We consider a class of multivariate recurrences frequently arising in the worst case analysis of Davis-Putnam-style exponential time backtracking algorithms for NP-hard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to reduce the problem to that of solving univariate linear recurrences; show how to use quasiconvex programming to determine the weight function yielding the smallest upper bound; and prove that the resulting upper bounds are within a polynomial factor of the true asymptotics of the recurrence. We develop and implement a multiple-gradient descent algorithm for the resulting quasiconvex programs, using a real-number arithmetic package for guaranteed accuracy of the computed worst case time bounds.
The journal version uses the longer title "Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms".
Studies the resilience of distributed computation networks against adversarial and random fault models; shows that, in both models, certain networks can withstand constant fault probabilities and still contain a large subnetwork with similar expansion to the original.
We give a linear time algorithm for pruning a node-weighted tree to maximize the average node weight of the pruned subtree; this problem was previously studied under the less obvious name "The Fractional Prize-Collecting Steiner Tree Problem on Trees".
We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface).
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
This follows on to our previous paper on using graph matching to cover a triangulated polyhedral model with a single triangle strip by extending the algorithms to models with boundaries. We provide two methods: one is based on using an algorithm for the Chinese Postman problem to find a small set of triangles to split in order to find a perfect matching in the dual mesh, while the other augments the model with virtual triangles to remove the boundaries and merges the strips formed by our previous algorithm on this augmented model. We implement the algorithms and report some preliminary experimental results.
The problem is to quickly find, in a set of sites with weights, the site maximizing the product of its weight with its distance from the query point. Our solution combines known results on core-sets with a reduction from the weighted to the unweighted problem that works in any metric space. This leads to fast approximation algorithms for the constrained minimum dilation star problem in any fixed dimension.
The problem is to place as few wedges as possible in the plane such that a desired polygon can be formed as some monotone Boolean combination of the wedges. The motivation is for wireless devices to prove that they are located within a target area by their ability to communicate with a subset of base stations (the wedges). We provide upper and lower bounds on the number of wedges needed for several classes of polygons.
We find efficient constant factor approximation algorithms for hierarchically clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition with approximately minimum total length.
We consider graph drawing algorithms for learning spaces, a type of st-oriented partial cube derived from antimatroids and used to model states of knowledge of students. We show how to draw any st-planar learning space so all internal faces are convex quadrilaterals with the bottom side horizontal and the left side vertical, with one minimal and one maximal vertex. Conversely, every such drawing represents an $st$-planar learning space. We also describe connections between these graphs and arrangements of translates of a quadrant.
We consider drawings of trees which, if the leaf edges were extended to infinite rays, would form partitions of the plane into unbounded convex polygons. For such a drawing the edges may be chosen independently without any possibility of edge crossing. We show how to choose the angles of such drawings to optimize the angular resolution of the drawing.
We show how to choose colors for the vertices of a graph drawing in such a way that all colors are easily distinguishable, but such that adjacent vertices have especially dissimilar colors, by considering the problem as one of embedding the graph into a three-dimensional color space.
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.
We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L1 metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.
Years – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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