Domagoj Matijevic

Goal-directed shortest-path queries using precomputed cluster distances

We demonstrate how Dijkstras algorithm for shortest path queries can be accelerated by using precomputed shortest path distances. Our approach allows a completely flexible tradeoff between query time and space consumption for precomputed distances. In particular, sublinear space is sufficient to give the search a strong “sense of direction”. We evaluate our approach experimentally using large, real-world road networks.

Goal directed shortest path queries using precomputed cluster distances

We demonstrate how Dijkstras algorithm for shortest path queries can be accelerated by using precomputed shortest path distances. Our approach allows a completely flexible tradeoff between query time and space consumption for precomputed distances. In particular, sublinear space is sufficient to give the search a strong “sense of direction”. We evaluate our approach experimentally using large, real-world road networks.

Approximating Energy Efficient Paths in Wireless Multi-hop Networks

Given the positions of n sites in a radio network we consider the problem of finding routes between any pair of sites that minimize energy consumption and do not use more than some constant number k of hops. Known exact algorithms for this problem required Ω(n log n) per query pair (p,q). In this paper we relax the exactness requirement and only compute approximate (1 + e) solutions which allows us to guarantee constant query time using linear space and O(n log n) preprocessing time. The dependence on e is polynomial in 1/e.

Improved Approximations for Guarding 1.5-Dimensional Terrains

We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (J. King, 2006). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.

Energy-aware stage illumination

Consider the following illumination problem: given a stage represented by a line segment L and a set of lightsources represented by a set of points S in the plane, assign powers to the lightsources such that every point on the stage receives a sufficient amount -- lets say one unit -- of light while minimizing the overall power consumption. By assuming that the amount of light arriving from a fixed lightsource decreases rapidly with the distance from the lightsource, this becomes an interesting optimization problem.We propose to reconsider the classical illumination problems as known from computational geometry literature (e.g. [12]) under this light attenuation model. This paper examines the simple problem introduced above and presents different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided.

Energy-Aware Stage Illumination

Consider the following illumination problem: given a stage represented by a line segment L and a set of light sources represented by a set of points S in the plane, assign powers to the light sources such that every point on the stage receives a sufficient amount – e.g. one unit – of light while minimizing the overall power consumption. Under the assumption that the amount of light arriving from a fixed light source decreases rapidly with the distance from the light source, this becomes an interesting optimization problem. We propose to reconsider the classical illumination problems as known from computational geometry literature under this light attenuation model. This paper examines the simple problem introduced above and presents different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided.

Guarding 1.5D terrains with demands

In this paper, we consider the 1.5D terrain guarding problem in which every point on the terrain that is to be covered has an integer demand associated with it. The goal is to find a minimum cardinality set of guards such that each point is guarded by a number of guards satisfying its demand. We present a first constant-factor approximation algorithm for the problem, that is, a (1+1/d min)-approximation algorithm, where d min is a minimum demand. The algorithm is based on solving appropriate subproblems established by a decomposition of the main problem due to linear programming relaxation of the corresponding covering problem.

A Fast Parallel Implementation of a PTAS for Fractional Packing and Covering Linear Programs

We present a parallel implementation of the randomized

Conic nearest neighbor queries and approximate Voronoi diagrams

(1+\varepsilon )