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 investigate three-dimensional visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering ray-shooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths.

/pdf/visibility-with-a-moving-point-of-view-j7n082h83k.pdf

Visibility with a moving point of view

We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.

Parameterized Complexity of 1-Planarity

We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness >= t.

Separating Geometric Thickness from Book Thickness

Author(s): Eppstein, David | Abstract: We consider the problem of finding pairs of vertex-disjoint paths in a DAG, either connecting two given nodes to a common ancestor, or connecting two given pairs of terminals. It was known how to find a single pair of paths, for either type of input, in polynomial time. We show how to find the k pairs with shortest combined length, in time O(mn + k). We also show how to count all such pairs of paths in O(mn) arithmetic operations. These results can be extended to finding or counting tuples of d disjoint paths, in time O(mn^(d-1 + k). We give further results on finding the subset of the DAG involved in pairs of disjoint paths, and on finding disjoint paths in linear space.

/pdf/finding-common-ancestors-and-disjoint-paths-in-dags-50ab64dgrq.pdf

Finding common ancestors and disjoint paths in DAGs

We solve the problem of one-dimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs. 
We then look at the impartial two-player game, proposed by Ravikumar, where two players take turns making peg moves, and whichever player is left without a move loses. We calculate some simple nim-values and discuss when the game separates into a disjunctive sum of smaller games. In the version where a series of hops can be made in a single move, we show that neither the P-positions nor the N-positions (i.e. wins for the previous or next player) are described by a regular or context-free language.

David Eppstein

Papers

Visibility with a moving point of view

Parameterized Complexity of 1-Planarity

Separating Geometric Thickness from Book Thickness

Finding common ancestors and disjoint paths in DAGs

One-Dimensional Peg Solitaire, and Duotaire