License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2018.16
URN: urn:nbn:de:0030-drops-88427
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8842/
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Chen, Lijie ; Duan, Ran ; Wang, Ruosong ; Zhang, Hanrui ; Zhang, Tianyi

An Improved Algorithm for Incremental DFS Tree in Undirected Graphs

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LIPIcs-SWAT-2018-16.pdf (0.7 MB)

Abstract

Depth first search (DFS) tree is one of the most well-known data structures for designing efficient graph algorithms. Given an undirected graph G=(V,E) with n vertices and m edges, the textbook algorithm takes O(n+m) time to construct a DFS tree. In this paper, we study the problem of maintaining a DFS tree when the graph is undergoing incremental updates. Formally, we show:
Given an arbitrary online sequence of edge or vertex insertions, there is an algorithm that reports a DFS tree in O(n) worst case time per operation, and requires O (min {m log n, n^2}) preprocessing time.
Our result improves the previous O(n log^3 n) worst case update time algorithm by Baswana et al. [Baswana et al., 2016] and the O(n log n) time by Nakamura and Sadakane [Nakamura and Sadakane, 2017], and matches the trivial Omega(n) lower bound when it is required to explicitly output a DFS tree.
Our result builds on the framework introduced in the breakthrough work by Baswana et al. [Baswana et al., 2016], together with a novel use of a tree-partition lemma by Duan and Zhang [Duan and Zhang, 2016], and the celebrated fractional cascading technique by Chazelle and Guibas [Chazelle and Guibas, 1986a; Chazelle and Guibas, 1986b].

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs:2018:8842,
  author =	{Lijie Chen and Ran Duan and Ruosong Wang and Hanrui Zhang and Tianyi Zhang},
  title =	{{An Improved Algorithm for Incremental DFS Tree in Undirected Graphs}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm  Theory (SWAT 2018)},
  pages =	{16:1--16:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{David Eppstein},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8842},
  URN =		{urn:nbn:de:0030-drops-88427},
  doi =		{10.4230/LIPIcs.SWAT.2018.16},
  annote =	{Keywords: DFS tree, fractional cascading, fully dynamic algorithm}
}

Keywords: DFS tree, fractional cascading, fully dynamic algorithm
Collection: 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
Issue Date: 2018
Date of publication: 04.06.2018

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