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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a metric space $(X,d_X)$, a $(\beta,s,\Delta)$-sparse cover is acollection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most$\Delta$, such that for every point $x\in X$, the ball$B_X(x,\frac\Delta\beta)$ is fully contained in some cluster $C\in\mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Ourmain contribution is to show that the shortest path metric of every $K_r$-minorfree graphs admits $(O(r),O(r^2),\Delta)$-sparse cover, and for every$\epsilon>0$, $(4+\epsilon,O(\frac1\epsilon)^r,\Delta)$-sparse cover (forarbitrary $\Delta>0$). We then use this sparse cover to show that every$K_r$-minor free graph embeds into$\ell_\infty^{\tilde{O}(\frac1\epsilon)^{r+1}\cdot\log n}$ with distortion$3+\eps$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ with distortion$O(r)$). Further, we provide applications of these sparse covers into paddeddecompositions, sparse partitions, universal TSP / Steiner tree, oblivious buyat bulk, name independent routing, and path reporting distance oracles.

On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and other applications