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The transitive closure of a graph is a new graph where every vertex is directly connected to all vertices to which it had a path in the original graph. Transitive closures are useful for reachability and relationship querying. Finding the transitive closure can be computationally expensive and requires a large memory footprint as the output is typically larger than the input. Some of the original research on transitive closures assumed that graphs were dense and used dense adjacency matrices. We have since learned that many real-world networks are extremely sparse, and the existing methods do not scale. In this work, we introduce a new algorithm called Anti-section Transitive Closure (ATC) for finding the transitive closure of a graph. We present a new parallel edges operation – anti-sections – for finding new edges to reachable vertices. ATC scales to massively multithreaded systems such as NVIDIA’s GPU with tens of thousands of threads. We show that the anti-section operation shares some traits with the triangle counting intersection operation in graph analysis. Lastly, we view the transitive closure problem as a dynamic graph problem requiring edge insertions. By doing this, our memory footprint is smaller. We also show a method for creating the batches in parallel using two different techniques: dual-round and hash. Using these techniques and the Hornet dynamic graph data structure, we show our new algorithm on an NVIDIA Titan V GPU. We compare with other packages such as NetworkX, SEI-GBTL, SuiteSparse, and cuSparse.
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Edge-transitive graphs and combinatorial designs
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. We present a complete classification of all connected edge-transitive graphs on less than or equal to 20 vertices. We investigate biregular bipartite edge-transitive graphs and present connections to combinatorial designs, and we show that the Cartesian products of complements of complete graphs give an additional family of edge-transitive graphs.
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- Award ID(s):
- 1659075
- PAR ID:
- 10126658
- Date Published:
- Journal Name:
- Involve
- Volume:
- 12
- Issue:
- 8
- ISSN:
- 1944-4184
- Page Range / eLocation ID:
- 1329-1341
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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