We describe an algorithm for finding all maximal cliques in a graph, in time O(dn3d/3) where n is the number of vertices and d is the degeneracy of the graph, a standard measure of its sparsity. This time bound matches the worst-case output size for these parameters. The algorithm modifies the Bron-Kerbosch algorithm for maximal cliques by ordering the vertices by degree in the outer recursive call of the algorithm.
For the journal version, see "Listing all maximal cliques in large sparse real-world graphs in near-optimal time," which combines results from this and a different conference paper.
Tutte's method of spring embeddings allows any triangulated planar graph to be drawn so that the outer face has any pre-specified convex shape, but it may place vertices exponentially close to each other. Alternative graph drawing methods provide polynomial-area straight line drawings but do not allow the outer face shape to be specified. We describe a drawing method that combines both properties: it has polynomial area, and can match any pre-specified shape of the outer face, even a shape in which some of the vertices have 180 degree angles. We apply our results to drawing polygonal schemas for graphs embedded on surfaces of positive genus.
We show how to draw any graph of maximum degree three in three dimensions with 120 degree angles at each vertex or bend, and any graph of maximum degree four in three dimensions with the angles of the diamond lattice at each vertex or bend. In each case there are no crossings and the number of bends per edge is a small constant.
Suppose that P is the intersection of n halfspaces in D dimensions, but that the bounded faces of P are at most d-dimensional, for some d that is much smaller than D. Then in this case we show that the number of vertices of P is O(nd), independent of D. We also investigate related bounds on the number of bounded faces of all dimensions of P, and algorithms for efficiently listing the vertices and bounded faces of P.
We study the recursive partitions of rectangles into sets of rectangles, and partitions of those rectangles into smaller rectangles, to form stylized visualizations of hierarchically subdivided geographic regions. There are several variations of varying difficulty depending on how much of the geographic information from the input we require the output to preserve.
We apply competitive analysis to the problem of deciding online which cell phone tower to change to when a phone moves out of the coverage region of the tower it is connected to. We show that, when the coverage regions have constant ply (at most a constant number of them overlap any point of the plane) it is possible to get within a constant factor of the minimum possible number of handovers that an offline algorithm could achieve.
We investigate greedy routing schemes for social networks, in which participants know categorical information about some other participants and use it to guide message delivery by forwarding messages to neighbors that have more categories in common with the eventual destination. We define the membership dimension of such a scheme to be the maximum number of categories that any individual belongs to, a natural measure of the cognitive load of greedy routing on its participants. And we show that membership dimension is closely related to the small world phenomenon: a social network can be given a category system with polylogarithmic membership dimension that supports greedy routing if, and only if, the network has polylogarithmic diameter.
We extend Lombardi drawing (in which each edge is a circular arc and the edges incident to a vertex must be equally spaced around it) to drawings in which edges are composed of multiple arcs, and we investigate the graphs that can be drawn in this more relaxed framework.
A confluent drawing of a graph is a set of points and curves in the plane with the property that two vertices are adjacent in the graph if and only if the corresponding points can be connected to each other by smooth paths in the drawing. We define a variant of confluent drawing, strict confluent drawing, in which a smooth path between two vertices (if it exists) must be unique. We show that it is NP-complete to test whether such drawings exist, in contrast to unrestricted confluence for which the complexity remains open. Additionally, we show that finding outerplanar drawings (in which the points are on the boundary of a disk and the curves are interior to it) with a fixed cyclic vertex ordering can be performed in polynomial time. We use circle packings to find nice versions of these drawings in which all tracks are represented by piecewise-circular curves.
This paper combines our theoretical results on clique-finding algorithms from ISAAC 2010 with our experimental results on the same algorithms from SEA 2011.
Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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