Near-Optimal Deterministic Vertex-Failure Connectivity Oracles
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We revisit the vertex-failure connectivity oracle problem. This is one of the most basic graph data structure problems under vertex updates, yet its complexity is still not well-understood. We essentially settle the complexity of this problem by showing a new data structure whose space, preprocessing time, update time, and query time are simultaneously optimal up to sub-polynomial factors assuming popular conjectures. Moreover, the data structure is deterministic. More precisely, for any integer d_⋆, the data structure preprocesses a graph G with n vertices and m edges in Ô(md_⋆) time and uses Õ(min{m,nd_⋆}) space. Then, given the vertex set D to be deleted where |D|=d≤ d_⋆, it takes Ô(d^2) updates time. Finally, given any vertex pair (u,v), it checks if u and v are connected in G∖ D in O(d) time. This improves the previously best deterministic algorithm by Duan and Pettie (SODA 2017) in both space and update time by a factor of d. It also significantly speeds up the Ω(min{mn,n^ω}) preprocessing time of all known (even randomized) algorithms with update time at most Õ(d^5).
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