On Explicit Constructions of Extremely Depth Robust Graphs

Authors Jeremiah Blocki , Mike Cinkoske, Seunghoon Lee , Jin Young Son

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Author Details

Jeremiah Blocki
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Mike Cinkoske
  • Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Seunghoon Lee
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Jin Young Son
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

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Jeremiah Blocki, Mike Cinkoske, Seunghoon Lee, and Jin Young Son. On Explicit Constructions of Extremely Depth Robust Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A directed acyclic graph G = (V,E) is said to be (e,d)-depth robust if for every subset S ⊆ V of |S| ≤ e nodes the graph G-S still contains a directed path of length d. If the graph is (e,d)-depth-robust for any e,d such that e+d ≤ (1-ε)|V| then the graph is said to be ε-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents running in time polylog|V| which takes as input a node v ∈ V and returns the set of v’s parents. We give the first explicit construction of locally navigable ε-extreme depth-robust graphs with indegree O(log |V|). Previous constructions of ε-extreme depth-robust graphs either had indegree ω̃(log² |V|) or were not explicit.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Combinatorics
  • Depth-Robust Graphs
  • Explicit Constructions
  • Data-Independent Memory Hard Functions
  • Proofs of Space and Replication


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