Flexible Graph Connectivity

Author(s):  
David Adjiashvili ◽  
Felix Hommelsheim ◽  
Moritz Mühlenthaler
Keyword(s):  
2020 ◽  
Vol 14 (4) ◽  
pp. 1-19
Author(s):  
Xiaofeng Zhu ◽  
Shichao Zhang ◽  
Jilian Zhang ◽  
Yonggang Li ◽  
Guangquan Lu ◽  
...  

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1227
Author(s):  
Shyam Sundar Santra ◽  
Prabhakaran Victor ◽  
Mahadevan Chandramouleeswaran ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher ◽  
...  

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of semiring valued graphs, also known as S-valued symmetry graphs, in 2015. Since then, works on S-valued symmetry graphs such as vertex dominating set, edge dominating set, regularity, etc. have been done. However, the connectivity of S-valued graphs has not been studied. Motivated by this, in this paper, the concept of connectivity in S-valued graphs has been studied. We have introduced the term vertex S-connectivity and edge S-connectivity and arrived some results for connectivity of a complete S-valued symmetry graph, S-path and S-star. Unlike the graph theory, we have observed that the inequality for connectivity κ(G)≤κ′(G)≤δ(G) holds in the case of S-valued graphs only when there is a symmetry of the graph as seen in Examples 3–5.


2016 ◽  
pp. 872-875
Author(s):  
Samir Khuller ◽  
Balaji Raghavachari
Keyword(s):  

2020 ◽  
Vol 14 (4) ◽  
pp. 653-667
Author(s):  
Laxman Dhulipala ◽  
Changwan Hong ◽  
Julian Shun

Connected components is a fundamental kernel in graph applications. The fastest existing multicore algorithms for solving graph connectivity are based on some form of edge sampling and/or linking and compressing trees. However, many combinations of these design choices have been left unexplored. In this paper, we design the ConnectIt framework, which provides different sampling strategies as well as various tree linking and compression schemes. ConnectIt enables us to obtain several hundred new variants of connectivity algorithms, most of which extend to computing spanning forest. In addition to static graphs, we also extend ConnectIt to support mixes of insertions and connectivity queries in the concurrent setting. We present an experimental evaluation of ConnectIt on a 72-core machine, which we believe is the most comprehensive evaluation of parallel connectivity algorithms to date. Compared to a collection of state-of-the-art static multicore algorithms, we obtain an average speedup of 12.4x (2.36x average speedup over the fastest existing implementation for each graph). Using ConnectIt, we are able to compute connectivity on the largest publicly-available graph (with over 3.5 billion vertices and 128 billion edges) in under 10 seconds using a 72-core machine, providing a 3.1x speedup over the fastest existing connectivity result for this graph, in any computational setting. For our incremental algorithms, we show that our algorithms can ingest graph updates at up to several billion edges per second. To guide the user in selecting the best variants in ConnectIt for different situations, we provide a detailed analysis of the different strategies. Finally, we show how the techniques in ConnectIt can be used to speed up two important graph applications: approximate minimum spanning forest and SCAN clustering.


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