Given a directed acyclic graph (DAG) G=(V,E), we say that G is (e,d)depthrobust (resp. (e,d)edgedepthrobust) 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 edgedepthrobust graphs are potentially easier to construct, many applications in cryptography require node depthrobust graphs with small indegree. We create a graph reduction that transforms an (e,d)edgedepthrobust graph with m edges into a (e/2,d)depthrobust graph with O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably (nloglognlogn,nlogn(logn)loglogn)depthrobust graph with constant indegree. Ourmore »
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On Explicit Constructions of Extremely Depth Robust Graphs
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 GS still contains a directed path of length d. If the graph is (e,d)depthrobust for any e,d such that e+d ≤ (1ε)V then the graph is said to be εextreme depthrobust. In the field of cryptography, (extremely) depthrobust graphs with low indegree have found numerous applications including the design of sidechannel resistant MemoryHard 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 polylogV 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 εextreme depthrobust graphs with indegree O(log V). Previous constructions of εextreme depthrobust graphs either had indegree ω̃(log² V) or were not explicit.
 Editors:
 Berenbrink, Petra and
 Publication Date:
 NSFPAR ID:
 10322470
 Journal Name:
 Leibniz international proceedings in informatics
 Volume:
 219
 ISSN:
 18688969
 Sponsoring Org:
 National Science Foundation
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