Sliding-window aggregation is a foundational stream processing primitive that efficiently summarizes recent data. The state-of-the-art algorithms for sliding-window aggregation are highly efficient when stream data items are evicted or inserted one at a time, even when some of the insertions occur out-of-order. However, real-world streams are often not only out-of-order but also bursty, causing data items to be evicted or inserted in larger bulks. This paper introduces a new algorithm for sliding-window aggregation with bulk eviction and bulk insertion. For the special case of single insert and evict, our algorithm matches the theoretical complexity of the best previous out-of-order algorithms. For the case of bulk evict, our algorithm improves upon the theoretical complexity of the best previous algorithm for that case and also outperforms it in practice. For the case of bulk insert, there are no prior algorithms, and our algorithm improves upon the naive approach of emulating bulk insert with a loop over single inserts, both in theory and in practice. Overall, this paper makes high-performance algorithms for sliding window aggregation more broadly applicable by efficiently handling the ubiquitous cases of out-of-order data and bursts.

We present new algorithms for$k$-means clustering on a data stream with a focus on providing fast responses to clustering queries. Compared to the state-of-the-art, our algorithms provide substantial improvements in the query time for cluster-center queries while retaining the desirable properties of provably small approximation error and low space usage. Our proposed clustering algorithms systematically reuse the “coresets” (summaries of data) computed for recent queries in answering the current clustering query, a novel technique which we refer to as coreset caching. We also present an algorithm calledOnlineCCthat integrates the coreset caching idea with a simple sequential streaming$k$-means algorithm. In practice,OnlineCCalgorithm can provide constant query time. We present both theoretical analysis and detailed experiments demonstrating the correctness, accuracy, and efficiency of all our proposed clustering algorithms.

Algorithms for dynamically maintaining minimum spanning trees (MSTs) have received much attention in both the parallel and sequential settings. While previous work has given optimal algorithms for dense graphs, all existing parallel batch-dynamic algorithms perform polynomial work per update in the worst case for sparse graphs. In this paper, we present the first work-efficient parallel batch-dynamic algorithm for incremental MST, which can insert l edges in O(l log(1+n/l) work in expectation and O(polylog(n)) span w.h.p. The key ingredient of our algorithm is an algorithm for constructing a compressed path tree of an edge-weighted tree, which is a smaller tree that contains all pairwise heaviest edges between a given set of marked vertices. Using our batch-incremental MST algorithm, we demonstrate a range of applications that become efficiently solvable in parallel in the sliding-window model, such as graph connectivity, approximate MSTs, testing bipartiteness, k-certificates, cycle-freeness, and maintaining sparsifiers.

Sliding-window aggregation derives a user-defined summary of the most-recent portion of a data stream. For in-order streams, each window change can be handled in O (1) time even when the aggregation operator is not invertible. But streaming data often arrive inherently out-of-order, e.g., due to clock drifts and communication delays. For such streams, prior work resorted to latency-prone buffering or spent O (log n ) time for every window change, where n is the instantaneous window size. This paper presents FiBA, a novel real-time sliding window aggregation algorithm that optimally handles streams of varying degrees of out-of-orderness. FiBA is as general as the state-of-the-art and supports variable-sized windows. An insert or evict takes amortized O (log d ) time, where d is the distance of the change to the window's boundary. This means O (1) time for in-order arrivals and nearly O (1) time for slightly out-of-order arrivals, tending to O (log n ) time for the most severely out-of-order arrivals. We also prove a matching lower bound, showing optimality. At its heart, the algorithm combines and extends finger searching, lazy rebalancing, and position-aware partial aggregates. Further, FiBA can answer range queries that aggregate subwindows for window sharing. Finally, our experiments show that FiBA performs well in practice and conforms to the theoretical findings, with significantly higher throughput than O (log n ) algorithms.

Summary The incremental graph connectivity (IGC) problem is to maintain a data structure that can quickly answer whether two given vertices in a graph are connected, while allowing more edges to be added to the graph. IGC is a fundamental problem and can be solved efficiently in the sequential setting using a solution to the classical union‐find problem. However, sequential solutions are not sufficient to handle modern‐day large, rapidly‐changing graphs where edge updates arrive at a very high rate. We present the first shared‐memory parallel data structure for union‐find (equivalently, IGC) that is both provably work‐efficient (ie, performs no more work than the best sequential counterpart) and has polylogarithmic parallel depth. We also present a simpler algorithm with slightly worse theoretical properties, but which is easier to implement and has good practical performance. Our experiments on large graph streams with various degree distributions show that it has good practical performance, capable of processing hundreds of millions of edges per second using a 20‐core machine.

We present methods for k-means clustering on a stream with a focus on providing fast responses to clustering queries. Compared to the current state-of-the-art, our methods provide substantial improvement in the query time for cluster centers while retaining the desirable properties of provably small approximation error and low space usage. Our algorithms rely on a novel idea of "coreset caching" that systematically reuses coresets (summaries of data) computed for recent queries in answering the current clustering query. We present both theoretical analysis and detailed experiments demonstrating their correctness and e ciency.

The single-source shortest path problem (SSSP) with nonnegative edge weights is notoriously difficult to solve efficiently in parallel---it is one of the graph problems said to suffer from the transitive-closure bottleneck. Yet, in practice, the Δ-stepping algorithm of Meyer and Sanders (J. Algorithms, 2003) often works efficiently but has no known theoretical bounds on general graphs. The algorithm takes a sequence of steps, each increasing the radius by a user-specified value Δ. Each step settles the vertices in its annulus but can take Θ(n) substeps, each requiring Θ(m) work (n vertices and m edges). Building on the success of Δ-stepping, this paper describes Radius Stepping, an algorithm with one of the best-known tradeoffs between work and depth bounds for SSSP with nearly-linear (~O(m)) work. The algorithm is a Δ-stepping-like algorithm but uses a variable instead of a fixed-size increase in radii, allowing us to prove a bound on the number of steps. In particular, by using what we define as a vertex k-radius, each step takes at most k+2 substeps. Furthermore, we define a (k, ρ)-graph property and show that if an undirected graph has this property, then the number of steps can be bounded by O(n/ρ log ρ L), for a total of O(kn/ρ log ρ L) substeps, each parallel. We describe how to preprocess a graph to have this property. Altogether, for an arbitrary input graph with n vertices and m edges, Radius Stepping, after preprocessing, takes O((m+nρ)log n) work and $O(n/ρ log n log (ρ L)) depth per source. The preprocessing step takes O(m log n + nρ2) work and O(ρlog ρ) depth, adding no more than O(nρ) edges.

Triangle counting and enumeration has emerged as a basic tool in large-scale network analysis, fueling the development of algorithms that scale to massive graphs. Most of the existing algorithms, however, are designed for the distributed-memory setting or the external-memory setting, and cannot take full advantage of a multicore machine, whose capacity has grown to accommodate even the largest of real-world graphs. This paper describes the design and implementation of simple and fast multicore parallel algorithms for exact, as well as approximate, triangle counting and other triangle computations that scale to billions of nodes and edges. Our algorithms are provably cache-friendly, easy to implement in a language that supports dynamic parallelism, such as Cilk Plus or OpenMP, and do not require parameter tuning. On a 40-core machine with two-way hyper-threading, our parallel exact global and local triangle counting algorithms obtain speedups of 17-50x on a set of real-world and synthetic graphs, and are faster than previous parallel exact triangle counting algorithms. We can compute the exact triangle count of the Yahoo Web graph (over 6 billion edges) in under 1.5 minutes. In addition, for approximate triangle counting, we are able to approximate the count for the Yahoo graph to within 99.6% accuracy in under 10 seconds, and for a given accuracy we are much faster than existing parallel approximate triangle counting implementations.

Stream processing is gaining importance as more data becomes available in the form of continuous streams and companies compete to promptly extract insights from them. In such applications, sliding-window aggregation is a central operator, and incremental aggregation helps avoid the performance penalty of re-aggregating from scratch for each window change. This paper presents Reactive Aggregator (RA), a new framework for incremental sliding-window aggregation. RA is general in that it does not require aggregation functions to be invertible or commutative, and it does not require windows to be FIFO. We implemented RA as a drop-in replacement for the Aggregate operator of a commercial streaming engine. Given m updates on a window of size n , RA has an algorithmic complexity of O ( m + m log ( n/m )), rivaling the best prior algorithms for any m . Furthermore, RA's implementation minimizes overheads from allocation and pointer traversals by using a single flat array.

Combining type theory, language design, and empirical work, we present techniques for computing with large and dynamically changing datasets. Based on lambda calculus, our techniques are suitable for expressing a diverse set of algorithms on large datasets and, via self-adjusting computation, enable computations to respond automatically to changes in their data. To improve the scalability of self-adjusting computation, we present a type system for precise dependency tracking that minimizes the time and space for storing dependency metadata. The type system eliminates an important assumption of prior work that can lead to recording spurious dependencies. We present a type-directed translation algorithm that generates correct self-adjusting programs without relying on this assumption. We then show a probabilistic-chunking technique to further decrease space usage by controlling the fundamental space-time tradeoff in self-adjusting computation. We implement and evaluate these techniques, showing promising results on challenging benchmarks involving large graphs.

We present efficient parallel streaming algorithms for fundamental frequency-based aggregates in both the sliding window and the infinite window settings. In the sliding window setting, we give a parallel algorithm for maintaining a space-bounded block counter (SBBC). Using SBBC, we derive algorithms for basic counting, frequency estimation, and heavy hitters that perform no more work than their best sequential counterparts. In the infinite window setting, we present algorithms for frequency estimation, heavy hitters, and count-min sketch. For both the infinite window and sliding window settings, our parallel algorithms process a "minibatch" of items using linear work and polylog parallel depth. We also prove a lower bound showing that the work of the parallel algorithm is optimal in the case of heavy hitters and frequency estimation. To our knowledge, these are the first parallel algorithms for these problems that are provably work efficient and have low depth.

The number of triangles in a graph is a fundamental metric widely used in social network analysis, link classification and recommendation, and more. In these applications, modern graphs of interest tend to both large and dynamic. This paper presents the design and implementation of a fast parallel algorithm for estimating the number of triangles in a massive undirected graph whose edges arrive as a stream. Our algorithm is designed for shared-memory multicore machines and can make efficient use of parallelism and the memory hierarchy. We provide theoretical guarantees on performance and accuracy, and our experiments on real-world datasets show accurate results and substantial speedups compared to an optimized sequential implementation.

This paper presents a new space-efficient algorithm for counting and sampling triangles--and more generally, constant-sized cliques--in a massive graph whose edges arrive as a stream. Compared to prior work, our algorithm yields significant improvements in the space and time complexity for these fundamental problems. Our algorithm is simple to implement and has very good practical performance on large graphs.

This paper presents parallel algorithms for embedding an arbitrary n-point metric space into a distribution of dominating trees with O(log n) expected stretch. Such embedding has proved useful in the design of many approximation algorithms in the sequential setting. We give a parallel algorithm that runs in O(n2 log n) work and O(log2 n) depth---these bounds are independent of Δ = (maxx,y d(x,y))/(minx≠ y d(x,y)), the ratio of the largest to smallest distance. Moreover, when Δ is exponentially bounded (Δ ≤ 2O(n)), our algorithm can be improved to O(n2) work and O(log2 n) depth. Using these results, we give an RNC O(log k)-approximation algorithm for k-median and an RNC O(log n)-approximation for buy-at-bulk network design. The k-median algorithm is the first RNC algorithm with non-trivial guarantees for arbitrary values of k, and the buy-at-bulk result is the first parallel algorithm for the problem.

This paper presents the design, analysis, and implementation of parallel and sequential I/O-efficient algorithms for set cover, tying together the line of work on parallel set cover and the line of work on efficient set cover algorithms for large, disk-resident instances.

This paper studies the problem of finding a (1+ε)-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. At FOCS'11, Jain and Yao gave an NC algorithm that requires O(t 1/ε13 log13 m log n) iterations on input n constraint matrices of dimension m-by-m, where each iteration performs at least Ω(mω) work since it involves computing the spectral decomposition. We present a simpler NC parallel algorithm that on input with n constraint matrices, requires O(1/ε4 log4 n log(1/ε)) iterations, each of which involves only simple matrix operations and computing the trace of the product of a matrix exponential and a positive semidefinite matrix. Further, given a positive SDP in a factorized form, the total work of our algorithm is nearly-linear in the number of non-zero entries in the factorization. Our algorithm can be viewed as a generalization of Young's algorithm and analysis techniques for positive linear programs (Young, FOCS'01) to the semidefinite programming setting.

This announcement describes the problem based benchmark suite (PBBS). PBBS is a set of benchmarks designed for comparing parallel algorithmic approaches, parallel programming language styles, and machine architectures across a broad set of problems. Each benchmark is defined concretely in terms of a problem specification and a set of input distributions. No requirements are made in terms of algorithmic approach, programming language, or machine architecture. The goal of the benchmarks is not only to compare runtimes, but also to be able to compare code and other aspects of an implementation (e.g., portability, robustness, determinism, and generality). As such the code for an implementation of a benchmark is as important as its runtime, and the public PBBS repository will include both code and performance results.

This paper presents the design and analysis of a near linear-work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD n-by-n matrix A with m non-zero entries and a vector b, our algorithm computes a vector x such that Ax - A+b ≤ ε • A+b in O(m logO(1) n log 1/ε) work and O(m1/3+θ log 1/ε) depth for any fixed θ > 0.

We present parallel greedy approximation algorithms for set cover and related problems. These algorithms build on an algorithm for solving a graph problem we formulate and study called Maximal Nearly Independent Set (MaNIS)---a graph abstraction of a key component in existing work on parallel set cover.

Many modern systems exploit data redundancy to improve efficiency. These systems split data into chunks, generate identifiers for each of them, and compare the identifiers among other data items to identify duplicate chunks. As a result, chunk size becomes a critical parameter for the efficiency of these systems: it trades potentially improved similarity detection (smaller chunks) with increased overhead to represent more chunks. Unfortunately, the similarity between files increases unpredictably with smaller chunk sizes, even for data of the same type. Existing systems often pick one chunk size that is "good enough'' for many cases because they lack efficient techniques to determine the benefits at other chunk sizes. This paper addresses this deficiency via two contributions: (1) we present multi-resolution (MR) handprinting, an application-independent technique that efficiently estimates similarity between data items at different chunk sizes using a compact, multi-size representation of the data; (2) we then evaluate the application of MR handprints to workloads from peer-to-peer, file transfer, and storage systems, demonstrating that the chunk size selection enabled by MR handprints can lead to real improvements over using a fixed chunk size in these systems.

Self-adjusting computation provides an evaluation model where computations can respond automatically to modifications to their data by using a mechanism for propagating modifications through the computation. Current approaches to self-adjusting computation guarantee correctness by recording dependencies in a trace at the granularity of individual memory operations. Tracing at the granularity of memory operations, however, has some limitations: it can be asymptotically inefficient (\eg, compared to optimal solutions) because it cannot take advantage of problem-specific structure, it requires keeping a large computation trace (often proportional to the runtime of the program on the current input), and it introduces moderately large constant factors in practice.

Memory bandwidth is a major limiting factor in the scalability of parallel iterative algorithms that rely on sparse matrix-vector multiplication (SpMV). This paper introduces Hierarchical Diagonal Blocking (HDB), an approach which we believe captures many of the existing optimization techniques for SpMV in a common representation. Using this representation in conjuction with precision-reduction techniques, we develop and evaluate high-performance SpMV kernels. We also study the implications of using our SpMV kernels in a complete iterative solver. Our method of choice is a Combinatorial Multigrid solver that can fully utilize our fastest reduced-precision SpMV kernel without sacrificing the quality of the solution. We provide extensive empirical evaluation of the effectiveness of the approach on a variety of benchmark matrices, demonstrating substantial speedups on all matrices considered.

This paper presents the design and analysis of parallel approximation algorithms for facility-location problems, including NC and RNC algorithms for (metric) facility location, k-center, k-median, and k-means. These problems have received considerable attention during the past decades from the approximation algorithms community, which primarily concentrates on improving the approximation guarantees. In this paper, we ask: Is it possible to parallelize some of the beautiful results from the sequential setting?.Our starting point is a small, but diverse, subset of results in approximation algorithms for facility-location problems, with a primary goal of developing techniques for devising their efficient parallel counterparts. We focus on giving algorithms with low depth, near work efficiency (compared to the sequential versions), and low cache complexity.

Recent work on adaptive functional programming (AFP) developed techniques for writing programs that can respond to modifications to their data by performing change propagation . To achieve this, executions of programs are represented with dynamic dependence graphs (DDGs) that record data dependences and control dependences in a way that a change-propagation algorithm can update the computation as if the program were from scratch, by re-executing only the parts of the computation affected by the changes. Since change-propagation only re-executes parts of the computation, it can respond to certain incremental modifications asymptotically faster than recomputing from scratch, potentially offering significant speedups. Such asymptotic speedups, however, are rare: for many computations and modifications, change propagation is no faster than recomputing from scratch. In this article, we realize a duality between dynamic dependence graphs and memoization, and combine them to give a change-propagation algorithm that can dramatically increase computation reuse. The key idea is to use DDGs to identify and re-execute the parts of the computation that are affected by modifications, while using memoization to identify the parts of the computation that remain unaffected by the changes. We refer to this approach as self-adjusting computation. Since DDGs are imperative, but (traditional) memoization requires purely functional computation, reusing computation correctly via memoization becomes a challenge. We overcome this challenge with a technique for remembering and reusing not just the results of function calls (as in conventional memoization), but their executions represented with DDGs. We show that the proposed approach is realistic by describing a library for self-adjusting computation, presenting efficient algorithms for realizing the library, and describing and evaluating an implementation. Our experimental evaluation with a variety of applications, ranging from simple list primitives to more sophisticated computational geometry algorithms, shows that the approach is effective in practice: compared to recomputing from-scratch; self-adjusting programs respond to small modifications to their data orders of magnitude faster.

In many optimization problems, a solution can be viewed as ascribing a ``cost\'\' to each client, and the goal is to optimize some aggregation of the per-client costs. We often optimize some $L_p$-norm (or some other symmetric convex function or norm) of the vector of costs---though different applications may suggest different norms to use. Ideally, we could obtain a solution that optimizes several norms simultaneously. In this paper, we examine approximation algorithms that simultaneously perform well on all norms, or on all $L_p$ norms. A natural problem in this framework is the $L_p$ Set Cover problem, which generalizes \textsc{Set Cover} and \textsc{Min-Sum Set Cover}. We show that the greedy algorithm \emph{simultaneously gives a $(p + \ln p + O(1))$-approximation for all $p$, and show that this approximation ratio is optimal up to constants} under reasonable complexity-theoretic assumptions. We additionally show how to use our analysis techniques to give similar results for the more general \emph{submodular set cover}, and prove some results for the so-called \emph{pipelined set cover} problem. We then go on to examine approximation algorithms in the ``all-norms\'\' and the ``all-$L_p$-norms\'\' frameworks more broadly, and present algorithms and structural results for other problems such as $k$-facility-location, TSP, and average flow-time minimization, extending and unifying previously known results.

No abstract available.

Dependence graphs and memoization can be used to efficiently update the output of a program as the input changes dynamically. Recent work has studied techniques for combining these approaches to effectively dynamize a wide range of applications. Toward this end various theoretical results were given. In this paper we describe the implementation of a library based on these ideas, and present experimental results on the efficiency of this library on a variety of applications. The results of the experiments indicate that the approach is effective in practice, often requiring orders of magnitude less time than recomputing the output from scratch. We believe this is the first experimental evidence that incremental computation of any type is effective in practice for a reasonably broad set of applications.