摘要：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 âS, V (resp. S âS 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 (kâ,, kâ,,)-ST-Robust if we can remove any kâ, nodes and there exists a subgraph containing at least kâ,, inputs and kâ,, outputs such that each of the kâ,, inputs is connected to all of the kâ,, outputs. If the graph if (kâ,,n-kâ,)-ST-Robust for all kâ, ⤠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 ð." of ST-robust graphs and an arbitrary (e, d)-edge-depth-robust graph G we construct a new constant-indegree graph Reduce(G, ð.") by replacing each node in G with an ST-robust graph from ð.". We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.
