Showing papers by "Michael T. Goodrich published in 2004"

Efficient tree-based revocation in groups of low-state devices

Abstract: We study the problem of broadcasting confidential information to a collection of n devices while providing the ability to revoke an arbitrary subset of those devices (and tolerating collusion among the revoked devices). In this paper, we restrict our attention to low-memory devices, that is, devices that can store at most O(log n) keys. We consider solutions for both zero-state and low-state cases, where such devices are organized in a tree structure T. We allow the group controller to encrypt broadcasts to any subtree of T, even if the tree is based on an multi-way organizational chart or a severely unbalanced multicast tree.

Efficient Tree-Based Revocation in Groups of Low-State Devices

Abstract: We study the problem of broadcasting confidential information to a collection of n devices while providing the ability to revoke an arbitrary subset of those devices (and tolerating collusion among the revoked devices). In this paper, we restrict our attention to low-memory devices, that is, devices that can store at most O(log n) keys. We consider solutions for both zero-state and low-state cases, where such devices are organized in a tree structure T. We allow the group controller to encrypt broadcasts to any subtree of T, even if the tree is based on an multi-way organizational chart or a severely unbalanced multicast tree.

A multi-dimensional approach to force-directed layouts of large graphs

Abstract: We present a novel hierarchical force-directed method for drawing large graphs. Given a graph G=(V.E), the algorithm produces an embedding for G in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Such projections typically result in drawings that are "smoother" and more symmetric than direct drawings in 2D and 3D. In order to obtain fast placement of the vertices of the graph our algorithm employs a multi-scale technique based on a maximal independent set filtration of vertices of the graph. While most existing force-directed algorithms begin with an initial random placement of all the vertices, our algorithm attempts to place vertices "intelligently", close to their final positions. Other notable features of our approach include a fast energy function minimization strategy and efficient memory management. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a 550 MHz Pentium PC.

Contour interpolation by straight skeletons

Abstract: In this paper we present an efficient method for interpolating a piecewise-linear surface between two parallel slices, each consisting of an arbitrary number of (possibly nested) polygons that define 'material' and 'non-material' regions. This problem has applications to medical imaging, geographic information systems, etc. Our method is fully automatic and is guaranteed to produce non-self-intersecting surfaces in all cases regardless of the number of contours in each slice, their complexity and geometry, and the depth of their hierarchy of nesting. The method is based on computing cells in the overlay of the slices that form the symmetric difference between them. Then, the straight skeletons of the selected cells guide the triangulation of each face of the skeletons. Finally, the resulting triangles are lifted up in space to form an interpolating surface. We provide some experimental results on various complex examples to show the good and robust performance of our algorithm.

Deterministic sampling and range counting in geometric data streams

Abstract: We present memory-efficient deterministic algorithms for constructing ∈-nets and ∈-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.

Three-Dimensional Layers of Maxima

Abstract: We present an O(n log n)-time algorithm to solve the three-dimensional layers-of-maxima problem. This is an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah et al. has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.

Drawing Planar Graphs with Large Vertices and Thick Edges

Abstract: We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.

Confluent layered drawings

Abstract: We combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.