Computational geometry

Models and Methods in Social Network Analysis presents the most important developments in quantitative models and methods for analyzing social network data that have appeared during the 1990s. Intended as a complement to Wasserman and Faust’s Social Network Analysis: Methods and Applications, it is a collection of original articles by leading methodologists reviewing recent advances in their particular areas of network methods. Reviewed are advances in network measurement, network sampling, the analysis of centrality, positional analysis or blockmodeling, the analysis of diffusion through networks, the analysis of affiliation or “two-mode” networks, the theory of random graphs, dependence graphs, exponential families of random graphs, the analysis of longitudinal network data, graphic techniques for exploring network data, and software for the analysis of social networks.

/pdf/models-and-methods-in-social-network-analysis-46f9pqfov7.pdf

Models and methods in social network analysis

In this paper we explore the use of weak B-trees to represent sorted lists. In weak B-trees each node has at least a and at most b sons where 2a ≤ b. We analyse the worst case cost of sequences of insertions and deletions in weak B-trees. This leads to a new data structure (level-linked weak B-trees) for representing sorted lists when the access pattern exhibits a (time-varying) locality of reference. Our structure is substantially simpler than the one proposed by Guibas, McCreight, Plass and Roberts, yet it has many of its properties. Our structure is as simple as the one proposed by Brown/Tarjan, but our structure can treat arbitrary sequences of insertions and deletions whilst theirs can only treat non-interacting insertions and deletions.

A New Data Structure for Representing Sorted Lists

Invited Lectures.- Flexible Path Planning Using Corridor Maps.- A Bridging Model for Multi-core Computing.- Contributed Papers.- Robust Kinetic Convex Hulls in 3D.- On Dominance Reporting in 3D.- Stabbing Convex Polygons with a Segment or a Polygon.- An Efficient Algorithm for 2D Euclidean 2-Center with Outliers.- A Near-Tight Bound for the Online Steiner Tree Problem in Graphs of Bounded Asymmetry.- Cache-Oblivious Red-Blue Line Segment Intersection.- The Complexity of Bisectors and Voronoi Diagrams on Realistic Terrains.- Space-Time Tradeoffs for Proximity Searching in Doubling Spaces.- A Scaling Algorithm for the Maximum Node-Capacitated Multiflow Problem.- Linear Time Planarity Testing and Embedding of Strongly Connected Cyclic Level Graphs.- Straight Skeletons of Three-Dimensional Polyhedra.- Randomized Competitive Analysis for Two-Server Problems.- Decompositions and Boundary Coverings of Non-convex Fat Polyhedra.- Approximating Multi-criteria Max-TSP.- An Integer Programming Algorithm for Routing Optimization in IP Networks.- A Constant-Approximate Feasibility Test for Multiprocessor Real-Time Scheduling.- Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree.- Engineering Tree Labeling Schemes: A Case Study on Least Common Ancestors.- A Practical Quicksort Algorithm for Graphics Processors.- Bloomier Filters: A Second Look.- Coupled Path Planning, Region Optimization, and Applications in Intensity-Modulated Radiation Therapy.- A New Approach to Exact Crossing Minimization.- A Characterization of 2-Player Mechanisms for Scheduling.- A Local-Search 2-Approximation for 2-Correlation-Clustering.- The Alcuin Number of a Graph.- Time-Dependent SHARC-Routing.- Detecting Regular Visit Patterns.- Improved Approximation Algorithms for Relay Placement.- Selfish Bin Packing.- Improved Randomized Results for That Interval Selection Problem.- Succinct Representations of Arbitrary Graphs.- Edge Coloring and Decompositions of Weighted Graphs.- The Complexity of Sorting with Networks of Stacks and Queues.- Faster Steiner Tree Computation in Polynomial-Space.- Fitting a Step Function to a Point Set.- Faster Swap Edge Computation in Minimum Diameter Spanning Trees.- The Partial Augment-Relabel Algorithm for the Maximum Flow Problem.- An Optimal Dynamic Spanner for Doubling Metric Spaces.- RFQ: Redemptive Fair Queuing.- Range Medians.- Locality and Bounding-Box Quality of Two-Dimensional Space-Filling Curves.- Probabilistic Analysis of Online Bin Coloring Algorithms Via Stochastic Comparison.- On the Complexity of Optimal Hotlink Assignment.- Oblivious Randomized Direct Search for Real-Parameter Optimization.- Path Minima in Incremental Unrooted Trees.- Improved Competitive Performance Bounds for CIOQ Switches.- Two-Stage Robust Network Design with Exponential Scenarios.- An Optimal Incremental Algorithm for Minimizing Lateness with Rejection.- More Robust Hashing: Cuckoo Hashing with a Stash.- Better and Simpler Approximation Algorithms for the Stable Marriage Problem.- Edit Distances and Factorisations of Even Permutations.- Speed Scaling Functions for Flow Time Scheduling Based on Active Job Count.- Facility Location in Dynamic Geometric Data Streams.- The Effects of Local Randomness in the Adversarial Queueing Model.- Parallel Imaging Problem.- An Online Algorithm for Finding the Longest Previous Factors.- Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions.- Improved BDD Algorithms for the Simulation of Quantum Circuits.- Mobile Route Planning.- How Reliable Are Practical Point-in-Polygon Strategies?.- Fast Divide-and-Conquer Algorithms for Preemptive Scheduling Problems with Controllable Processing Times - A Polymatroid Optimization Approach.- Approximability of Average Completion Time Scheduling on Unrelated Machines.- Relative Convex Hulls in Semi-dynamic Subdivisions.- An Experimental Analysis of Robinson-Foulds Distance Matrix Algorithms.- On the Size of the 3D Visibility Skeleton: Experimental Results.- An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions.- Deterministic Sampling Algorithms for Network Design.

Algorithms - ESA 2008 : 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008 : proceedings

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the $(\le k)$-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk2) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries and an algorithm that constructs the order-k Voronoi diagram in O(n log n+nk log k) expected time.

Sampling, Halfspace Range Reporting, and Construction of (<= k)-Levels in Three Dimensions.

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).

/pdf/optimal-deterministic-algorithms-for-2-d-and-3-d-shallow-yxwzuw4pi8.pdf

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).

We consider the dynamic two-dimensional maxima query problem. Let P be a set of n points in the plane. A point is maximal if it is not dominated by any other point in P. We describe two data structures that support the reporting of the t maximal points that dominate a given query point, and allow for insertions and deletions of points in P. In the pointer machine model we present a linear space data structure with O(log n + t) worst case query time and O(log n) worst case update time. This is the first dynamic data structure for the planar maxima dominance query problem that achieves these bounds in the worst case. The data structure also supports the more general query of reporting the maximal points among the points that lie in a given 3-sided orthogonal range unbounded from above in the same complexity. We can support 4-sided queries in O(log2 n+t) worst case time, and O(log2 n) worst case update time, using O(n log n) space, where t is the size of the output. This improves the worst case deletion time of the dynamic rectangular visibility query problem from O(log3 n) to O(log2 n). We adapt the data structure to the RAM model with word size w, where the coordinates of the points are integers in the range U={0,..., 2w-1}. We present a linear space data structure that supports 3-sided range maxima queries in O(log n/log log n + t) worst case time and updates in O(logn/log log n) worst case time. These are the first sublogarithmic worst case bounds for all operations in the RAM model.

/pdf/dynamic-planar-range-maxima-queries-11nuwa02mf.pdf

Dynamic planar range maxima queries

We present I/O-efficient fully persistent B-Trees that support range searches at any version in O(logBn + t/B) I/Os and updates at any version in O(logBn + log2B) amortized I/Os, using space O(m/B) disk blocks. By n we denote the number of elements in the accessed version, by m the total number of updates, by t the size of the query's output, and by B the disk block size. The result improves the previous fully persistent B-Trees of Lanka and Mays by a factor of O(logBm) for the range query complexity and O(logBn) for the update complexity. To achieve the result, we first present a new B-Tree implementation that supports searches and updates in O(logBn) I/Os, using O(n/B) blocks of space. Moreover, every update makes in the worst case a constant number of modifications to the data structure. We make these B-Trees fully persistent using an I/O-efficient method for full persistence that is inspired by the node-splitting method of Driscoll et al. The method we present is interesting in its own right and can be applied to any external memory pointer based data structure with maximum in-degree din bounded by a constant and out-degree bounded by O(B), where every node occupies a constant number of blocks on disk. The I/O-overhead per modification to the ephemeral structure is O(din log2B) amortized I/Os, and the space overhead is O(din/B) amortized blocks. Access to a field of an ephemeral block is supported in O(log2din) worst case I/Os.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973099.51

Fully persistent B-trees

We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(nlogn + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((nlogn + k)loglogn) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Patrascu [SoCG’11] that attains the same time complexity but in expectation.

/pdf/deterministic-rectangle-enclosure-and-offline-dominance-1dl6a8gpqd.pdf

Konstantinos Tsakalidis

Papers

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

Dynamic planar range maxima queries

Fully persistent B-trees

Deterministic rectangle enclosure and offline dominance reporting on the RAM