In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the k-level of the arrangement A(F) is O(k 3+" /(n=k)), for any " ? 0, where /(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F . This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [51], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static...

In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

In this paper, we show the existence of small coresets for the problems of computing k-median and k-means clustering for points in low dimension. In other words, we show that given a point set P in IR , one can compute a weighted set S P , of size log n), such that one can compute the k-median/means clustering on S instead of on P , and get an (1 + ")-approximation.

The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n-gon with r reflex vertices in time O(n 1+" +n 8=11+" r 9=11+" ), for any fixed " ? 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in IR 3 and answer queries asking which triangle would be first hit by a query ray, and (2) maintain a changing set of rays in IR 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a ...

We describe a machine model for automatically dynamizing static algorithms and apply it to historyindependent data structures. Static programs expressed in this model are dynamized automatically by keeping track of dependences between code and data in the form of a dynamic dependence graph. To study the performance of such automatically dynamized algorithms we present an analysis technique based on trace stability. As an example of the use of the model, we dynamize the Parallel Tree Contraction Algorithm of Miller and Reif to obtain a history-independent data structure for the dynamic trees problem of Sleator and Tarjan. 1

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 perform linear programming optimizations on the intersection of k polyhedra in R³, represented by their outer recursive decompositions, in expected time O(k log k log n + √k log k log³ n). We use this result to derive efficient algorithms for dynamic linear programming problems in which constraints are inserted and deleted, and queries must optimize specified objective functions. As an application, we describe an improved solution to the planar 2-center and 2-median problems.

We examine the problem of efficiently computing sum/count/avg aggregates over...

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangent-finding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...