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

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with triangular faces may not be unfoldable no matter how they are cut.

Ununfoldable polyhedra with convex faces

We describe efficient PRAM algorithms for constructing unbalanced quadtrees, balanced quadtrees, and quadtree-based finite element meshes. Our algorithms take time O(log n) for point set input and O(log n log k) time for planar straight-line graphs, using O(n+k/log n) processors, where n measures input size and k output size.

/pdf/parallel-construction-of-quadtrees-and-quality-1076kjfa27.pdf

Parallel construction of quadtrees and quality triangulations

We show that, for any n-vertex graph G and integer parameter k, there are at most 34k-n4n-3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time O(34k-n4n-3k). These bounds are tight when n/4 ≤ k ≤ n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 34/3/4)n) ≅ 2.4150n, improving a previous O((1 + 31/3)n) ≅ 2.4422n algorithm of Lawler (1976).

/pdf/small-maximal-independent-sets-and-faster-exact-graph-4jj666rovh.pdf

Small Maximal Independent Sets and Faster Exact Graph Coloring

In this paper, we study the straggler identification problem, in which an algorithm must determine the identities of the remaining members of a set after it has had a large number of insertion and deletion operations performed on it, and now has relatively few remaining members. The goal is to do this in o(n) space, where n is the total number of identities. Straggler identification has applications, for example, in determining the unacknowledged packets in a high-bandwidth multicast data stream. We provide a deterministic solution to the straggler identification problem that uses only O(d log n) bits, based on a novel application of Newton's identities for symmetric polynomials. This solution can identify any subset of d stragglers from a set of n O(log n)-bit identifiers, assuming that there are no false deletions of identities not already in the set. Indeed, we give a lower bound argument that shows that any small-space deterministic solution to the straggler identification problem cannot be guaranteed to handle false deletions. Nevertheless, we provide a simple randomized solution, using O(d log n log (1/∈)) bits that can maintain a multiset and solve the straggler identification problem, tolerating false deletions, where ∈ > 0 is a user-defined parameter bounding the probability of an incorrect response. This randomized solution is based on a new type of Bloom filter, which we call the invertible Bloom filter.

Straggler Identification in Round-Trip Data Streams via Newton's Identities and Invertible Bloom Filters

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. These layouts are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whe...

/pdf/area-universal-and-constrained-rectangular-layouts-5bac8fx8mr.pdf

David Eppstein

Papers

Ununfoldable polyhedra with convex faces

Parallel construction of quadtrees and quality triangulations

Small Maximal Independent Sets and Faster Exact Graph Coloring

Straggler Identification in Round-Trip Data Streams via Newton's Identities and Invertible Bloom Filters

Area-universal and constrained rectangular layouts ∗