Facility location problems in the constant work-space read-only memory model

TL;DR

This paper introduces space-efficient algorithms for fundamental facility location problems in a read-only memory model with constant work-space, achieving sub-quadratic time solutions for large data and small devices.

Contribution

It presents the first sub-quadratic algorithms for key facility location problems in the constant-work-space model, extending prune-and-search techniques to this setting.

Findings

Algorithms for Euclidean 1-center, weighted 1-center, and 2-center in O(n poly(log n)) time.

First sub-quadratic solutions for these problems in the constant-work-space model.

Linear time algorithms for centroid and weighted median in the same model.

Abstract

Facility location problems are captivating both from theoretical and practical point of view. In this paper, we study some fundamental facility location problems from the space-efficient perspective. Here the input is considered to be given in a read-only memory and only constant amount of work-space is available during the computation. This {\em constant-work-space model} is well-motivated for handling big-data as well as for computing in smart portable devices with small amount of extra-space. First, we propose a strategy to implement prune-and-search in this model. As a warm up, we illustrate this technique for finding the Euclidean 1-center constrained on a line for a set of points in $\IR^2$. This method works even if the input is given in a sequential access read-only memory. Using this we show how to compute (i) the Euclidean 1-center of a set of points in $\IR^2$, and (ii) the weighted 1-center and weighted 2-center of a tree network. The running time of all these algorithms are $O(n~poly(\log n))$. While the result of (i) gives a positive answer to an open question asked by Asano, Mulzer, Rote and Wang in 2011, the technique used can be applied to other problems which admit solutions by prune-and-search paradigm. For example, we can apply the technique to solve two and three dimensional linear programming in $O(n~poly(\log n))$ time in this model. To the best of our knowledge, these are the first sub-quadratic time algorithms for all the above mentioned problems in the constant-work-space model. We also present optimal linear time algorithms for finding the centroid and weighted median of a tree in this model.