Succinct Dynamic Cardinal Trees

Abstract

Cardinal trees (or tries of degree $k$) are a fundamental data structure, in particular for many text-processing algorithms. Each node in a cardinal tree has at most $k$ children, each labeled with a symbol from the alphabet $\{1,\ldots , k\}$. In this paper we introduce succinct representations for dynamic cardinal trees on $n$ nodes, requiring $2n + n\lg {k} + o(n\lg {k})$ bits of space. These are the first dynamic cardinal tree representations that support a (fairly) complete set of operations while using almost optimal space. For $k= \mathcal {O}(\mathrm{polylog}(n))$, we show how the navigational and query operations on the tree can be supported in $\mathcal {O}(1)$ time, while supporting insertions and deletions of leaves in $\mathcal {O}(1)$ amortized time. For $k= \omega ({\mathrm{polylog}(n)})$ [and $\mathcal {O}(n)$], we show that the same set of operations can be supported in $\mathcal {O}(\lg {k}/\lg {\lg {k}})$ time (amortized in the case of insertions/deletions). We also show how to associate $b$-bit satellite data to the tree nodes using $bn+o(n)$ extra bits. Finally, we show how the machinery introduced for dynamic cardinal trees can be adapted to represent dynamic binary trees using $2n+o(n)$ bits of space, so that the tree operations are supported in $\mathcal {O}(1)$ time (amortized in the case of insert/delete operations). We support adding satellite data to the tree nodes using $bn+o(n)$ extra bits [(vs. $bn+o(bn)$ extra bits of the fastest previous dynamic representation from the literature], while providing several trade-offs for accessing/modifying the data.