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 study the problem of folding a polyomino P into a polycube Q, allowing faces of Q to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) mu...

/pdf/folding-polyominoes-into-poly-cubes-wgpwaflhrf.pdf

Folding polyominoes into (poly)cubes

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n 2/4+i¾?n for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n 2/4-i¾?n, smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size Onlog O1 n, which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.

/pdf/superpatterns-and-universal-point-sets-4y89k01198.pdf

Superpatterns and Universal Point Sets

The single triangle-strip loop generation algorithm on a triangulated two-manifold presented by Gopi and Eppstein (2004) is based on the guaranteed existence of a perfect matching in its dual graph. However, such a perfect matching is not guaranteed in the dual graph of triangulated manifolds with boundaries. In this paper, we present algorithms that suitably modify the results of the dual graph matching to generate a single strip loop on manifolds with boundaries. Further, the algorithm presented by Gopi and Eppstein (2004) can produce only strip loops, but not linear strips. We present an algorithm that does topological surgery to construct linear strips, with user-specified start and end triangles, on manifolds with or without boundaries. The main contributions of this paper include graph algorithms to handle unmatched triangles, reduction of the number of Steiner vertices introduced to create strip loops, and finally a novel method to generate single linear strips with arbitrary start and end positions

/pdf/single-triangle-strip-and-loop-on-manifolds-with-boundaries-45uswcm024.pdf

Single Triangle Strip and Loop on Manifolds with Boundaries

We give linear-time quasiconvex programming algorithms for finding a Moebius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and brain flat mapping.

Optimal Moebius Transformations for Information Visualization and Meshing

We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2^{n/3}) ~= 1.260^n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2^{3n/8}) ~= 1.297^n. We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4)^{n/3}) ~= 1.890^n and O((27/4+epsilon)^{n/3}) respectively. Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle. Our cycle listing algorithm shows that every degree three graph has O(2^{3n/8}) Hamiltonian cycles; we also exhibit a family of graphs with 2^{n/3} Hamiltonian cycles per graph.

David Eppstein

Papers

Folding polyominoes into (poly)cubes

Superpatterns and Universal Point Sets

Single Triangle Strip and Loop on Manifolds with Boundaries

Optimal Moebius Transformations for Information Visualization and Meshing

The traveling salesman problem for cubic graphs