English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 $S\subset V$ (resp. $S \subseteq E$) of at most $|S| \leq e$ nodes (resp. edges)the graph $G-S$ contains a directed path of length $d$. While edge-depth-robustgraphs are potentially easier to construct many applications in cryptographyrequire node depth-robust graphs with small indegree. We create a graphreduction that transforms an $(e, d)$-edge-depth-robust graph with $m$ edgesinto 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 aprovably $(\frac{n \log \log n}{\log n}, \frac{n}{(\log n)^{1 + \log \logn}})$-depth-robust graph with constant indegree, where previous constructionsfor $e =\frac{n \log \log n}{\log n}$ had $d = O(n^{1-\epsilon})$. Ourreduction crucially relies on ST-Robust graphs, a new graph property weintroduce which may be of independent interest. We say that a directed, acyclicgraph with $n$ inputs and $n$ outputs is $(k_1, k_2)$-ST-Robust if we canremove any $k_1$ nodes and there exists a subgraph containing at least $k_2$inputs and $k_2$ outputs such that each of the $k_2$ inputs is connected to allof the $k_2$ outputs. If the graph if $(k_1,n-k_1)$-ST-Robust for all $k_1 \leqn$ we say that the graph is maximally ST-robust. We show how to constructmaximally ST-robust graphs with constant indegree and $O(n)$ nodes. Given afamily $ \mathbb{M}$ of ST-robust graphs and an arbitrary $(e,d)$-edge-depth-robust graph $G$ we construct a new constant-indegree graph $\mathrm{Reduce}(G, \mathbb{M})$ by replacing each node in $G$ with an ST-robustgraph from $ \mathbb{M}$. We also show that ST-robust graphs can be used toconstruct (tight) proofs-of-space and (asymptotically) improved wide-blocklabeling functions.

A New Connection Between Node and Edge Depth Robust Graphs