can split it into the convex hulls of the individual color classes in time O(n(log log n) 2 ). The results (i)-(iii) generalize to higher dimensions. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearest-neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.