We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC.
For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.
In the k-arc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least k-arc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)-approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. [GGTW09]. Similar to the 2-approximation algorithm for this problem [FJ81], our algorithm simply takes the union of a k in-arborescence and a k out-arborescence. The main difference is in the sele…
In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random $d$-in/$d$-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows …
This paper investigates parallel random sampling from a potentially-unending data stream whose elements are revealed in a series of element sequences (minibatches). While sampling from a stream was extensively studied sequentially, not much has been explored in the parallel context, with prior parallel random-sampling algorithms focusing on the static batch model. We present parallel algorithms for minibatch-stream sampling in two settings: (1) sliding window, which draws samples from a prespecified number of most-recently observed elements, and (2) infinite window, which draws samples from al…
A deterministic BSP algorithm for constructing the suffix array of a given string is presented, based on a technique which we call accelerated sampling. It runs in optimal O(n/p) local computation and communication, and requires a near optimal O(log log p) synchronisation steps. The algorithm provides an improvement over the synchronisation costs of existing algorithms, and reinforces the importance of the sampling technique.
We consider the problem of sampling $n$ numbers from the range $\{1,\ldots,N\}$ without replacement on modern architectures. The main result is a simple divide-and-conquer scheme that makes sequential algorithms more cache efficient and leads to a parallel algorithm running in expected time $\mathcal{O}(n/p+\log p)$ on $p$ processors, i.e., scales to massively parallel machines even for moderate values of $n$. The amount of communication between the processors is very small (at most $\mathcal{O}(\log p)$) and independent of the sample size. We also discuss modifications needed for load balanci…
Data structures for efficient sampling from a set of weighted items are an important building block of many applications. However, few parallel solutions are known. We close many of these gaps both for shared-memory and distributed-memory machines. We give efficient, fast, and practicable parallel algorithms for building data structures that support sampling single items (alias tables, compressed data structures). This also yields a simplified and more space-efficient sequential algorithm for alias table construction. Our approaches to sampling $k$ out of $n$ items with/without replacement and…
The uniform sampling of simple graphs matching a prescribed degree sequence is an important tool in network science, e.g. to construct graph generators or null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice. Given an arbitrary simple graph with the required degree sequence, ES-MC carries out a large number of small changes, called edge switches, to eventually obtain a uniform sample. In practice, reasonably short runs efficiently yield approximate uniform samples. In this work, we study the problem of executing edge switches in parallel. We discuss parallelizations of…
\emph{Sampling} constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the \emph{Markov Chain Monte Carlo} method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the \emph{Glauber dynamics}, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficie…
Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an $n\times n$ SDDM matrix ${\bf \mathbf{M}}$, and a constant $-1 \leq p \leq 1$, our algorithm gives efficient access to a sparse $n\times n$ linear operator $\tilde{\mathbf{C}}$ such that $${\mathbf{M}}^{p} \approx \tilde{\mathbf{C}} \tilde{\mathbf{C}}^\top.$$ The solution is based on factoring ${\bf \mathbf{M}}$ into a product of simple and sparse matrices using squaring and spectral sparsificatio…
The k-center problem is one of several classic NP-hard clustering questions. For contemporary massive data sets, RAM-based algorithms become impractical. And although there exist good sequential algorithms for k-center, they are not easily parallelizable. In this paper, we design and implement parallel approximation algorithms for this problem. We observe that Gonzalez's greedy algorithm can be efficiently parallelized in several MapReduce rounds; in practice, we find that two rounds are sufficient, leading to a 4-approximation. We contrast this with an existing parallel algorithm for k-center…
We examine the problem of almost-uniform sampling proper $q$-colorings of a graph whose maximum degree is $Δ$. A famous result, discovered independently by Jerrum(1995) and Salas and Sokal(1997), is that, assuming $q > (2+δ) Δ$, the Glauber dynamics (a.k.a. single-site dynamics) for this problem has mixing time $O(n \log n)$, where $n$ is the number of vertices, and thus provides a nearly linear time sampling algorithm for this problem. A natural question is the extent to which this algorithm can be parallelized. Previous work Feng, Sun and Yin [PODC'17] has shown that a $O(Δ\log n)$ time para…
Betweenness centrality is one of the most popular vertex centrality measures in network analysis. Hence, many (sequential and parallel) algorithms to compute or approximate betweenness have been devised. Recent algorithmic advances have made it possible to approximate betweenness very efficiently on shared-memory architectures. Yet, the best shared-memory algorithms can still take hours of running time for large graphs, especially for graphs with a high diameter or when a small relative error is required. In this work, we present an MPI-based generalization of the state-of-the-art shared-memor…
