Fully-functional static and dynamic succinct trees

We propose new succinct representations of ordinal trees, which have been studied exten-sively. It is known that any n-node static tree can be represented in 2n + o(n) bits and 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 range min-max tree, that reduces the large num-ber of relevant tree operations considered in the literature to a few primitives that are carried out in constant time on sufficiently small trees. The result is extended to trees of arbitrary size, achieving 2n + O(n/polylog(n)) bits of space, which is optimal for some operations. The redundancy is significantly lower than any previous proposal. For the dynamic case, where insertion/deletion of nodes is allowed, the existing data structures support very limited oper-ations. Our data structure builds on the range min-max tree to achieve 2n + O(n / log n) bits of space and O(log n) time for all the operations. We also propose an improved data structure using 2n + O(n log log n / log n) bits and improving the time to the optimal O(log n / log log n) for most operations. We extend our support to forests, where whole subtrees can be attached to or detached from others, in time O(log1+ n) for any > 0. Our techniques are of independent interest. An immediate derivation gives improved solution to range minimum/maximum queries where consecutive elements differ by ±1, achieving O(n+ n/polylog(n)) bits of space. A second one stores an array of numbers supporting operations sum and search and limited updates, in optimal time O(log n / log log n). A third one allows representing dynamic bitmaps and sequences supporting rank/select and indels, within zero-order entropy bounds and optimal time O(log n / log log n) for all operations on bitmaps and polylog-sized alphabets, and O(log n log σ/(log log n)2) on larger alphabet sizes σ. 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.