Tag Archives: graph traversal

On Designing Structure-Aware High-Performance Graph Algorithms (PhD Thesis)

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Mohsen Koohi Esfahani
Supervisors: Hans Vandierendonck and Peter Kilpatrick

Thesis in PDF format
Thesis on QUB Pure Portal

Graph algorithms find several usages in industry, science, humanities, and technology. The fast-growing size of graph datasets in the context of the processing model of the current hardware has resulted in different bottlenecks such as memory locality, work-efficiency, and load-balance that degrade the performance. To tackle these limitations, high-performance computing considers different aspects of the execution in order to design optimized algorithms through efficient usage of hardware resources.

The main idea in this thesis is to analyze the structure of graphs to exploit special features that are key to introduce new graph algorithms with optimized performance.

First, we study the structure of real-world graph datasets with skewed degree distribution and the applicability of graph relabeling algorithms as the main restructuring tools to improve performance and memory locality. To that end, we introduce novel locality metrics including Cache Miss Rate Degree Distribution, Effective Cache Size, Push Locality and Pull Locality, and Degree Range Decomposition.

Based on this structural analysis, we introduce the Uniform Memory Demands strategy that (i) recognizes diverse memory demands and behaviours as a source of performance inefficiency, (ii) separates contrasting memory demands into groups with uniform behaviours across each group, and (iii) designs bespoke data structures and algorithms for each group in order to satisfy memory demands with the lowest overhead.

We apply the Uniform Memory Demands strategy to design three graph algorithms with optimized performance: (i) the SAPCo Sort algorithm as a parallel counting sort algorithm that is faster than comparison-based sorting algorithms in degree-ordering of power-law graphs, (ii) the iHTL algorithm that optimizes locality in Sparse Matrix-Vector (SpMV) Multiplication graph algorithms by extracting dense subgraphs containing incoming edges to in-hubs and processing them in the push direction, and (iii) the LOTUS algorithm that optimizes locality in Triangle Counting by separating different caching demands and deploying specific data structure and algorithm for each of them.

Bibtex

@phdthesis{ODSAGA-ethos.874822,
  title  = {On Designing Structure-Aware High-Performance Graph Algorithms},
  author = {Mohsen Koohi Esfahani},
  year   = 2022,
  url    = {https://blogs.qub.ac.uk/DIPSA/On-Designing-Structure-Aware-High-Performance-Graph-Algorithms-PhD-Thesis/},
  school = {Queen's University Belfast},
  EThOSID = {uk.bl.ethos.874822}
}

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MASTIFF: Structure-Aware Minimum Spanning Tree/Forest – ICS’22

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36th ACM International Conference on Supercomputing 2022
June 27-30, 2022
Acceptance Rate: 25%

DOI: 10.1145/3524059.3532365
Authors’ Copy (PDF Format)

The Minimum Spanning Forest (MSF) problem finds usage in many different applications. While theoretical analysis shows that linear-time solutions exist, in practice, parallel MSF algorithms remain computationally demanding due to the continuously increasing size of data sets.

In this paper, we study the MSF algorithm from the perspective of graph structure and investigate the implications of the power-law degree distribution of real-world graphs
on this algorithm.

We introduce the MASTIFF algorithm as a structure-aware MSF algorithm that optimizes work efficiency by (1) dynamically tracking the largest forest component of each graph component and exempting them from processing, and (2) by avoiding topology-related operations such as relabeling and merging neighbour lists.

The evaluations on 2 different processor architectures with up to 128 cores and on graphs of up to 124 billion edges, shows that Mastiff is 3.4–5.9× faster than previous works.

Code Availability
The source-code of MASTIFF is available on LaganLighter Repository (alg3_mastiff.c and msf.c files). A sample execution of this source code for “Twitter-MPI” graph is shown in the following:

BibTex

@INPROCEEDINGS{10.1145/3524059.3532365,
author = {Koohi Esfahani, Mohsen and Kilpatrick, Peter and Vandierendonck, Hans},
title = {{MASTIFF}: Structure-Aware Minimum Spanning Tree/Forest},
year = {2022},
isbn = {},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3524059.3532365},
doi = {10.1145/3524059.3532365},
booktitle = {Proceedings of the 36th ACM International Conference on Supercomputing},
numpages = {13}
}

LaganLighter

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Software-Defined Floating-Point Number Formats and Their Application to Graph Processing – ICS’22

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DOI: 10.1145/3524059.3532360

This paper proposes software-defined floating-point number formats for graph processing workloads, which can improve performance in irregular workloads by reducing cache misses. Efficient arithmetic on software-defined number formats is challenging, even when based on conversion to wider, hardware-supported formats. We derive efficient conversion schemes that are tuned to the IA64 and AVX512 instruction sets.

We demonstrate that: (i) reduced-precision number formats can be applied to graph processing without loss of accuracy; (ii) conversion of floating-point values is possible
with minimal instructions; (iii) conversions are most efficient when utilizing vectorized instruction sets, specifically on IA64 processors.

Experiments on twelve real-world graph data sets demonstrate that our techniques result in speedups up to 89% for PageRank and Accelerated PageRank, and up to 35% for Single-Source Shortest Paths. The same techniques help to accelerate the integer-based maximal independent set problem by up to 262%.

Locality Analysis of Graph Reordering Algorithms – IISWC’21

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2021 IEEE International Symposium on Workload Characterization (IISWC’21)
November 7-9, 2021
Acceptance Rate: 39.5%
DOI: 10.1109/IISWC53511.2021.00020

Authors’ Copy (PDF Format)

Graph reordering algorithms try to improve locality of graph algorithms by assigning new IDs to vertices that ultimately changes the order of random memory accesses. While graph relabeling algorithms such as SlashBurn, GOrder, and Rabbit-Order provide better locality, it is not clear how they affect graph processing and different graph datasets , mainly, for three reasons:
(1) The large size of datasets,
(2) The lack of suitable measurement tools, and
(3) Disparate characteristics of graphs.
The paucity of analysis has also inhibited the design of more efficient RAs.

This paper introduces a number of metrics and tools to investigate the functionality of graph reordering algorithms and their effects on different real-world graph datasets:
(1) We introduce the Cache Miss Rate Degree Distribution and Degree Distribution of Neighbour to Neighbour Average Distance ID (N2N AID) to show how reordering algorithms affect different vertices,
(2) We introduce the Effective Cache Size as a metric to measure how much of cache capacity is used by reordered graphs for satisfying random memory accesses,
(3) We introduce the Assymetricity Degree Distribution and Neighbourhood Decomposition to explain the composition of neighbourhood of vertices to explain structural differences between web graphs and social networks.
(4) We investigate the effects of the structure of real-world graphs on the locality and performance of traversing graphs in pull and push directions by introducing Push Locality and Pull Locality.

Finally, we present improvements to graph reordering algorithms and propose other suggestions based on the new insights and features of real-world graphs introduced by this paper.

BibTex

@INPROCEEDINGS{10.1109/IISWC53511.2021.00020,
  author={Koohi Esfahani, Mohsen and Kilpatrick, Peter and Vandierendonck, Hans},
  booktitle={2021 IEEE International Symposium on Workload Characterization (IISWC'21)},  
  title={Locality Analysis of Graph Reordering Algorithms}, 
  year={2021},
  volume={},
  number={},
  pages={101-112},
  publisher={IEEE Computer Society},
  doi={10.1109/IISWC53511.2021.00020}
}

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Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing – ICPP’21

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ICPP'21

50th International Conference on Parallel Processing (ICPP’21)
August 9-12, 2021
Acceptance Rate: 26.4%

DOI:10.1145/3472456.3472462
ACM Digital Library
PDF Version (Authors’ Copy)

This paper investigates the implications made by the structure of real-world graphs with power-law degree distribution on the locality of SpMV graph analytics, and by considering the efficacy of locality optimizing graph reordering algorithms (such as SlashBurn, GOrder, and Rabbit-Order) shows that irregular datasets requires special traversals in order to improve locality for hub vertices that dedicate a large portion of the processing time to themselves.

We introduce in-Hub Temporal Locality (iHTL) as a structure-aware and cache-friendly graph traversal that optimizes locality in pull traversal. iHTL identifies different blocks in the adjacency matrix of a graph and applies a suitable traversal direction (push or pull) for each block based on its contents. In other words, iHTL optimizes locality of one traversal of all edges of the graph by:

(1) applying push direction for flipped blocks containing edges to in-hubs, and
(2) applying pull direction for processing sparse block containing edges to non-hubs.

Moreover, iHTL introduces a new algorithm to efficiently identify the number of flipped blocks by investigating connection between hub vertices of the graph. This allows iHTL to create flipped blocks as much as the graph structure requires and makes iHTL suitable for a wide range of different real-world graph datasets like social networks and web graphs.

iHTL is 1.5× – 2.4× faster than pull and 4.8× – 9.5× faster than push in state-of-the-art graph processing frameworks. More importantly, iHTL is 1.3× – 1.5× faster than pull traversal of state-of-the-art locality optimizing reordering algorithms such as SlashBurn, GOrder, and Rabbit-Order while reduces the preprocessing time by 780×, on average.

  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : Outline
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : Introduction
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : Pull vs Push
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : Is Pull A Suitable Direction
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : iHTL: in-Hub Temporal Locality
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : iHTL Graph Structure
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : SpMV in iHTL
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph Processing : Evaluation
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph : Conclusion
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph : Thanks
  • Exploiting in-Hub Temporal Locality in SpMV-based Graph : A Gift From QUB

Code Availability
The source-code will be published soon.

BibTex


@INPROCEEDINGS{10.1145/3472456.3472462,
author = {Koohi Esfahani, Mohsen and Kilpatrick, Peter and Vandierendonck, Hans},
title = {Exploiting In-Hub Temporal Locality In SpMV-Based Graph Processing},
year = {2021},
isbn = {9781450390682},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3472456.3472462},
doi = {10.1145/3472456.3472462},
booktitle = {50th International Conference on Parallel Processing},
numpages = {10},
location = {Lemont, IL, USA},
series = {ICPP 2021}
}

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How Do Graph Relabeling Algorithms Improve Memory Locality? ISPASS’21 (Poster)

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IEEE Xplore (DOI: 10.1109/ISPASS51385.2021.00023)
ISPASS-2021
2021 IEEE International Symposium on Performance Analysis of Systems and Software
March 28-30, 2021

Authors’ Copy (PDF Format)

For a complete version of this article, please refer to Locality Analysis of Graph Reordering Algorithms and also Chapter 3 of the On Designing Structure-Aware High-Performance Graph Algorithms thesis.

Relabeling (reordering) algorithms aim to improve the poor memory locality of graph processing by changing the order of vertices. This paper analyses the functionality of three state-of-the-art relabeling algorithms: SlashBurn, GOrder, and Rabbit-Order for real-world graphs.

We use a number of techniques to explain how locality is affected by relabeling algorithms and how locality of different datasets (like social networks and web graphs) is enhanced by relabeling algorithms.

We use last level cache simulation to study miss rate degree distribution. We also use the degree distribution of Giant Connected Component (GCC) in SlashBurn iterations to see if real-world graphs follow the assumption that “power-law graphs are created/destroyed recursively” [SlashBurn]. We represent SlashBurn++ as an enhanced version of SlashBurn with lower preprocessing time and better locality.

Using cache simulation, we count the number of misses for accessing vertices data of high-degree vertices. This helps to explain how GOrder provides better temporal locality by managing cache space. Average ID Distance (AID) is a spatial locality metric introduced in this paper to explain how clustering relabeling algorithms like Rabbit-Order provide better spatial locality.

This paper also investigates why push and pull traversals of different datasets show different performances by introducing Push Locality and Pull Locality.

  • How Do Graph Relabeling Algorithms Improve Memory Locality?
  • Outline
  • What is Locality?
  • What is Graph Relabeling (Reordering) ?
  • Why is Relabeling Important for Graph Processing?
  • Why is Relabeling Important for Graph Processing?
  • Graph Traversals
  • Performance of Relabeling Algorithms
  • Performance of Relabeling Algorithms
  • Performance of Relabeling Algorithms
  • SlashBurn: Community Detection for Power-law Graphs
  • SlashBurn: Community Detection for Power-law Graphs
  • SlashBurn++
  • GOrder: Cache Simulation for Pull Traversal
  • GOrder: Cache Simulation for Pull Traversal
  • Rabbit-Order: Parallel Community Detection
  • Rabbit-Order: Parallel Community Detection
  • Rabbit-Order: Parallel Community Detection
  • Push Locality vs. Pull Locality
  • Push Locality vs. Pull Locality
  • Conclusion
  • Thanks
  • QUB

Code Availability
The source-code of LaganLighter is available on LaganLighter Repository.

BibTex

@INPROCEEDINGS{10.1109/ISPASS51385.2021.00023,
  author={Koohi Esfahani, Mohsen and Kilpatrick, Peter and Vandierendonck, Hans},
  booktitle={2021 IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS)}, 
  title={How Do Graph Relabeling Algorithms Improve Memory Locality?}, 
  year={2021},
  volume={},
  number={},
  pages={84-86},
  publisher={IEEE Computer Society},
  doi={10.1109/ISPASS51385.2021.00023}
}

ISPASS’21
ISPASS’21 Final Program
LaganLighter

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The GraphGrind Framework: Fast Graph Analytics on Large Shared-Memory Systems (PhD Thesis)

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Thesis on QUB Pure Portal
Thesis in PDF Format

Author: Jiawen Sun, https://www.linkedin.com/in/jiawen-sun-33b368103/

As shared memory systems support terabyte-sized main memory, they provide an opportunity to perform efficient graph analytics on a single machine. Graph analytics is characterised by frequent synchronisation, which is addressed in part by shared memory systems. However, performance is limited by load imbalance and poor memory locality, which originate in the irregular structure of small-world graphs.
This dissertation demonstrates how graph partitioning can be used to optimise (i) load balance, (ii) Non-Uniform Memory Access (NUMA) locality and (iii) temporal locality of graph partitioning in shared memory systems. The developed techniques are implemented in GraphGrind, a new shared memory graph analytics framework.

At first, this dissertation shows that heuristic edge-balanced partitioning results in an imbalance in the number of vertices per partition. Thus, load imbalance exists between partitions, either for loops iterating over vertices, or for loops iterating over edges. To address this issue, this dissertation introduces a classification of algorithms to distinguish whether they algorithmically benefit from edge-balanced or vertex-balanced partitioning. This classification supports the adaptation of partitions to the characteristics of graph algorithms. Evaluation in GraphGrind, shows that this outperforms state-of-the-art graph analytics frameworks for shared memory including Ligra by 1.46x on average, and Polymer by 1.16x on average, using a variety of graph algorithms and datasets.

Secondly, this dissertation demonstrates that increasing the number of graph partitions is effective to improve temporal locality due to smaller working sets.
However, the increasing number of partitions results in vertex replication in some graph data structures. This dissertation resorts to using a graph layout that is immune to vertex replication and an automatic graph traversal algorithm that extends the previously established graph traversal heuristics to a 3-way graph layout choice is designed. This new algorithm furthermore depends upon the classification of graph algorithms introduced in the first part of the work. These techniques achieve an average speedup of 1.79x over Ligra and 1.42x over Polymer.

Finally, this dissertation presents a graph ordering algorithm to challenge the widely accepted heuristic to balance the number of edges per partition and minimise edge or vertex cut. This algorithm balances the number of edges per partition as well as the number of unique destinations of those edges. It balances edges and vertices for graphs with a power-law degree distribution. Moreover, this dissertation shows that the performance of graph ordering depends upon the characteristics of graph analytics frameworks, such as NUMA-awareness. This graph ordering algorithm achieves an average speedup of 1.87x over Ligra and 1.51x over Polymer.