Showing papers by "Jeff Erickson published in 2006"

Minimum-cost coverage of point sets by disks

Abstract: We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (tj) and radii (rj) that cover a given set of demand points Y∈R2 at the smallest possible cost. We consider cost functions of the form Ejf(rj), where f(r)=rα is the cost of transmission to radius r. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers tj, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points tj on a given line in order to cover demand points Y∈R2; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in R2 and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

Minimum-Cost Coverage of Point Sets by Disks

Abstract: We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

An h-adaptive spacetime-discontinuous Galerkin method for linear elastodynamics

Abstract: We present an h-adaptive version of the spacetime-discontinuous Galerkin (SDG) finite element method for linearized elastodynamics (Abedi et al., 2006). The adaptive version inherits key properties of the basic SDG formulation, including element-wise balance of linear and angular momentum, complexity that is linear in the number of elements and oscillationfree shock capturing. Unstructured spacetime grids allow simultaneous adaptation in space and time. A localized patch-by-patch solution process limits the cost of reanalysis when the error indicator calls for more refinement. Numerical examples demonstrate the method’s performance and shock-capturing capabilities.

Tightening non-simple paths and cycles on surfaces

Abstract: We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity n, genus g ≥ 2, and no boundary, we construct in O(n2 log n) time a tight octagonal decomposition of the surface---a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in O(gnk) time, or the shortest cycle homotopic to a given cycle of complexity k in O(gnk log(nk)) time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdiere and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.

Necklaces, convolutions, and X + Y

Abstract: We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the lp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p=2, and p=∞. For p=2, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and (median,+) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n2) time, whereas the obvious algorithms for these problems run in Θ (n2) time.

Splitting (complicated) surfaces is hard

Abstract: Let M be an orientable combinatorial surface without boundary. A cycle on M is splitting if it has no self-intersections and it partitions M into two components, neither of which is homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus. Specifically, we describe an algorithm to compute the shortest splitting cycle in gO(g)n log n time.

On the Least Median Square Problem

Abstract: We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in ${\Bbb R}^d$ and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding whether n given points in ${\Bbb R}^d$ are affinely non-degenerate requires Ω(nd) time.

Scheduling and admission control

Abstract: We present algorithms and hardness results for three resource allocation problems. The first is an abstract admission control problem where the system receives a series of requests and wants to satisfy as many as possible, but has bounded resources. Algorithms can have performance guarantees for this problem with respect to either acceptances or rejections. These types of guarantees are incomparable and algorithms having different types of guarantee can have nearly opposite behavior. We give two procedures for combining one algorithm of each type into a single algorithm having both types of guarantee simultaneously. The second problem we consider is scheduling with rejections, a combination of scheduling and admission control. For this problem, each job comes with a rejection cost in addition to the standard scheduling parameters. The system schedules a subset of the jobs and schedule quality is total flow plus the cost of rejected jobs. We give lower bounds on the competitive ratio that can be achieved by any deterministic algorithm. We also give an optimal offline algorithm for unit-length jobs with arbitrary rejection costs and two 2-competitive online algorithms for unit-length jobs. Finally, we show that the offline problem is NP-hard even when each job's rejection cost is proportional to its processing time. Our third problem is power-aware scheduling, where the processor can run at different speeds, with its energy consumption depending on the speeds selected. If schedule quality is measured with makespan, we give linear-time algorithms to compute all non-dominated solutions for the general uniprocessor problem and for the multiprocessor problem when every job requires the same amount of work. We also show that the multiprocessor problem becomes NP-hard when jobs can require different amounts of work. If schedule quality is measured with total flow time, we show that finding the optimal schedule for a particular energy budget is impossible. We do, however, give an arbitrarily-good approximation for scheduling equal-work jobs on a multiprocessor.