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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Two sets <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula>, whose elements fulfill a total order on operator <inline-formula> <tex-math notation="LaTeX">$\le $ </tex-math></inline-formula>, can have a binary relation <inline-formula> <tex-math notation="LaTeX">$R \subseteq A \times B$ </tex-math></inline-formula> represented by the k2-tree compact data structure, which greatly improves storage space. Currently, <inline-formula> <tex-math notation="LaTeX">$Count$ </tex-math></inline-formula> query is managed by either using <inline-formula> <tex-math notation="LaTeX">$Range$ </tex-math></inline-formula> query or to modify the structure to have aggregate information, implying additional time or space in order to perform the query. This article presents <italic>Compact Count</italic>, which exploits the k2-tree properties to reduce the paths to be scanned to count the numbers in a range <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>, thus ensuring an expected runtime of <inline-formula> <tex-math notation="LaTeX">$O(\log _{k} r \log _{k} n)$ </tex-math></inline-formula> and storage of <inline-formula> <tex-math notation="LaTeX">$O(\log _{k} r)$ </tex-math></inline-formula> with the k2-tree parameters <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. Our algorithm was compared through a series of experiments that consider both synthetic data with different distributions and real data, with a solution based on the <italic>Range</italic> algorithm. Experimental results show that <italic>Compact Count</italic> is 250 to 1,000 times faster than <italic>Range</italic> on synthetic and real data, respectively, with a small additional storage cost, as expected by the theoretical analysis.

An Efficient Algorithm to Count the Relations in a Range of Binary Relations Represented by a k²-Tree