License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ITCS.2021.64
URN: urn:nbn:de:0030-drops-136038
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Blocki, Jeremiah ; Cinkoske, Mike

A New Connection Between Node and Edge Depth Robust Graphs

LIPIcs-ITCS-2021-64.pdf (0.5 MB)


Given a directed acyclic graph (DAG) G = (V,E), we say that G is (e,d)-depth-robust (resp. (e,d)-edge-depth-robust) if for any set S ⊂ V (resp. S ⊆ E) of at most |S| ≤ e nodes (resp. edges) the graph G-S contains a directed path of length d. While edge-depth-robust graphs are potentially easier to construct many applications in cryptography require node depth-robust graphs with small indegree. We create a graph reduction that transforms an (e, d)-edge-depth-robust graph with m edges into a (e/2,d)-depth-robust graph with O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably ((n log log n)/log n, n/{(log n)^{1 + log log n}})-depth-robust graph with constant indegree, where previous constructions for e = (n log log n)/log n had d = O(n^{1-ε}). Our reduction crucially relies on ST-Robust graphs, a new graph property we introduce which may be of independent interest. We say that a directed, acyclic graph with n inputs and n outputs is (k₁, k₂)-ST-Robust if we can remove any k₁ nodes and there exists a subgraph containing at least k₂ inputs and k₂ outputs such that each of the k₂ inputs is connected to all of the k₂ outputs. If the graph if (k₁,n-k₁)-ST-Robust for all k₁ ≤ n we say that the graph is maximally ST-robust. We show how to construct maximally ST-robust graphs with constant indegree and O(n) nodes. Given a family 𝕄 of ST-robust graphs and an arbitrary (e, d)-edge-depth-robust graph G we construct a new constant-indegree graph Reduce(G, 𝕄) by replacing each node in G with an ST-robust graph from 𝕄. We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.

BibTeX - Entry

  author =	{Jeremiah Blocki and Mike Cinkoske},
  title =	{{A New Connection Between Node and Edge Depth Robust Graphs}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{64:1--64:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{James R. Lee},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-136038},
  doi =		{10.4230/LIPIcs.ITCS.2021.64},
  annote =	{Keywords: Depth robust graphs, memory hard functions}

Keywords: Depth robust graphs, memory hard functions
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021

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