Fully Functional Static and Dynamic Succinct Trees

We propose new succinct representations of ordinal trees and match various space/time lower bounds. It is known that any <i>n</i>-node static tree can be represented in 2<i>n</i> + <i>o</i>(<i>n</i>) bits so that a number of operations on the tree can be supported in constant time under the word-RAM model. However, the data structures are complicated and difficult to dynamize. We propose a simple and flexible data structure, called the <i>range min-max tree</i>, that reduces the large number of relevant tree operations considered in the literature to a few primitives that are carried out in constant time on polylog-sized trees. The result is extended to trees of arbitrary size, retaining constant time and reaching 2<i>n</i> + <i>O</i>(<i>n</i>/polylog(<i>n</i>)) bits of space. This space is optimal for a core subset of the operations supported and significantly lower than in any previous proposal. For the dynamic case, where insertion/deletion (indels) of nodes is allowed, the existing data structures support a very limited set of operations. Our data structure builds on the range min-max tree to achieve 2<i>n</i> + <i>O</i>(<i>n</i>/log <i>n</i>) bits of space and <i>O</i>(log <i>n</i>) time for all operations supported in the static scenario, plus indels. We also propose an improved data structure using 2<i>n</i> + <i>O</i>(<i>n</i>log log <i>n</i>/log <i>n</i>) bits and improving the time to the optimal <i>O</i>(log <i>n</i>/log log <i>n</i>) for most operations. We extend our support to forests, where whole subtrees can be attached to or detached from others, in time <i>O</i>(log^1+ε <i>n</i>) for any ε > 0. Such operations had not been considered before. Our techniques are of independent interest. An immediate derivation yields an improved solution to range minimum/maximum queries where consecutive elements differ by ± 1, achieving <i>n</i> + <i>O</i>(<i>n</i>/polylog(<i>n</i>)) bits of space. A second one stores an array of numbers supporting operations <i>sum</i> and <i>search</i> and limited updates, in optimal time <i>O</i>(log <i>n</i>/log log <i>n</i>). A third one allows representing dynamic bitmaps and sequences over alphabets of size σ, supporting rank/select and indels, within zero-order entropy bounds and time <i>O</i>(log <i>n</i> log σ/(log log <i>n</i>)²) for all operations. This time is the optimal <i>O</i>(log <i>n</i>/log log <i>n</i>) on bitmaps and polylog-sized alphabets. This improves upon the best existing bounds for entropy-bounded storage of dynamic sequences, compressed full-text self-indexes, and compressed-space construction of the Burrows-Wheeler transform.