Parallel Symmetry-Breaking in Sparse Graphs

Goldberg, Andrew V.; Plotkin, Serge; Shannon, Gregory E.

doi:10.1137/0401044

Cited by 174 publications

(77 citation statements)

References 16 publications

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“…A (∆ + 1)-coloring can be computed in time O(log n + ∆) [30] or O(∆ log n) [31]. In [20], Maraco et al proposed a distributed algorithm that computed an O(∆)-coloring in time O(log n).…”

Section: Performance Evaluation Of Schedulingmentioning

confidence: 99%

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Wang

et al. 2008

IEEE Trans. Parallel Distrib. Syst.

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Abstract-We study efficient interference-aware joint routing and TDMA link scheduling for a multihop wireless network to maximize its throughput. Efficient link scheduling can greatly reduce the interference effect of close-by transmissions. Unlike the previous studies that often assume a unit disk graph model, we assume that different terminals could have different transmission ranges and different interference ranges. In our model, it is also possible that a communication link may not exist due to barriers or is not used by a predetermined routing protocol, while the transmission of a node always result interference to all non-intended receivers within its interference range.Using a mathematical formulation, we develop interference aware joint routing and synchronized TDMA link schedulings that optimize the networking throughput subject to various constraints. Our linear programming formulation will find a flow routing whose achieved throughput is at least a constant fraction of the optimum, and the achieved fairness is also a constant fraction of the requirement. Then, by assuming known link capacities and link traffic loads, we study link scheduling under the RTS/CTS interference model and the protocol interference model with fixed transmission power. For both models, we present both efficient centralized and distributed algorithms that use time slots within a constant factor of the optimum. We also present efficient distributed algorithms whose performances are still comparable with optimum, but with much less communications. We prove that the time-slots needed by our faster distributed algorithms are only at most O(min(log n, log ψ)) for RTS/CTS interference model and protocol interference model. Our theoretical results are corroborated by extensive simulation studies.

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Section: Performance Evaluation Of Schedulingmentioning

confidence: 99%

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Wang

et al. 2008

IEEE Trans. Parallel Distrib. Syst.

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“…Goldberg et al [30] describe an algorithm for finding a greedy coloring of O(1)-degree graphs in O(lg n) time in the EREW PRAM model using a linear number of processors. They observe that their technique can be applied recursively to color ∆-degree graphs in O(∆ lg ∆ lg n) time.…”

Section: Related Workmentioning

confidence: 99%

“…Parallel coloring algorithms have been explored extensively in the distributed computing domain [3,5,30,31,35,38,39,41]. These algorithms are evaluated in the message-passing model, where nodes are allowed unlimited local computation and exchange messages through a sequence of synchronized rounds.…”

Section: Related Workmentioning

confidence: 99%

Ordering heuristics for parallel graph coloring

Hasenplaugh¹,

Kaler²,

Schardl³

et al. 2014

Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures

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This paper introduces the largest-log-degree-first (LLF) and smallest-log-degree-last (SLL) ordering heuristics for parallel greedy graph-coloring algorithms, which are inspired by the largest-degree-first (LF) and smallest-degree-last (SL) serial heuristics, respectively. We show that although LF and SL, in practice, generate colorings with relatively small numbers of colors, they are vulnerable to adversarial inputs for which any parallelization yields a poor parallel speedup. In contrast, LLF and SLL allow for provably good speedups on arbitrary inputs while, in practice, producing colorings of competitive quality to their serial analogs.We applied LLF and SLL to the parallel greedy coloring algorithm introduced by Jones and Plassmann, referred to here as JP. Jones and Plassman analyze the variant of JP that processes the vertices of a graph in a random order, and show that on an O(1)-degree graph G = (V, E), this JP-R variant has an expected parallel running time of O(lgV / lg lgV ) in a PRAM model. We improve this bound to show, using work-span analysis, that JP-R, augmented to handle arbitrary-degree graphs, colors a graph G = (V, E) with degree ∆ using Θ(V + E) work and O(lgV + lg ∆ · min{ √ E, ∆ + lg ∆ lgV / lg lgV }) expected span. We prove that JP-LLF and JP-SLL-JP using the LLF and SLL heuristics, respectivelyexecute with the same asymptotic work as JP-R and only logarithmically more span while producing higher-quality colorings than JP-R in practice.We engineered an efficient implementation of JP for modern shared-memory multicore computers and evaluated its performance on a machine with 12 Intel Core-i7 (Nehalem) processor cores. Our implementation of JP-LLF achieves a geometric-mean speedup of 7.83 on eight real-world graphs and a geometric-mean speedup of 8.08 on ten synthetic graphs, while our implementation using SLL achieves a geometric-mean speedup of 5.36 on these real-world graphs and a geometric-mean speedup of 7.02 on these synthetic graphs. Furthermore, on one processor, JP-LLF is slightly faster than a well-engineered serial greedy algorithm using LF, and likewise, JP-SLL is slightly faster than the greedy algorithm using SL.

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“…Although it is NP-complete to find either an optimal coloring of a graph [40] -a coloring that uses the smallest possible number of colors -or a O(V ε )-approximation of the optimal col- oring [77], as Section 6 discusses, an optimal coloring is not necessary for PRISM to perform well in practice, as long as the data graph is colored with sufficiently few colors. Many parallel coloring algorithms exist that satisfy the needs of PRISM (see, for example, [3,5,45,46,52,59,61,62,73,94]). In fact, if the data-graph computation performs sufficiently many updates, a Θ(V + E)-work greedy coloring algorithm, such as that introduced by Welsh and Powell [96], can suffice as well.…”

Section: The Prism Algorithmmentioning

confidence: 99%

Executing Dynamic Data-Graph Computations Deterministically Using Chromatic Scheduling

Kaler

Hasenplaugh

Schardl

et al. 2016

ACM Trans. Parallel Comput.

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A data-graph computation -popularized by such programming systems as Galois, Pregel, GraphLab, PowerGraph, and GraphChi -is an algorithm that performs local updates on the vertices of a graph. During each round of a data-graph computation, an update function atomically modifies the data associated with a vertex as a function of the vertex's prior data and that of adjacent vertices. A dynamic data-graph computation updates only an active subset of the vertices during a round, and those updates determine the set of active vertices for the next round.This paper introduces PRISM, a chromatic-scheduling algorithm for executing dynamic data-graph computations. PRISM uses a vertex-coloring of the graph to coordinate updates performed in a round, precluding the need for mutual-exclusion locks or other nondeterministic data synchronization. A multibag data structure is used by PRISM to maintain a dynamic set of active vertices as an unordered set partitioned by color. We analyze PRISM using work-span analysis. Let G = (V, E) be a degree-∆ graph colored with χ colors, and suppose that Q ⊆ V is the set of active vertices in a round. Define size(Q) = |Q| + v∈Q deg(v), which is proportional to the space required to store the vertices of Q using a sparsegraph layout. We show that a P-processor execution of PRISM performs updates in Q using O(χ(lg(Q/χ) + lg ∆) + lg P) span and Θ(size(Q) + χ + P) work. These theoretical guarantees are matched by good empirical performance. We modified GraphLab to incorporate PRISM and studied seven application benchmarks on a 12-core multicore machine. PRISM executes the benchmarks 1.2-2.1 times faster than GraphLab's nondeterministic lock-based scheduler while providing deterministic behavior. This paper also presents PRISM-R, a variation of PRISM that executes dynamic data-graph computations deterministically even when updates modify global variables with associative operations. PRISM-R satisfies the same theoretical bounds as PRISM, but its implementation is more involved, incorporating a multivector data structure to maintain an ordered set of vertices partitioned by color.

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Parallel Symmetry-Breaking in Sparse Graphs

Cited by 174 publications

References 16 publications

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Interference-Aware Joint Routing and TDMA Link Scheduling for Static Wireless Networks

Ordering heuristics for parallel graph coloring

Executing Dynamic Data-Graph Computations Deterministically Using Chromatic Scheduling

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