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# Connectivity Queries Under Vertex Failures: Not Optimal, but Practical

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## Acknowledgements

I want to thank my advisor, Loukas Georgiadis, for helpful comments on this manuscript. I also want to thank the anonymous reviewers for their useful suggestions.

## Cite As

Evangelos Kosinas. Connectivity Queries Under Vertex Failures: Not Optimal, but Practical. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 75:1-75:13, Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.75

## Abstract

We revisit once more the problem of designing an oracle for answering connectivity queries in undirected graphs in the presence of vertex failures. Specifically, given an undirected graph G with n vertices and m edges and an integer d_β βͺ n, the goal is to preprocess the graph in order to construct a data structure π such that, given a set of vertices F with |F| = d β€ d_β, we can derive an oracle from π that can efficiently answer queries of the form "is x connected with y in Gβ§΅F?". Very recently, Long and Saranurak (FOCS 2022) provided a solution to this problem that is almost optimal with respect to the preprocessing time, the space usage, the update time, and the query time. However, their solution is highly complicated, and it seems very difficult to be implemented efficiently. Furthermore, it does not settle the complexity of the problem in the regime where d_β is a constant. Here, we provide a much simpler solution to this problem, that uses only textbook data structures. Our algorithm is deterministic, it has preprocessing time and space complexity O(d_β m log n), update time O(dβ΄ log n), and query time O(d). These bounds compare very well with the previous best, especially considering the simplicity of our approach. In fact, if we assume that d_β is a constant (d_β β₯ 4), then our algorithm provides some trade-offs that improve the state of the art in some respects. Finally, the data structure that we provide is flexible with respect to d_β: it can be adapted to increases and decreases, in time and space that are almost proportional to the change in d_β and the size of the graph.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing β Paths and connectivity problems
• Theory of computation β Graph algorithms analysis
##### Keywords
• Graphs
• Connectivity
• Fault-Tolerant
• Oracles

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