Abstract

Fault tolerant distance preservers are sparse subgraphs that preserve distances between given pairs of nodes under edge or vertex failures. In this paper, we present the first non-trivial constructions of subset distance preservers, which preserve all distances among a subset of nodes S, that can handle either an edge or a vertex fault.
- For an n-vertex undirected weighted graph or weighted DAG G = (V,E) and S ⊆ V, we present a construction of a subset preserver with Õ(|S|n) edges that is resilient to a single fault. In the single pair case (|S| = 2), the bound improves to O(n). We further provide a nearly-matching lower bound of Ω(|S|n) in either setting, and we show that the same lower bound holds conditionally even if attention is restricted to unweighted graphs.
- For an n-vertex directed unweighted graph G = (V,E) and r ∈ V, S ⊆ V, we present a construction of a preserver of distances in {r} × S with Õ(n^{4/3} |S|^{5/6}) edges that is resilient to a single fault. In the case |S| = 1 the bound improves to O(n^{4/3}), and for this case we provide another matching conditional lower bound.
- For an n-vertex directed weighted graph G = (V, E) and r ∈ V, S ⊆ V, we present a construction of a preserver of distances in {r} × S with Õ(n^{3/2} |S|^{3/4}) edges that is resilient to a single vertex fault. (It was proved in [Greg Bodwin et al., 2017] that the bound improves to O(n^{3/2}) when |S| = 1, and that this is conditionally tight.)

@InProceedings{bodwin_et_al:LIPIcs:2020:12422,
  author =	{Greg Bodwin and Keerti Choudhary and Merav Parter and Noa Shahar},
  title =	{{New Fault Tolerant Subset Preservers}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12422},
  URN =		{urn:nbn:de:0030-drops-124222},
  doi =		{10.4230/LIPIcs.ICALP.2020.15},
  annote =	{Keywords: Subset Preservers, Distances, Fault-tolerance}
}