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

We present a data structure representing a dynamic set S of w-bit integers on a w-bit word RAM. With &amp;#x7C;S&amp;#x7C; = n and w &amp;ge; log n and space O(n), we support the following standard operations in O(log n/log w) time: insert(x) sets S = S + {x}. delete(x) sets S = S {x}. predecessor(x) returns max{y &amp;#x2208; S &amp;#x7C; y &amp;lt; x}. rank(x) returns #{y &amp;#x2208; S &amp;#x7C; y &amp;lt; x}. select (i) returns y &amp;#x2208; S with rank (y) = i, if any. Our O(log n/log w) bound is optimal for dynamic rank and select, matching a lower bound of Fredman and Saks [STOC'89]. When the word length is large, our time bound is also optimal for dynamic predecessor, matching a static lower bound of Beame and Fich [STOC'99] whenever log n/log w = O(log w/log log w). Technically, the most interesting aspect of our data structure is that it supports all the above operations in constant time for sets of size n = w O(1). This resolves a main open problem of Ajtai, Komlos, and Fredman [FOCS&amp;#129;f83]. Ajtai et al. presented such a data structure in Yao&amp;#129;fs abstract cell-probe model with w-bit cells/words, but pointed out that the functions used could not be implemented. As a partial solution to the problem, Fredman and Willard [STOC&amp;#129;f90] introduced a fusion node that could handle queries in constant time, but used polynomial time on the updates. We call our small set data structure a dynamic fusion node as it does both queries and updates in constant time.

Dynamic Integer Sets with Optimal Rank, Select, and Predecessor Search