We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in amortized time O(n 1/2 log 2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n # ) per update. Our algorithm uses a novel construction, the ordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. 1 Introduction A dynamic geometric data structure is one that maintains the solution to some problem, defined on a geometric input such as a point set, as the input undergoes update operations such as insertions or deletions of single points. Dynamic algorithms have been studied for many geometric optimization problems, including closest pairs [7, 23, 25, 26], diameter [7, 26], width [4], convex hulls [15, 22], linear ...

We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a non-degenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d 3, the expected time bound is O(n ), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linear-programming-type problems; some other applications of this technique are briefly mentioned.

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions take O(log 6 n) expected amortized time, and extreme-point queries take O(log 2 n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n ε)-time method by Agarwal and Matouˇsek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane). 1

Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log 3 f) where n is the number of sites and f is the number of output cells; and (2) a deterministic parallel algorithm in the EREW PRAM model for computing an algebraic planar Voronoi diagram (in which bisectors between sites are simple curves consisting of a constant number of algebraic pieces of constant degree) that runs in time O(log 2 n) using optimal O(n log n) work. The first result implies an algorithm for the problems of computing the convex hull of a point set and the intersection of a set of halfspaces in R 5, and computing the Euclidean Voronoi diagram in R 4. The second result implies both sequential and parallel work-optimal deterministic algorithms for a number of Voronoi diagram problems (including line segments in the plane), and other non-Voronoi diagram problems that can fit in the framework (including the intersection of equal radius balls in R 3 and some lower envelope problems in R 3). 1

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n 1−1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε) preprocessing time for any fixed ε> 0. An earlier method by Matouˇsek (SoCG’91) requires O(n log n) preprocessing time but O(n1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1−1/d) query time with high probability. Our method has several advantages: • It is conceptually simpler than Matouˇsek’s SoCG’92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children’s cells at each node). • It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). • A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n 1−1/⌊d/2 ⌋ ) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points,...). 1

Fitting a curve of a certain type to a given set of points in the plane is a basic problem in statistics and has numerous applications. We consider fitting a polyline with k joints under the min-sum criteria with respect to L1- and L2-metrics, which are more appropriate measures than uniform and Hausdorff metrics in statistical context. We present efficient algorithms for the 1-joint versions of the problem and fully polynomial-time approximation schemes for the general k-joint versions. 1

We show how to maintain the width of a set of n planar points subject to insertions and deletions of points in O( n) amortized time per update. Previously, no fully dynamic algorithm with a guaranteed sublinear time bound was known.

The ability to represent and query continuously moving objects is important in many applications of spatio-temporal database systems. In this paper we develop data structures for answering various queries on moving objects, including range and proximity queries, and study tradeoffs between various performance measures—query time, data structure size, and accuracy of results. 1

This paper gives an algorithm for polytope covering: let L and U be sets of points in R d , comprising n points altogether. A cover for L from U is a set C ae U with L a subset of the convex hull of C. Suppose c is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than c(5d ln c), for c large enough. The algorithm requires O(c 2 n 1+ffi ) expected time. 1 More exactly, the time bound is O(cn 1+ffi + c(nc) 1=(1+fl=(1+ffi)) ); where fl j 1=bd=2c. The previous best bounds were cO(log n) cover size in O(n d ) time.[MS92b] A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error ffl requires c = O(1=ffl) (d\Gamma1)=2 vertices, and the algorithm gives an approximation with c(5d 3 ln(1=ffl)) vertices. The algorithms apply ideas previously used for small-dimensional linear programm...

We describe dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function. Using these data structures, we obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree.