Super-Fast Distributed Algorithms for Metric Facility Location

This paper presents a distributed O(1)-approximation algorithm, with expected-$O(\log \log n)$ running time, in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-$n$ clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that appeared in ICALP 201

A Super-Fast Distributed Algorithm for Bipartite Metric Facility Location

The \textit{facility location} problem consists of a set of \textit{facilities} $\mathcal{F}$, a set of \textit{clients} $\mathcal{C}$, an \textit{opening cost} $f_i$ associated with each facility $x_i$, and a \textit{connection cost} $D(x_i,y_j)$ between each facility $x_i$ and client $y_j$. The goal is to find a subset of facilities to \textit{open}, and to connect each client to an open facility, so as to minimize the total facility opening costs plus connection costs. This paper presents the first expected-sub-logarithmic-round distributed O(1)-approximation algorithm in the $\mathcal{CONGEST}$ model for the \textit{metric} facility location problem on the complete bipartite network with parts $\mathcal{F}$ and $\mathcal{C}$. Our algorithm has an expected running time of $O((\log \log n)^3)$ rounds, where $n = |\mathcal{F}| + |\mathcal{C}|$. This result can be viewed as a continuation of our recent work (ICALP 2012) in which we presented the first sub-logarithmic-round distributed O(1)-approximation algorithm for metric facility location on a \textit{clique} network. The bipartite setting presents several new challenges not present in the problem on a clique network. We present two new techniques to overcome these challenges. (i) In order to deal with the problem of not being able to choose appropriate probabilities (due to lack of adequate knowledge), we design an algorithm that performs a random walk over a probability space and analyze the progress our algorithm makes as the random walk proceeds. (ii) In order to deal with a problem of quickly disseminating a collection of messages, possibly containing many duplicates, over the bipartite network, we design a probabilistic hashing scheme that delivers all of the messages in expected-$O(\log \log n)$ rounds.Comment: 22 pages. This is the full version of a paper that appeared in DISC 201

Scalable Facility Location for Massive Graphs on Pregel-like Systems

We propose a new scalable algorithm for facility location. Facility location is a classic problem, where the goal is to select a subset of facilities to open, from a set of candidate facilities F , in order to serve a set of clients C. The objective is to minimize the total cost of opening facilities plus the cost of serving each client from the facility it is assigned to. In this work, we are interested in the graph setting, where the cost of serving a client from a facility is represented by the shortest-path distance on the graph. This setting allows to model natural problems arising in the Web and in social media applications. It also allows to leverage the inherent sparsity of such graphs, as the input is much smaller than the full pairwise distances between all vertices. To obtain truly scalable performance, we design a parallel algorithm that operates on clusters of shared-nothing machines. In particular, we target modern Pregel-like architectures, and we implement our algorithm on Apache Giraph. Our solution makes use of a recent result to build sketches for massive graphs, and of a fast parallel algorithm to find maximal independent sets, as building blocks. In so doing, we show how these problems can be solved on a Pregel-like architecture, and we investigate the properties of these algorithms. Extensive experimental results show that our algorithm scales gracefully to graphs with billions of edges, while obtaining values of the objective function that are competitive with a state-of-the-art sequential algorithm

Lessons from the Congested Clique Applied to MapReduce

The main results of this paper are (I) a simulation algorithm which, under quite general constraints, transforms algorithms running on the Congested Clique into algorithms running in the MapReduce model, and (II) a distributed $O(\Delta)$-coloring algorithm running on the Congested Clique which has an expected running time of (i) $O(1)$ rounds, if $\Delta \geq \Theta(\log^4 n)$; and (ii) $O(\log \log n)$ rounds otherwise. Applying the simulation theorem to the Congested-Clique $O(\Delta)$-coloring algorithm yields an $O(1)$-round $O(\Delta)$-coloring algorithm in the MapReduce model. Our simulation algorithm illustrates a natural correspondence between per-node bandwidth in the Congested Clique model and memory per machine in the MapReduce model. In the Congested Clique (and more generally, any network in the $\mathcal{CONGEST}$ model), the major impediment to constructing fast algorithms is the $O(\log n)$ restriction on message sizes. Similarly, in the MapReduce model, the combined restrictions on memory per machine and total system memory have a dominant effect on algorithm design. In showing a fairly general simulation algorithm, we highlight the similarities and differences between these models.Comment: 15 page

Super-Fast 3-Ruling Sets

A $t$-ruling set of a graph $G = (V, E)$ is a vertex-subset $S \subseteq V$ that is independent and satisfies the property that every vertex $v \in V$ is at a distance of at most $t$ from some vertex in $S$. A \textit{maximal independent set (MIS)} is a 1-ruling set. The problem of computing an MIS on a network is a fundamental problem in distributed algorithms and the fastest algorithm for this problem is the $O(\log n)$-round algorithm due to Luby (SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago. Since then the problem has resisted all efforts to yield to a sub-logarithmic algorithm. There has been recent progress on this problem, most importantly an $O(\log \Delta \cdot \sqrt{\log n})$-round algorithm on graphs with $n$ vertices and maximum degree $\Delta$, due to Barenboim et al. (Barenboim, Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS 2012). We approach the MIS problem from a different angle and ask if O(1)-ruling sets can be computed much more efficiently than an MIS? As an answer to this question, we show how to compute a 2-ruling set of an $n$-vertex graph in $O((\log n)^{3/4})$ rounds. We also show that the above result can be improved for special classes of graphs such as graphs with high girth, trees, and graphs of bounded arboricity. Our main technique involves randomized sparsification that rapidly reduces the graph degree while ensuring that every deleted vertex is close to some vertex that remains. This technique may have further applications in other contexts, e.g., in designing sub-logarithmic distributed approximation algorithms. Our results raise intriguing questions about how quickly an MIS (or 1-ruling sets) can be computed, given that 2-ruling sets can be computed in sub-logarithmic rounds