Faster Worst Case Deterministic Dynamic Connectivity

Kejlberg-Rasmussen, Casper; Kopelowitz, Tsvi; Pettie, Seth; Thorup, Mikkel

doi:10.4230/lipics.esa.2016.53

Cited by 5 publications

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“…Kapron et al [31] proposed an algorithm with complexity O(log 5 n), but it turned out that it can produce false negatives. In 2016, Kejlberg-Rasmussen et al [32] improved the complexity to O( n(log log n) 2 /log n). Henzinger and King were the first to look at amortized costs and achieve polynomial logarithmic amortized complexity.…”

Section: Related Workmentioning

confidence: 99%

Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

et al. 2022

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Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.

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Section: Related Workmentioning

confidence: 99%

Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

et al. 2022

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Connectivity Oracles for Graphs Subject to Vertex Failures

Duan¹,

Pettie²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d failed vertices inÕ(d 3 ) time and thereafter answers connectivity queries in O(d) time. It occupies space O(d m log n). We develop a randomized Monte Carlo version of our data structure with update timeÕ(d 2 ), query time O(d), and spaceÕ(m) for any d . This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures.Our data structures are based on a new decomposition theorem for an undirected graph G = (V, E), which is of independent interest. It states that for any terminal set U ⊆ V we can remove a set B of |U |/(s − 2) vertices such that the remaining graph contains a Steiner forest for U − B with maximum degree s. IntroductionThe dynamic subgraph model [19,21,36,39,37,42,65] is a constrained dynamic graph model. Rather than allow the graph to evolve in completely arbitrary ways (via an unbounded sequence of edge insertions and deletions), there is assumed to be a fixed ideal graph G = (V, E) that can be preprocessed in advance. The ideal graph is susceptible only to the failure of edges/vertices and their subsequent recovery, possibly with a bound d on the number of failures at one time. Queries naturally answer questions about the current failure-free subgraph. This model is useful because it more accurately represents the behavior of many real-world networks: changes to the underlying topology are relatively rare but transient failures very common. More importantly, this model offers the algorithm designer the freedom to explore exotic graph rep-

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Fully Dynamic Connectivity in O(log n(log log n)²) Amortized Expected Time

Huang

Kopelowitz

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms.We present a new randomized dynamic connectivity structure with O(log n(log log n)2 ) amortized expected update time and O(log n/ log log log n) query time, which comes within an O((log log n) 2 ) factor of a lower bound due to Pǎtraşcu and Demaine. The new structure is based on a dynamic connectivity algorithm proposed by Thorup in an extended abstract at STOC 2000, which left out some important details.

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Faster Worst Case Deterministic Dynamic Connectivity

Cited by 5 publications

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Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

Connectivity Oracles for Graphs Subject to Vertex Failures

Fully Dynamic Connectivity in O(log n(log log n)²) Amortized Expected Time

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Faster Worst Case Deterministic Dynamic Connectivity

Cited by 5 publications

References 0 publications

Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

Dynamic spanning trees for connectivity queries on fully-dynamic undirected graphs

Connectivity Oracles for Graphs Subject to Vertex Failures

Fully Dynamic Connectivity in O(log n(log log n)2) Amortized Expected Time

Contact Info

Product

Resources

About

Fully Dynamic Connectivity in O(log n(log log n)²) Amortized Expected Time