Parallel Wavelet Tree Construction

We present parallel algorithms for wavelet tree construction with polylogarithmic depth, improving upon the linear depth of the recent parallel algorithms by Fuentes-Sepulveda et al. We experimentally show on a 40-core machine with two-way hyper-threading that we outperform the existing parallel algorithms by 1.3--5.6x and achieve up to 27x speedup over the sequential algorithm on a variety of real-world and artificial inputs. Our algorithms show good scalability with increasing thread count, input size and alphabet size. We also discuss extensions to variants of the standard wavelet tree.Comment: This is a longer version of the paper that appears in the Proceedings of the IEEE Data Compression Conference, 201

Connected Spatial Networks over Random Points and a Route-Length Statistic

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fast Arrays: Atomic Arrays with Constant Time Initialization

Some algorithms require a large array, but only operate on a small fraction of its indices. Examples include adjacency matrices for sparse graphs, hash tables, and van Emde Boas trees. For such algorithms, array initialization can be the most time-consuming operation. Fast arrays were invented to avoid this costly initialization. A fast array is a software implementation of an array, such that the entire array can be initialized in just constant time. While algorithms for sequential fast arrays have been known for a long time, to the best of our knowledge, there are no previous algorithms for concurrent fast arrays. We present the first such algorithms in this paper. Our first algorithm is linearizable and wait-free, uses only linear space, and supports all operations - initialize, read, and write - in constant time. Our second algorithm enhances the first to additionally support all the read-modify-write operations available in hardware (such as compare-and-swap) in constant time

Parallel Index-Based Structural Graph Clustering and Its Approximation

SCAN (Structural Clustering Algorithm for Networks) is a well-studied, widely used graph clustering algorithm. For large graphs, however, sequential SCAN variants are prohibitively slow, and parallel SCAN variants do not effectively share work among queries with different SCAN parameter settings. Since users of SCAN often explore many parameter settings to find good clusterings, it is worthwhile to precompute an index that speeds up queries. This paper presents a practical and provably efficient parallel index-based SCAN algorithm based on GS*-Index, a recent sequential algorithm. Our parallel algorithm improves upon the asymptotic work of the sequential algorithm by using integer sorting. It is also highly parallel, achieving logarithmic span (parallel time) for both index construction and clustering queries. Furthermore, we apply locality-sensitive hashing (LSH) to design a novel approximate SCAN algorithm and prove guarantees for its clustering behavior. We present an experimental evaluation of our algorithms on large real-world graphs. On a 48-core machine with two-way hyper-threading, our parallel index construction achieves 50--151$\times$ speedup over the construction of GS*-Index. In fact, even on a single thread, our index construction algorithm is faster than GS*-Index. Our parallel index query implementation achieves 5--32$\times$ speedup over GS*-Index queries across a range of SCAN parameter values, and our implementation is always faster than ppSCAN, a state-of-the-art parallel SCAN algorithm. Moreover, our experiments show that applying LSH results in faster index construction while maintaining good clustering quality

Parallel Algorithms for Hierarchical Nucleus Decomposition

Nucleus decompositions have been shown to be a useful tool for finding dense subgraphs. The coreness value of a clique represents its density based on the number of other cliques it is adjacent to. One useful output of nucleus decomposition is to generate a hierarchy among dense subgraphs at different resolutions. However, existing parallel algorithms for nucleus decomposition do not generate this hierarchy, and only compute the coreness values. This paper presents a scalable parallel algorithm for hierarchy construction, with practical optimizations, such as interleaving the coreness computation with hierarchy construction and using a concurrent union-find data structure in an innovative way to generate the hierarchy. We also introduce a parallel approximation algorithm for nucleus decomposition, which achieves much lower span in theory and better performance in practice. We prove strong theoretical bounds on the work and span (parallel time) of our algorithms. On a 30-core machine with two-way hyper-threading on real-world graphs, our parallel hierarchy construction algorithm achieves up to a 58.84x speedup over the state-of-the-art sequential hierarchy construction algorithm by Sariyuce et al. and up to a 30.96x self-relative parallel speedup. On the same machine, our approximation algorithm achieves a 3.3x speedup over our exact algorithm, while generating coreness estimates with a multiplicative error of 1.33x on average

ConnectIt: A Framework for Static and Incremental Parallel Graph Connectivity Algorithms

Connected components is a fundamental kernel in graph applications due to its usefulness in measuring how well-connected a graph is, as well as its use as subroutines in many other graph algorithms. The fastest existing parallel multicore algorithms for connectivity are based on some form of edge sampling and/or linking and compressing trees. However, many combinations of these design choices have been left unexplored. In this paper, we design the ConnectIt framework, which provides different sampling strategies as well as various tree linking and compression schemes. ConnectIt enables us to obtain several hundred new variants of connectivity algorithms, most of which extend to computing spanning forest. In addition to static graphs, we also extend ConnectIt to support mixes of insertions and connectivity queries in the concurrent setting. We present an experimental evaluation of ConnectIt on a 72-core machine, which we believe is the most comprehensive evaluation of parallel connectivity algorithms to date. Compared to a collection of state-of-the-art static multicore algorithms, we obtain an average speedup of 37.4x (2.36x average speedup over the fastest existing implementation for each graph). Using ConnectIt, we are able to compute connectivity on the largest publicly-available graph (with over 3.5 billion vertices and 128 billion edges) in under 10 seconds using a 72-core machine, providing a 3.1x speedup over the fastest existing connectivity result for this graph, in any computational setting. For our incremental algorithms, we show that our algorithms can ingest graph updates at up to several billion edges per second. Finally, to guide the user in selecting the best variants in ConnectIt for different situations, we provide a detailed analysis of the different strategies in terms of their work and locality

Parallel Five-Cycle Counting Algorithms

Counting the frequency of subgraphs in large networks is a classic research question that reveals the underlying substructures of these networks for important applications. However, subgraph counting is a challenging problem, even for subgraph sizes as small as five, due to the combinatorial explosion in the number of possible occurrences. This paper focuses on the five-cycle, which is an important special case of five-vertex subgraph counting and one of the most difficult to count efficiently. We design two new parallel five-cycle counting algorithms and prove that they are work-efficient and achieve polylogarithmic span. Both algorithms are based on computing low out-degree orientations, which enables the efficient computation of directed two-paths and three-paths, and the algorithms differ in the ways in which they use this orientation to eliminate double-counting. We develop fast multicore implementations of the algorithms and propose a work scheduling optimization to improve their performance. Our experiments on a variety of real-world graphs using a 36-core machine with two-way hyper-threading show that our algorithms achieves 10-46x self-relative speed-up, outperform our serial benchmarks by 10-32x, and outperform the previous state-of-the-art serial algorithm by up to 818x

PECANN: Parallel Efficient Clustering with Graph-Based Approximate Nearest Neighbor Search

This paper studies density-based clustering of point sets. These methods use dense regions of points to detect clusters of arbitrary shapes. In particular, we study variants of density peaks clustering, a popular type of algorithm that has been shown to work well in practice. Our goal is to cluster large high-dimensional datasets, which are prevalent in practice. Prior solutions are either sequential, and cannot scale to large data, or are specialized for low-dimensional data. This paper unifies the different variants of density peaks clustering into a single framework, PECANN, by abstracting out several key steps common to this class of algorithms. One such key step is to find nearest neighbors that satisfy a predicate function, and one of the main contributions of this paper is an efficient way to do this predicate search using graph-based approximate nearest neighbor search (ANNS). To provide ample parallelism, we propose a doubling search technique that enables points to find an approximate nearest neighbor satisfying the predicate in a small number of rounds. Our technique can be applied to many existing graph-based ANNS algorithms, which can all be plugged into PECANN. We implement five clustering algorithms with PECANN and evaluate them on synthetic and real-world datasets with up to 1.28 million points and up to 1024 dimensions on a 30-core machine with two-way hyper-threading. Compared to the state-of-the-art FASTDP algorithm for high-dimensional density peaks clustering, which is sequential, our best algorithm is 45x-734x faster while achieving competitive ARI scores. Compared to the state-of-the-art parallel DPC-based algorithm, which is optimized for low dimensions, we show that PECANN is two orders of magnitude faster. As far as we know, our work is the first to evaluate DPC variants on large high-dimensional real-world image and text embedding datasets