arXiv subjects

Laxman Dhulipala

Publications and source records attributed to Laxman Dhulipala.

63 records · Page 4Linked to original sources

arXiv2020

Parallel Batch-Dynamic Graph Connectivity

In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The most well known sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves $O(\log^2 n)$ amortized time per edge insertion or deletion, and $O(\log n / \log\log n)$ time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where $Δ$ is the average batch size of all deletions, our algorithm achieves $O(\log n \log(1 + n / Δ))$ expected amortized work per edge insertion and deletion and $O(\log^3 n)$ depth w.h.p. Our algorithm answers a batch of $k$ connectivity queries in $O(k \log(1 + n/k))$ expected work and $O(\log n)$ depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

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arXiv2020

Parallel Batch-dynamic Trees via Change Propagation

The dynamic trees problem is to maintain a forest subject to edge insertions and deletions while facilitating queries such as connectivity, path weights, and subtree weights. Dynamic trees are a fundamental building block of a large number of graph algorithms. Although traditionally studied in the single-update setting, dynamic algorithms capable of supporting batches of updates are increasingly relevant today due to the emergence of rapidly evolving dynamic datasets. Since processing updates on a single processor is often unrealistic for large batches of updates, designing parallel batch-dynamic algorithms that achieve provably low span is important for many applications. In this work, we design the first work-efficient parallel batch-dynamic algorithm for dynamic trees that is capable of supporting both path queries and subtree queries, as well as a variety of non-local queries. To achieve this, we propose a framework for algorithmically dynamizing static round-synchronous algorithms that allows us to obtain parallel batch-dynamic algorithms with good bounds on their work and span. In our framework, the algorithm designer can apply the technique to any suitably defined static algorithm. We then obtain theoretical guarantees for algorithms in our framework by defining the notion of a computation distance between two executions of the underlying algorithm. Our dynamic trees algorithm is obtained by applying our dynamization framework to the parallel tree contraction algorithm of Miller and Reif, and then performing a novel analysis of the computation distance of this algorithm under batch updates. We show that $k$ updates can be performed in $O(k \log(1+n/k))$ work in expectation, which matches an existing algorithm of Tseng et al. while providing support for a substantially larger number of queries and applications.

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arXiv2020

Near-Optimal Massively Parallel Graph Connectivity

Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is $n^δ$ for some desirably small constant $δ\in (0, 1)$. We present an algorithm that for graphs with diameter $D$ in the wide range $[\log^ε n, n]$, takes $O(\log D)$ rounds to identify the connected components and takes $O(\log \log n)$ rounds for all other graphs. The algorithm is randomized, succeeds with high probability, does not require prior knowledge of $D$, and uses an optimal total space of $O(m)$. We complement this by showing a conditional lower-bound based on the widely believed TwoCycle conjecture that $Ω(\log D)$ rounds are indeed necessary in this setting. Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni et al. [FOCS 2018] who presented an algorithm with $O(\log D \cdot \log \log n)$ round-complexity. Our algorithm improves this result for the whole range of values of $D$ and almost settles the problem due to the conditional lower-bound. Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of the (CRCW) PRAM in asymptotically the same number of rounds.

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arXiv2020

Optimizing Ordered Graph Algorithms with GraphIt

Many graph problems can be solved using ordered parallel graph algorithms that achieve significant speedup over their unordered counterparts by reducing redundant work. This paper introduces a new priority-based extension to GraphIt, a domain-specific language for writing graph applications, to simplify writing high-performance parallel ordered graph algorithms. The extension enables vertices to be processed in a dynamic order while hiding low-level implementation details from the user. We extend the compiler with new program analyses, transformations, and code generation to produce fast implementations of ordered parallel graph algorithms. We also introduce bucket fusion, a new performance optimization that fuses together different rounds of ordered algorithms to reduce synchronization overhead, resulting in $1.2\times$--3$\times$ speedup over the fastest existing ordered algorithm implementations on road networks with large diameters. With the extension, GraphIt achieves up to 3$\times$ speedup on six ordered graph algorithms over state-of-the-art frameworks and hand-optimized implementations (Julienne, Galois, and GAPBS) that support ordered algorithms.

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arXiv2019

Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable

There has been significant recent interest in parallel graph processing due to the need to quickly analyze the large graphs available today. Many graph codes have been designed for distributed memory or external memory. However, today even the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) can fit in the memory of a single commodity multicore server. Nevertheless, most experimental work in the literature report results on much smaller graphs, and the ones for the Hyperlink graph use distributed or external memory. Therefore, it is natural to ask whether we can efficiently solve a broad class of graph problems on this graph in memory. This paper shows that theoretically-efficient parallel graph algorithms can scale to the largest publicly-available graphs using a single machine with a terabyte of RAM, processing them in minutes. We give implementations of theoretically-efficient parallel algorithms for 20 important graph problems. We also present the optimizations and techniques that we used in our implementations, which were crucial in enabling us to process these large graphs quickly. We show that the running times of our implementations outperform existing state-of-the-art implementations on the largest real-world graphs. For many of the problems that we consider, this is the first time they have been solved on graphs at this scale. We have made the implementations developed in this work publicly-available as the Graph-Based Benchmark Suite (GBBS).

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arXiv2019

Parallel Batch-Dynamic Graphs: Algorithms and Lower Bounds

In this paper we study the problem of dynamically maintaining graph properties under batches of edge insertions and deletions in the massively parallel model of computation. In this setting, the graph is stored on a number of machines, each having space strongly sublinear with respect to the number of vertices, that is, $n^ε$ for some constant $0 < ε< 1$. Our goal is to handle batches of updates and queries where the data for each batch fits onto one machine in constant rounds of parallel computation, as well as to reduce the total communication between the machines. This objective corresponds to the gradual buildup of databases over time, while the goal of obtaining constant rounds of communication for problems in the static setting has been elusive for problems as simple as undirected graph connectivity. We give an algorithm for dynamic graph connectivity in this setting with constant communication rounds and communication cost almost linear in terms of the batch size. Our techniques combine a new graph contraction technique, an independent random sample extractor from correlated samples, as well as distributed data structures supporting parallel updates and queries in batches. We also illustrate the power of dynamic algorithms in the MPC model by showing that the batched version of the adaptive connectivity problem is $\mathsf{P}$-complete in the centralized setting, but sub-linear sized batches can be handled in a constant number of rounds. Due to the wide applicability of our approaches, we believe it represents a practically-motivated workaround to the current difficulties in designing more efficient massively parallel static graph algorithms.

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arXiv2019

Massively Parallel Computation via Remote Memory Access

We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms using MapReduce and a distributed hash table service. This extension allows us to give new graph algorithms with much lower round complexities compared to the best known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in $O(1)$ rounds and connectivity/minimum spanning tree in $O(\log\log_{m/n} n)$ rounds both using $O(n^δ)$ space per machine for constant $δ< 1$. In the same memory regime for MPC, the best known algorithms for these problems require polylog $n$ rounds. Our results imply that the 2-Cycle conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.

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arXiv2019

Low-Latency Graph Streaming Using Compressed Purely-Functional Trees

Due to the dynamic nature of real-world graphs, there has been a growing interest in the graph-streaming setting where a continuous stream of graph updates is mixed with arbitrary graph queries. In principle, purely-functional trees are an ideal choice for this setting due as they enable safe parallelism, lightweight snapshots, and strict serializability for queries. However, directly using them for graph processing would lead to significant space overhead and poor cache locality. This paper presents C-trees, a compressed purely-functional search tree data structure that significantly improves on the space usage and locality of purely-functional trees. The key idea is to use a chunking technique over trees in order to store multiple entries per tree-node. We design theoretically-efficient and practical algorithms for performing batch updates to C-trees, and also show that we can store massive dynamic real-world graphs using only a few bytes per edge, thereby achieving space usage close to that of the best static graph processing frameworks. To study the efficiency and applicability of our data structure, we designed Aspen, a graph-streaming framework that extends the interface of Ligra with operations for updating graphs. We show that Aspen is faster than two state-of-the-art graph-streaming systems, Stinger and LLAMA, while requiring less memory, and is competitive in performance with the state-of-the-art static graph frameworks, Galois, GAP, and Ligra+. With Aspen, we are able to efficiently process the largest publicly-available graph with over two hundred billion edges in the graph-streaming setting using a single commodity multicore server with 1TB of memory.

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arXiv2016

Compressing Graphs and Indexes with Recursive Graph Bisection

Graph reordering is a powerful technique to increase the locality of the representations of graphs, which can be helpful in several applications. We study how the technique can be used to improve compression of graphs and inverted indexes. We extend the recent theoretical model of Chierichetti et al. (KDD 2009) for graph compression, and show how it can be employed for compression-friendly reordering of social networks and web graphs and for assigning document identifiers in inverted indexes. We design and implement a novel theoretically sound reordering algorithm that is based on recursive graph bisection. Our experiments show a significant improvement of the compression rate of graph and indexes over existing heuristics. The new method is relatively simple and allows efficient parallel and distributed implementations, which is demonstrated on graphs with billions of vertices and hundreds of billions of edges.

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