B-Treaps Revised: Write Efficient Randomized Block Search Trees with High Load

Abstract

Uniquely represented data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of $n$ keys from a totally ordered universe in this context. We introduce a two-layer data structure called $(\alpha,\varepsilon)$-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory. Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze the {\em expected} search, update, and storage, efficiency in terms of block-reads, block-writes, and blocks stored. We prove that searches have $O(\varepsilon^{-1} + \log_\alpha n)$ block-reads, that $(\alpha, \varepsilon)$-RBSTs have an asymptotic load-factor of at least $(1-\varepsilon)$ for every $\varepsilon \in (0,1/2]$, and that dynamic updates perform $O(\varepsilon^{-1} + \log_\alpha(n)/\alpha)$ block-writes, i.e. $O(1/\varepsilon)$ writes if $\alpha=\Omega(\frac{\log n}{\log \log n} )$. Thus $(\alpha, \varepsilon)$-RBSTs provide improved search, storage-, and write-efficiency bounds in regard to the known, uniquely represented B-Treap [Golovin; ICALP'09].