The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on highdimensional spaces and present a new randomized algorithm which efficiently computes approximate rnets with respect to Euclidean distance. For any fixed > 0, the approximation factor is 1 + and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSHbased) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on , provided that is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + )-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

No abstract

We study the multistage K-facility reallocation problem on the real line, where we maintain K facility locations over T stages, based on the stage-dependent locations of n agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. K-facility reallocation was introduced by de Keijzer and Wojtczak [10], where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider online K-facility reallocation, where the algorithm becomes aware of agent locations in a stage-by-stage fashion. By exploiting an interesting connection to the classical K-server problem, we present a constant-competitive algorithm for K = 2 facilities. 1 consecutive timesteps. The stability of the solutions is modeled by introducing an additional moving cost (or switching cost), which has a different definition depending on the particular setting. Model and Motivation.In this work, we study the multistage K-facility reallocation problem on the real line, introduced by de Keijzer and Wojtczak [10]. In K-facility reallocation, K facilities are initially located at (x 0 1 , . . . , x 0 K ) on the real line. Facilities are meant to serve n agents for the next T days. At each day, each agent connects to the facility closest to its location and incurs a connection cost equal to this distance. The locations of the agents may change every day, thus we have to move facilities accordingly in order to reduce the connection cost. Naturally, moving a facility is not for free, but comes with the price of the distance that the facility was moved. Our goal is to specify the exact positions of the facilities at each day so that the total connection cost plus the total moving cost is minimized over all T days. In the online version of the problem, the positions of the agents at each stage t are revealed only after determining the locations of the facilities at stage t − 1.For a motivating example, consider a company willing to advertise its products. To this end, it organizes K advertising campaigns at different locations of a large city for the next T days. Based on planned events, weather forecasts, etc., the company estimates a population distribution over the locations of the city for each day. Then, the company decides to compute the best possible campaign reallocation with K campaigns over all days (see also [10] for more examples).de Keijzer and Wojtczak [10] fully characterized the optimal offline and online algorithms for the special case of a single facility and presented a dynamic programming algorithm for K ≥ 1 facilities with running time exponential in K. Despite the practical significance and the interesting theoretical properties of Kfacility reallocation, its computationa...

The Dynamic Facility Location problem is a generalization of the classic Facility Location problem, in which the distance metric between clients and facilities changes over time. Such metrics that develop as a function of time are usually called "evolving metrics", thus Dynamic Facility Location can be alternatively interpreted as a Facility Location problem in evolving metrics. The objective in this time-dependent variant is to balance the trade-off between optimizing the classic objective function and the stability of the solution, which is modeled by charging a switching cost when a client’s assignment changes from one facility to another. In this paper, we study the online variant of Dynamic Facility Location. We present a randomized O(logm+logn)-competitive algorithm, where m is the number of facilities and n is the number of clients. In the first step, our algorithm produces a fractional solution, in each timestep, to the objective of Dynamic Facility Location involving a regularization function. This step is an adaptation of the generic algorithm proposed by Buchbinder et al. in their work "Competitive Analysis via Regularization." Then, our algorithm rounds the fractional solution of this timestep to an integral one with the use of exponential clocks. We complement our result by proving a lower bound of Ω(m) for deterministic algorithms and lower bound of Ω(logm) for randomized algorithms. To the best of our knowledge, these are the first results for the online variant of the Dynamic Facility Location problem.

No abstract