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DOI: 10.4230/LIPIcs.SEA.2022.11

A Fast Data Structure for Dynamic Graphs Based on Hash-Indexed Adjacency Blocks

Authors: Alexander van der Grinten, Maria Predari, and Florian Willich

Published in: LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)

Abstract

Several dynamic graph data structures have been proposed in literature. Yet, these data structures either offer limited support for arbitrary graph algorithms or they are designed as part of specific frameworks (e.g., for GPUs or specialized hardware). Such frameworks are difficult to adopt to arbitrary graph computations and lead practitioners to fall back to less sophisticated solutions when dealing with dynamic graphs. In this work, we propose a new "dynamic hashed blocks" (DHB) data structure for sparse dynamic graphs and matrices on general-purpose CPU architectures. DHB combines an efficient block-based memory layout to store incident edges with an additional per-vertex hash index for high degree vertices. This hash index allows us to quickly insert edges without introducing duplicates, while the block-based memory layout retains advantageous cache locality properties of traditional adjacency arrays. Experiments show that DHB outperforms competing dynamic graph structures for edge insertions, updates, deletions, and traversal operations. Compared to static CSR layouts, DHB exhibits only a small overhead in traversal performance. DHB’s interface is similar to general-purpose abstract graph data types and can be easily used as a drop-in replacement for traditional adjacency arrays. To demonstrate that, we modify the well-known NetworKit framework to use DHB instead of its own dynamic graph representation. Experiments show that this modification only slightly penalizes the performance of graph algorithms while considerably boosting update rates.

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Alexander van der Grinten, Maria Predari, and Florian Willich. A Fast Data Structure for Dynamic Graphs Based on Hash-Indexed Adjacency Blocks. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vandergrinten_et_al:LIPIcs.SEA.2022.11,
  author =	{van der Grinten, Alexander and Predari, Maria and Willich, Florian},
  title =	{{A Fast Data Structure for Dynamic Graphs Based on Hash-Indexed Adjacency Blocks}},
  booktitle =	{20th International Symposium on Experimental Algorithms (SEA 2022)},
  pages =	{11:1--11:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-251-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{233},
  editor =	{Schulz, Christian and U\c{c}ar, Bora},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.11},
  URN =		{urn:nbn:de:0030-drops-165453},
  doi =		{10.4230/LIPIcs.SEA.2022.11},
  annote =	{Keywords: dynamic graph data structures, sparse matrix layout, dynamic algorithms, parallel algorithms, graph analysis}
}

Document

DOI: 10.4230/LIPIcs.ESA.2020.6

Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis

Authors: Eugenio Angriman, Maria Predari, Alexander van der Grinten, and Henning Meyerhenke

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract

The ubiquity of massive graph data sets in numerous applications requires fast algorithms for extracting knowledge from these data. We are motivated here by three electrical measures for the analysis of large small-world graphs G = (V, E) - i. e., graphs with diameter in O(log |V|), which are abundant in complex network analysis. From a computational point of view, the three measures have in common that their crucial component is the diagonal of the graph Laplacian’s pseudoinverse, L^+. Computing diag(L^+) exactly by pseudoinversion, however, is as expensive as dense matrix multiplication - and the standard tools in practice even require cubic time. Moreover, the pseudoinverse requires quadratic space - hardly feasible for large graphs. Resorting to approximation by, e. g., using the Johnson-Lindenstrauss transform, requires the solution of O(log |V| / ε²) Laplacian linear systems to guarantee a relative error, which is still very expensive for large inputs. In this paper, we present a novel approximation algorithm that requires the solution of only one Laplacian linear system. The remaining parts are purely combinatorial - mainly sampling uniform spanning trees, which we relate to diag(L^+) via effective resistances. For small-world networks, our algorithm obtains a ± ε-approximation with high probability, in a time that is nearly-linear in |E| and quadratic in 1 / ε. Another positive aspect of our algorithm is its parallel nature due to independent sampling. We thus provide two parallel implementations of our algorithm: one using OpenMP, one MPI + OpenMP. In our experiments against the state of the art, our algorithm (i) yields more accurate approximation results for diag(L^+), (ii) is much faster and more memory-efficient, and (iii) obtains good parallel speedups, in particular in the distributed setting.

Cite as

Eugenio Angriman, Maria Predari, Alexander van der Grinten, and Henning Meyerhenke. Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{angriman_et_al:LIPIcs.ESA.2020.6,
  author =	{Angriman, Eugenio and Predari, Maria and van der Grinten, Alexander and Meyerhenke, Henning},
  title =	{{Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{6:1--6:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.6},
  URN =		{urn:nbn:de:0030-drops-128723},
  doi =		{10.4230/LIPIcs.ESA.2020.6},
  annote =	{Keywords: Laplacian pseudoinverse, electrical centrality measures, uniform spanning tree, effective resistance, parallel sampling}
}

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Documents authored by Predari, Maria

A Fast Data Structure for Dynamic Graphs Based on Hash-Indexed Adjacency Blocks

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Approximation of the Diagonal of a Laplacian’s Pseudoinverse for Complex Network Analysis

Abstract

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