Low depth cache-oblivious algorithms
13 Jun 2010-pp 189-199
TL;DR: This paper describes several cache-oblivious algorithms with optimal work, polylogarithmic depth, and sequential cache complexities that match the best sequential algorithms, including the first such algorithms for sorting and for sparse-matrix vector multiply on matrices with good vertex separators.
Abstract: In this paper we explore a simple and general approach for developing parallel algorithms that lead to good cache complexity on parallel machines with private or shared caches. The approach is to design nested-parallel algorithms that have low depth (span, critical path length) and for which the natural sequential evaluation order has low cache complexity in the cache-oblivious model. We describe several cache-oblivious algorithms with optimal work, polylogarithmic depth, and sequential cache complexities that match the best sequential algorithms, including the first such algorithms for sorting and for sparse-matrix vector multiply on matrices with good vertex separators.Using known mappings, our results lead to low cache complexities on shared-memory multiprocessors with a single level of private caches or a single shared cache. We generalize these mappings to multi-level cache hierarchies of private or shared caches, implying that our algorithms also have low cache complexities on such hierarchies. The key factor in obtaining these low parallel cache complexities is the low depth of the algorithms we propose.
Citations
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23 Feb 2013
TL;DR: This paper presents a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write and significantly more efficient than previously reported results using graph frameworks on machines with many more cores.
Abstract: There has been significant recent interest in parallel frameworks for processing graphs due to their applicability in studying social networks, the Web graph, networks in biology, and unstructured meshes in scientific simulation. Due to the desire to process large graphs, these systems have emphasized the ability to run on distributed memory machines. Today, however, a single multicore server can support more than a terabyte of memory, which can fit graphs with tens or even hundreds of billions of edges. Furthermore, for graph algorithms, shared-memory multicores are generally significantly more efficient on a per core, per dollar, and per joule basis than distributed memory systems, and shared-memory algorithms tend to be simpler than their distributed counterparts.In this paper, we present a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write. The framework has two very simple routines, one for mapping over edges and one for mapping over vertices. Our routines can be applied to any subset of the vertices, which makes the framework useful for many graph traversal algorithms that operate on subsets of the vertices. Based on recent ideas used in a very fast algorithm for breadth-first search (BFS), our routines automatically adapt to the density of vertex sets. We implement several algorithms in this framework, including BFS, graph radii estimation, graph connectivity, betweenness centrality, PageRank and single-source shortest paths. Our algorithms expressed using this framework are very simple and concise, and perform almost as well as highly optimized code. Furthermore, they get good speedups on a 40-core machine and are significantly more efficient than previously reported results using graph frameworks on machines with many more cores.
816 citations
13 Apr 2015
TL;DR: 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, and is much faster than existing parallel approximate triangle counting implementations.
Abstract: 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.
143 citations
25 Feb 2012
TL;DR: The main contribution is to demonstrate that for this wide body of problems, there exist efficient internally deterministic algorithms, and moreover that these algorithms are natural to reason about and not complicated to code.
Abstract: The virtues of deterministic parallelism have been argued for decades and many forms of deterministic parallelism have been described and analyzed. Here we are concerned with one of the strongest forms, requiring that for any input there is a unique dependence graph representing a trace of the computation annotated with every operation and value. This has been referred to as internal determinism, and implies a sequential semantics---i.e., considering any sequential traversal of the dependence graph is sufficient for analyzing the correctness of the code. In addition to returning deterministic results, internal determinism has many advantages including ease of reasoning about the code, ease of verifying correctness, ease of debugging, ease of defining invariants, ease of defining good coverage for testing, and ease of formally, informally and experimentally reasoning about performance. On the other hand one needs to consider the possible downsides of determinism, which might include making algorithms (i) more complicated, unnatural or special purpose and/or (ii) slower or less scalable.In this paper we study the effectiveness of this strong form of determinism through a broad set of benchmark problems. Our main contribution is to demonstrate that for this wide body of problems, there exist efficient internally deterministic algorithms, and moreover that these algorithms are natural to reason about and not complicated to code. We leverage an approach to determinism suggested by Steele (1990), which is to use nested parallelism with commutative operations. Our algorithms apply several diverse programming paradigms that fit within the model including (i) a strict functional style (no shared state among concurrent operations), (ii) an approach we refer to as deterministic reservations, and (iii) the use of commutative, linearizable operations on data structures. We describe algorithms for the benchmark problems that use these deterministic approaches and present performance results on a 32-core machine. Perhaps surprisingly, for all problems, our internally deterministic algorithms achieve good speedup and good performance even relative to prior nondeterministic solutions.
141 citations
Cites methods from "Low depth cache-oblivious algorithm..."
...Comparison Sort: We use a low-depth cache-efficient sample sort [9]....
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09 Jun 2012
TL;DR: It is demonstrated that the otherwise uncontrolled growth of the Ninja gap can be contained and offer a more stable and predictable performance growth over future architectures, offering strong evidence that radical language changes are not required.
Abstract: Current processor trends of integrating more cores with wider SIMD units, along with a deeper and complex memory hierarchy, have made it increasingly more challenging to extract performance from applications. It is believed by some that traditional approaches to programming do not apply to these modern processors and hence radical new languages must be discovered. In this paper, we question this thinking and offer evidence in support of traditional programming methods and the performance-vs-programming effort effectiveness of common multi-core processors and upcoming many-core architectures in delivering significant speedup, and close-to-optimal performance for commonly used parallel computing workloads. We first quantify the extent of the "Ninja gap", which is the performance gap between naively written C/C++ code that is parallelism unaware (often serial) and best-optimized code on modern multi-/many-core processors. Using a set of representative throughput computing benchmarks, we show that there is an average Ninja gap of 24X (up to 53X) for a recent 6-core Intel® Core™ i7 X980 Westmere CPU, and that this gap if left unaddressed will inevitably increase. We show how a set of well-known algorithmic changes coupled with advancements in modern compiler technology can bring down the Ninja gap to an average of just 1.3X. These changes typically require low programming effort, as compared to the very high effort in producing Ninja code. We also discuss hardware support for programmability that can reduce the impact of these changes and even further increase programmer productivity. We show equally encouraging results for the upcoming Intel® Many Integrated Core architecture (Intel® MIC) which has more cores and wider SIMD. We thus demonstrate that we can contain the otherwise uncontrolled growth of the Ninja gap and offer a more stable and predictable performance growth over future architectures, offering strong evidence that radical language changes are not required.
87 citations
Cites methods from "Low depth cache-oblivious algorithm..."
...There have been various techniques proposed to address these algorithmic changes, either using compiler assisted optimization [27], using cache-oblivious algorithms [6] or specialized languages like Sequoia [21]....
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04 Jun 2011
TL;DR: The parallel cache-oblivious (PCO) model is presented, a relatively simple modification to the CO model that can be used to account for costs on a broad range of cache hierarchies, and a new scheduler is described, which attains provably good cache performance and runtime on parallel machine models with hierarchical caches.
Abstract: For nested-parallel computations with low depth (span, critical path length) analyzing the work, depth, and sequential cache complexity suffices to attain reasonably strong bounds on the parallel runtime and cache complexity on machine models with either shared or private caches. These bounds, however, do not extend to general hierarchical caches, due to limitations in (i) the cache-oblivious (CO) model used to analyze cache complexity and (ii) the schedulers used to map computation tasks to processors. This paper presents the parallel cache-oblivious (PCO) model, a relatively simple modification to the CO model that can be used to account for costs on a broad range of cache hierarchies. The first change is to avoid capturing artificial data sharing among parallel threads, and the second is to account for parallelism-memory imbalances within tasks. Despite the more restrictive nature of PCO compared to CO, many algorithms have the same asymptotic cache complexity bounds.The paper then describes a new scheduler for hierarchical caches, which extends recent work on "space-bounded schedulers" to allow for computations with arbitrary work imbalance among parallel subtasks. This scheduler attains provably good cache performance and runtime on parallel machine models with hierarchical caches, for nested-parallel computations analyzed using the PCO model. We show that under reasonable assumptions our scheduler is "work efficient" in the sense that the cost of the cache misses are evenly balanced across the processors---i.e., the runtime can be determined within a constant factor by taking the total cost of the cache misses analyzed for a computation and dividing it by the number of processors. In contrast, to further support our model, we show that no scheduler can achieve such bounds (optimizing for both cache misses and runtime) if work, depth, and sequential cache complexity are the only parameters used to analyze a computation.
84 citations
Cites background from "Low depth cache-oblivious algorithm..."
...However, all future references .t into cache until reaching a supertask that does not .t in cache, at which point the Problem Span Cache Complexity Q * Scan (pre.x sums, etc.) O(log n) O(ln/Bl) Matrix Transpose (n × m matrix) [20] v v O(log(n + m))v O(lnm/Bl)v Matrix Multiplication ( n × n matrix) [20] v v O( n)v O(ln 1.5/Bl/ M + 1) v Matrix Inversion ( n × n matrix) O( n) O(ln 1.5/Bl/ M + 1) Quicksort [22] O(log2 n) O(ln/Bl(1 + logln/(M + 1)l)) Sample Sort [10] O(log2 n) O(ln/BlllogM+2 nl) Sparse-Matrix Vector Multiply [10] (m nonzeros, nE edge separators) O(log2 n) O(lm/B + n/(M + 1)1-El) Convex Hull (e.g., see [8]) O(log2 n) O(ln/BlllogM+2 nl) Barnes Hut tree (e.g., see [8]) O(log2 n) O(ln/Bl(1 + logln/(M + 1)l)) Table 1: Cache complexities of some algorithms analyzed in the PCO model....
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...Unfortunately, current dynamic parallelism approaches have important limitations: they either apply to hierarchies of only private or only shared caches [1,9,10,16,21], require some strict balance criteria [7,15], or require a joint algorithm/scheduler analysis [7, 13–16]....
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...Sample Sort [10] O(log(2) n) O(dn/BedlogM+2 ne) Sparse-Matrix Vector Multiply [10] O(log(2) n) O(dm/B + n/(M + 1)1− e) (m nonzeros, n edge separators)...
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...A pair of common abstract measures for capturing parallel cache based locality are the number of misses given a sequential ordering of a parallel computation [1, 9, 10, 21], and the depth (span, critical path length) of the computation....
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References
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12 Oct 2005
TL;DR: A modern object-oriented programming language, X10, is designed for high performance, high productivity programming of NUCC systems and an overview of the X10 programming model and language, experience with the reference implementation, and results from some initial productivity comparisons between the X 10 and Java™ languages are presented.
Abstract: It is now well established that the device scaling predicted by Moore's Law is no longer a viable option for increasing the clock frequency of future uniprocessor systems at the rate that had been sustained during the last two decades. As a result, future systems are rapidly moving from uniprocessor to multiprocessor configurations, so as to use parallelism instead of frequency scaling as the foundation for increased compute capacity. The dominant emerging multiprocessor structure for the future is a Non-Uniform Cluster Computing (NUCC) system with nodes that are built out of multi-core SMP chips with non-uniform memory hierarchies, and interconnected in horizontally scalable cluster configurations such as blade servers. Unlike previous generations of hardware evolution, this shift will have a major impact on existing software. Current OO language facilities for concurrent and distributed programming are inadequate for addressing the needs of NUCC systems because they do not support the notions of non-uniform data access within a node, or of tight coupling of distributed nodes.We have designed a modern object-oriented programming language, X10, for high performance, high productivity programming of NUCC systems. A member of the partitioned global address space family of languages, X10 highlights the explicit reification of locality in the form of places}; lightweight activities embodied in async, future, foreach, and ateach constructs; a construct for termination detection (finish); the use of lock-free synchronization (atomic blocks); and the manipulation of cluster-wide global data structures. We present an overview of the X10 programming model and language, experience with our reference implementation, and results from some initial productivity comparisons between the X10 and Java™ languages.
1,469 citations
"Low depth cache-oblivious algorithm..." refers background in this paper
...The key factor in obtaining these low parallel cache complexities is the low depth of the algorithms we propose....
[...]
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.
1,312 citations
01 Oct 1977
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A,B,C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A,B,C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2\sqrt{2}\sqrt{2}$ vertices. We exhibit an algorithm which finds such a partition A,B,C in O(n) time.
1,264 citations
"Low depth cache-oblivious algorithm..." refers background in this paper
...G = (V, E) in S with n vertices can be partitioned into three sets of vertices Va, Vs, Vb such that |Vs| ≤ βf(n), |Va|, |Vb| ≤ αn, and {(u, v) ∈ E|(u ∈ Va ∧ v ∈ Vb) ∨ (u ∈ Vb ∧ v ∈ Va)} = ∅ [43]....
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...For planar graphs this can be done in linear time [43]....
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TL;DR: This paper gives the first provably good work-stealing scheduler for multithreaded computations with dependencies, and shows that the expected time to execute a fully strict computation on P processors using this scheduler is 1:1.
Abstract: This paper studies the problem of efficiently schedulling fully strict (i.e., well-structured) multithreaded computations on parallel computers. A popular and practical method of scheduling this kind of dynamic MIMD-style computation is “work stealing,” in which processors needing work steal computational threads from other processors. In this paper, we give the first provably good work-stealing scheduler for multithreaded computations with dependencies.Specifically, our analysis shows that the expected time to execute a fully strict computation on P processors using our work-stealing scheduler is T1/P + O(T ∞ , where T1 is the minimum serial execution time of the multithreaded computation and (T ∞ is the minimum execution time with an infinite number of processors. Moreover, the space required by the execution is at most S1P, where S1 is the minimum serial space requirement. We also show that the expected total communication of the algorithm is at most O(PT ∞( 1 + nd)Smax), where Smax is the size of the largest activation record of any thread and nd is the maximum number of times that any thread synchronizes with its parent. This communication bound justifies the folk wisdom that work-stealing schedulers are more communication efficient than their work-sharing counterparts. All three of these bounds are existentially optimal to within a constant factor.
1,202 citations
17 Oct 1999
TL;DR: It is proved that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model can be simulated efficiently by LRU replacement.
Abstract: This paper presents asymptotically optimal algorithms for rectangular matrix transpose, FFT, and sorting on computers with multiple levels of caching. Unlike previous optimal algorithms, these algorithms are cache oblivious: no variables dependent on hardware parameters, such as cache size and cache-line length, need to be tuned to achieve optimality. Nevertheless, these algorithms use an optimal amount of work and move data optimally among multiple levels of cache. For a cache with size Z and cache-line length L where Z=/spl Omega/(L/sup 2/) the number of cache misses for an m/spl times/n matrix transpose is /spl Theta/(1+mn/L). The number of cache misses for either an n-point FFT or the sorting of n numbers is /spl Theta/(1+(n/L)(1+log/sub Z/n)). We also give an /spl Theta/(mnp)-work algorithm to multiply an m/spl times/n matrix by an n/spl times/p matrix that incurs /spl Theta/(1+(mn+np+mp)/L+mnp/L/spl radic/Z) cache faults. We introduce an "ideal-cache" model to analyze our algorithms. We prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model. Can be simulated efficiently by LRU replacement. We also provide preliminary empirical results on the effectiveness of cache-oblivious algorithms in practice.
789 citations