There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

/pdf/finding-community-structure-in-networks-using-the-4em5so8glh.pdf

Finding community structure in networks using the eigenvectors of matrices

We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

/pdf/approximate-nearest-neighbors-towards-removing-the-curse-of-27lmlz5prr.pdf

Approximate nearest neighbors: towards removing the curse of dimensionality

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Voronoi diagrams—a survey of a fundamental geometric data structure

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R2) or the skip octree (for point data in Rd, with constant d > 2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location, approximate range, and approximate nearest neighbor queries.

/pdf/the-skip-quadtree-a-simple-dynamic-data-structure-for-335viu1y6j.pdf

The skip quadtree: a simple dynamic data structure for multidimensional data

This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data. Unlike previous approaches, our study is not based on the assumption that road networks are planar graphs. Indeed, based on the a number of experiments we have performed on the road networks of the 50 United States and District of Columbia, we provide strong empirical evidence that road networks are quite non-planar. Our approach therefore instead is directed at finding algorithmically-motivated properties of road networks as non-planar geometric graphs, focusing on alternative properties of road networks that can still lead to efficient algorithms for such problems as shortest paths and Voronoi diagrams. In particular, we study road networks as multiscale-dispersed graphs, which is a concept we formalize in terms of disk neighborhood systems. This approach allows us to develop fast algorithms for road networks without making any additional assumptions about the distribution of edge weights. In fact, our algorithms can allow for non-metric weights.

Studying (non-planar) road networks through an algorithmic lens

Dynamic programming solutions to two recurrence equations, used to compute a sequence alignment from a set of matching fragments between two strings, and to predict RNA secondary structure, are considered. These recurrences are defined over a number of points that is quadratic in the input size; however, only a sparse set matters for the result. Efficient algorithms are given for solving these problems, when the cost of a gap in the alignment or a loop in the secondary structure is taken as a convex or concave function of the gap or loop length. The time complexity of our algorithms depends almost linearly on the number of points that need to be considered; when the problems are sparse, this results in a substantial speed-up over known algorithms.

/pdf/sparse-dynamic-programming-ii-convex-and-concave-cost-54db4gzneq.pdf

Sparse dynamic programming II: convex and concave cost functions

We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n1/2 log2n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in timeO(ne) per update. Our algorithm uses a novel construction, theordered 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.

/pdf/dynamic-euclidean-minimum-spanning-trees-and-extrema-of-583u571srv.pdf

Dynamic Euclidean minimum spanning trees and extrema of binary functions

We have described a number of algorithmic tools that have been found useful in the construction of parallel algorithms; among these are prefix computation, ranking, Euler tours, ear decomposition, and matrix calculations. We have also described some of the applications of these tools, and listed many other applications. These algorithms seem likely to be useful not only in their own right, but also as examples of ways to break up other problems into parts suitable for parallel solution.

David Eppstein

Papers

The skip quadtree: a simple dynamic data structure for multidimensional data

Studying (non-planar) road networks through an algorithmic lens

Sparse dynamic programming II: convex and concave cost functions

Dynamic Euclidean minimum spanning trees and extrema of binary functions

Parallel algorithmic techniques for combinatorial computation