The Maximum Weighted Clique(MWC) problem remains challenging due to its unfavourable time complexity.In this paper, we analyze the execution of exact search-based MWC algorithms and show that high-accuracy weighted cliques can be discovered in the early stages of the execution if searching the combinatorial space is performed systematically.
Based on this observation, we introduce QClique as an approximate MWC algorithm that processes the search space as long as better cliques are expected. QClique uses a tunable parameter to trade-off between accuracy vs. execution time and delivers 4.7-$82.3 time speedup in comparison to previous state-of-the-art MWC algorithms while providing 91.4% accuracy and achieves a parallel speedup of up to 56x on 128 threads.
Additionally, QClique accelerates the exact MWC computation by replacing the initial clique of the exact algorithm. For WLMC, an exact state-of-the-art MWC algorithm, this results in 3.3x on average.
Comprehensive evaluation is one of the basis of experimental science. In High-Performance Graph Processing, a thorough evaluation of contributions becomes more achievable by supporting common input formats over different frameworks. However, each framework creates its specific format, which may not support reading large-scale real-world graph datasets. This shows a demand for high-performance libraries capable of loading graphs to (i) accelerate designing new graph algorithms, (ii) to evaluate the contributions on a wide range of graph algorithms, and (iii) to facilitate easy and fast comparison over different graph frameworks.
To that end, we present ParaGrapher, a high-performance API and library for loading large-scale and compressed graphs. ParaGrapher supports different types of requests for accessing graphs in shared- and distributed-memory and out-of-core graph processing. We explain the design of ParaGrapher and present a performance model of graph decompression, which is used for evaluation of ParaGrapher over three storage types.
Our evaluation shows that by decompressing compressed graphs in WebGraph format, ParaGrapher delivers up to 3.2 times speedup in loading and up to 5.2 times speedup in end-to-end execution in comparison to the binary and textual formats.
@misc{paragrapher-arxiv,
title = { Selective Parallel Loading of Large-Scale
Compressed Graphs with ParaGrapher},
author = { {Mohsen} {Koohi Esfahani} and Marco D'Antonio and
Syed Ibtisam Tauhidi and Thai Son Mai and
Hans Vandierendonck},
year = {2024},
eprint = {2404.19735},
archivePrefix = {arXiv},
primaryClass = {cs.AR},
doi = {10.48550/arXiv.2404.19735}
}
The OS cache of storage contents have been dropped after each evaluation (sudo sh -c 'echo 3 >/proc/sys/vm/drop_caches'). The flushcache.c file (https://github.com/DIPSA-QUB/ParaGrapher/blob/main/test/flushcache.c) can be used with the same functionality for users without sudo access, however, it usually takes more time to be finished.
For LustreFS, we have repeated the evaluation of read and pread using O_DIRECT flag as this flag prevents client-side caching.
For HDD and SSD experiments, we have used a machine with Intel W-2295 3.00GHz CPU, 18 cores, 36 hyper-threads, 24MB L3 cache, 256 GB DDR4 2933Mhz memory, running Debian 12 Linux 6.1. For LustreFS, we have used a machine with 2TB 3.2GHz DDR4 memory, 2 AMD 7702 CPUs, in total, 128 cores, 256 threads.
The results of the evaluation using read_bandwidth.c are in the following table. The values are Bandwidth in MB/s. Also, 1-2 digits close to each number with a white background are are percentage of load imbalance between parallel threads.
Please click on the image to expand.
C vs. Java
We measure the bandwidth of SSD and HDD in C (mmap and pread) vs. Java (mmap and read). We use a machine with Intel W-2295 3.00GHz CPU, 18 cores, 36 hyper-threads, 24MB L3 cache, 256 GB DDR4 2933Mhz memory, running Debian 12 Linux 6.1 and the following codes:
To access the files, you need to register/login to IEEE DataPort and then visit the MS-BioGraphs page. By saving the page as an HTML file such as dp.html, you may download the datasets (as an example MS1) using the following script:
dsname="MS1"
html_file="dp.html"
urls=`cat $html_file | sed -e 's/\&/\&/g' | grep -Eo "(http|https)://[a-zA-Z0-9./?&=_%:-]*" | grep amazonaws | sort | uniq | grep -E "$dsname[-_\.]"`
for u in $urls; do
wget $u
if [ $? != 0 ]; then break; fi
done
# removing query strings
for f in $(find $1 -type f); do
if [ $f = ${f%%\?*} ]; then continue; fi
mv "${f}" "${f%%\?*}"
done
# liking offsets.bin to be found by ParaGrapher
ln -s ${dsname}_offsets.bin ${dsname}-underlying_offsets.bin
ParaGrapher source code for accessing WebGraphs have been published. The supported graph types are:
PARAGRAPHER_CSX_WG_400_AP: graphs compressed in WebGraph format with 4 Bytes ID per vertex. Graphs in this category: LAW web graphs (https://law.di.unimi.it/datasets.php) .
PARAGRAPHER_CSX_WG_404_AP: graphs compressed in WebGraph format with 4 Bytes ID per vertex and 4 Bytes integer weights per edge. Graphs in this category: MS-BioGraphs (https://blogs.qub.ac.uk/DIPSA/MS-BioGraphs/).
ParaGrapher uses its asynchronous and parallel API to implement these graph types. The user needs to implement a callback function that is called by the API upon completion of reading a block of edges. Poplar uses a shared memory for interaction between its C library and the Java library that deploys the WebGraph framework.
For further details, please refer to Poplar source code repository: https://github.com/DIPSA-QUB/ParaGrapher, particularly, src/webgraph.c and src/WG*.java files.
We provide a Shell script, validation.sh, and a Java program, EdgeBlockSHA.java, to verify the the correctness of the graphs. Each graph has a .ojson file whose shasum is verified by the value retreived from our server. Files such as offsets.bin, wcc.bin, n2o.bin, trans_offsets.bin, and edges_shas.txt have shasum records in the ojson file which is used for validation of these files.
The graph in WebGraph format has been compressed in MS??-underlying.* and MS??-weights.* files. In order to validate the compressed graph, the EdgeBlockSHA.java is used. It is a parallel Java code that uses the WebGraph library to traverse the graph and calculate the shasum of blocks of edges (endpoints and weights). Then, the calculated results are matched with the edges_shas.txt file of the graph.
It is also possible to validate some particular blocks by matching the calculated shasum with the relevant row in the edges_shas.txt file. This file has a format such as the following. Each block contains 64 Million consecutive edges. The start of each block is identified by a vertex ID and its edge index. The Column endpoint_sha is the shasum of the 64 Million endpoints when stored as an array of 4-Bytes elements in the binary format and in the little endian order. Similarly, Column weights_sha shows the shasum of weights (labels). We have separated weights from endpoints as in some applications weights are not needed and therefore it is not necessary to read and validate them.
Licensed under the GNU v3 General Public License, as published by the Free Software Foundation. You must not use this Software except in compliance with the terms of the License. Unless required by applicable law or agreed upon in writing, this Software is distributed on an “as is” basis, without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose, neither express nor implied.
Copyright 2022-2023 The Queen’s University of Belfast, Northern Ireland, UK
LaganLighter supports the following graph formats:
CSR/CSC graph in text format, for testing. This format has 4 lines: (i) number of vertices (|V|), (ii) number of edges (|E|), (iii) |V| space-separated numbers showing offsets of the vertices, and (iv) |E| space-separated numbers indicating edges.
In addition to execution time, we use the PAPI library to measure hardware counters such as L3 cache misses, hardware instructions, DTLB misses, and load and store memory instructions. ( papi_(init/start/reset/stop) and (print/reset)_hw_events functions defined in omp.c).
To measure load balance, we measure the total time of executing a loop and the time each thread spends in this loop (mt and ttimes in the following sample code). Using these values, PTIP macro (defined in omp.c) calculates the percentage of average idle time (as an indicator of load imbalance) and prints it with the total time (mt).
mt = - get_nano_time()
#pragma omp parallel
{
unsigned tid = omp_get_thread_num();
ttimes[tid] = - get_nano_time();
#pragma omp for nowait
for(unsigned int v = 0; v < g->vertices_count; v++)
{
// .....
}
ttimes[tid] += get_nano_time();
}
mt += get_nano_time();
PTIP("Step ... ");
As an example, the following execution of Thrifty, shows that the “Zero Planting” step has been performed in 8.98 milliseconds and with a 8.22% load imbalance, while processors have been idle for 72.22% of the execution time, on average, in the “Initial Push” step.
NUMA-Aware and Locality-Preserving Partitioning and Scheduling
In order to assign consecutive partitions (vertices and/or their edges) to each parallel processor, we initially divide partitions and assign a number of consecutive partitions to each thread. Then, we specify the order of victim threads in the work-stealing process. During the initialization of LaganLighter parallel processing environment (in initialize_omp_par_env() function defined in file omp.c), for each thread, we create a list of threads as consequent victims of stealing.
A thread, first, steals jobs (i.e., partitions) from consequent threads in the same NUMA node and then from the threads in consequent NUMA nodes. As an example, the following image shows the stealing order of a 24-core machine with 2 NUMA nodes. This shows that thread 1 steals from threads 2, 3, …,11, and ,0 running on the same NUMA socket and then from threads 13, 14, …, 23, and 12 running on the next NUMA socket.
We use dynamic_partitioning_...() functions (in file partitioning.c) to process partitions by threads in the specified order. A sample code is in the following:
As “we write bugs that in particular cases have been tested to work correctly”, we try to evaluate and validate the algorithms and their implementations. If you receive wrong results or you are suspicious about parts of the code, please contact us or submit an issue.
License
Licensed under the GNU v3 General Public License, as published by the Free Software Foundation. You must not use this Software except in compliance with the terms of the License. Unless required by applicable law or agreed upon in writing, this Software is distributed on an “as is” basis, without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose, neither express nor implied. For details see terms of the License.
Copyright 2022 The Queen’s University of Belfast, Northern Ireland, UK
Finally got around to this: publishing the Graptor source code. With time passing, the code has changed quite a bit compared to that used in the paper: Graptor: efficient pull and push style vectorized graph processing. The evolution of the code has advantages: it’s faster. There are also disadvantages: not all versions and variations of the code that were experimented with can still be compiled.
There will likely be issues (errors, lack of documentation, …) as this is experimental research code. Drop me a line if you need a hand h {a dot} vandierendonck {an at} qub {another dot} ac {the last dot} uk .
