This paper describes new methods for maintaining a point-location data structure for a dynamically-changing monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of k-edge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial point-location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point-lo...

In this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for object-oriented and temporal databases and for constraint logic programming. Part of the structure uses a novel weight-balancing technique for efficient worst-case manipulation of balanced trees of independent interest. The external interval tree, as well at our new balancing technique, have recently been used to develop several efficient external data structures.

We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O(logn) time, while updates take O(log² n) time (amortized for vertex insertion/deletion and worst-case for the others). The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.

Efficient data structures are given for the following two query problems: preprocess a set P of simple polygons with a total of n edges, so that all polygons of P intersected by a query segment can be reported efficiently, and (ii) preprocess a set S of n segments, so that the connected components of the arrangement of S intersected by a query segment can be reported quickly. In these problems we do not want to return the polygons or connected components explicitly (i.e., we do not wish to report the segments defining the polygon or the segments lying in the connected components). Instead, we assume that the polygons (or connected components) are labeled and we just want to report their labels. We present data structures of size O(n 1+ffl ) that can answer a query in time O(n 1=2+ffl +k), where k is the output size. If the edges of P (or the segments in S) are orthogonal, the query time can be improved to O(log n + k) using O(n log n) space. We also present data structures that c...

We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line ℓ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ. We present two data structures for this problem. The first data structure uses O(n 1+ε) space and preprocessing time, and answers queries in O(2 1/ε log n) time. The second data structure uses O(n log 3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n 2) space. In the process of developing the second data structure, we develop a new representation of nearestpoint and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make Θ(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries. 1

Abstract. We study the problem of securely searching for resources in p2p networks where a constant fraction of the peers may act maliciously. We present two novel hashing-based schemes that can be employed to reliably support resource location and content retrieval queries, limiting the ability of adversarial nodes to carry out attacks. Our schemes achieve scalability and load balancing and have small authentication overhead. In particular, for a network with n peers, resources are securely located with O(log 2 n) messages and content from a collection of m data items is securely retrieved with O(log nlog m) messages. Key words: peer-to-peer, overlay networks, distributed hash tables, one-way hash functions, digital signatures 1