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 prove that it is NP-complete, given a graph G and a parameter h, to determine whether G contains a complete graph Kh as a minor.

Finding Large Clique Minors is Hard

Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and OD maps In these cases the regular shapes of the grid cells allows easy representation of extra information about the regions The main challenge is to find an association that allows a user to find a region in the grid quickly We call the representation of a set of regions as a grid a grid map

https://fstaals.net/publications/visualization/gridmaps2013/gridmaps2013_local.pdf

Improved grid map layout by point set matching

We introduce new dynamic set intersection data structures, which we call 2-3 cuckoo filters and hash tables. These structures differ from the standard cuckoo hash tables and cuckoo filters in that they choose two out of three locations to store each item, instead of one out of two, ensuring that any item in an intersection of two structures will have at least one common location in both structures. We demonstrate the utility of these structures by using them in improved algorithms for listing triangles and answering set intersection queries in internal or external memory. For a graph G of n vertices and m edges, our internal-memory triangle listing algorithm runs in O(m⌈(α(G)log w)/w⌉ + k) expected time, where α(G) is the arboricity of G, w is the number of bits in a machine word, and k is the number of output triangles. Our external-memory algorithm uses O(sort(n,α(G))+ sort(m⌈(α(G)log w)/w⌉) + sort(k)) expected number of I/Os.

https://dl.acm.org/doi/pdf/10.1145/3034786.3056115

2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection

We consider the problem of placing a small number of angle guards inside a simple polygon P so asto provide efficient proofs that any given point is inside P. Each angle guard views an infinite wedge of the plane, and a point can prove membership in P if it is inside the wedges for a set of guards whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, which we call "sculpture garden" problems and for which we provide upper and lower bounds. In particular, we show there is a polygon P such that a "natural" angle-guard vertex placement cannot fully distinguish between pointson the inside and outside of P (even if we place a guard at every vertex of P), which implies that Steiner-point guards are sometimes necessary. More generally, we show that, for any polygon P, there is a set of n+2(h-1) angle guards that solve the sculpture garden problem for P, where h is the number of holes in P (so a simple polygon can be defined with n-2 guards). In addition, we show that, for any orthogonal polygon P, the sculpture garden problem can besolved using n/2 angle guards. We also give an example of a class of simple (non-general-position) polygons that have sculpture garden solutions using O(√n) guards, and we show this bound is optimal to within a constant factor. Finally, while optimizing the number of guards solving a sculpture garden problem for a particular P is of unknown complexity, we show how to find in polynomial time a guard placement whose size is within a factor of 2 of the optimal number for any particular polygon.

/pdf/guard-placement-for-efficient-point-in-polygon-proofs-3zgtx6dqt2.pdf

Guard placement for efficient point-in-polygon proofs

We show how to test the bipartiteness of an intersection graph of n line segments or simple polygons in the plane, or of an intersection graph of balls in d-dimensional Euclidean space, in time O(n log n). More generally, we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. Our algorithms for these problems return either a bipartition of the input or an odd cycle in its intersection graph. We also consider lower bounds for connectivity and k-colorability problems of geometric intersection graphs. For unit balls in d dimensions, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has the same lower bounds as Hopcroft's point-line incidence testing problem; therefore, for these problems, connectivity is unlikely to be solved as efficiently as bipartiteness. For line segments or planar disks, testing k-colorability of intersection graphs for k > 2 is NP-complete.

David Eppstein

Papers

Finding Large Clique Minors is Hard

Improved grid map layout by point set matching

2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection

Guard placement for efficient point-in-polygon proofs

Testing bipartiteness of geometric intersection graphs