Fully-Functional Static and Dynamic Succinct Trees (0905.0768v5)

Abstract: We propose new succinct representations of ordinal trees, which have been studied extensively. 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 number 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. The redundancy is significantly lower than any previous proposal. 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. Furthermore, we support sophisticated operations that allow attaching and detaching whole subtrees, in time $\Order(\log^{1+\epsilon} n / \log\log n)$. Our techniques are of independent interest. 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 $O(\log n \log \sigma / (\log\log n)^2)$ on larger alphabet sizes $\sigma$. 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.

• The paper introduces novel succinct data structures for static and dynamic ordinal trees using range min-max trees, significantly improving space and time efficiency.
• For static trees, the approach achieves space efficiency of $2n + O(n/\log^c n)$ bits with constant-time operations like parent, depth, and subtree size.
• For dynamic trees, the methods use $2n + O(n \log\log n/\log n)$ bits of space and support operations in $O(\log n / \log \log n)$ time, with flexible complex modifications.

Fully-Functional Static and Dynamic Succinct Trees

This paper introduces novel succinct representations for ordinal trees, both static and dynamic, with significant improvements in operational efficiency and space utilization. Ordinal trees are essential structures in computer science, and their succinct representations enable effective manipulation while minimizing space usage. The authors propose a range min-max tree as a unifying data structure that condenses numerous tree operations into a few primitives, allowing efficient computations and extensions to various tree sizes.

Static Succinct Tree Representation

For static trees, the paper achieves succinct representations using $2n + O(n/\log^c n)$ bits, where $n$ is the number of nodes. This improves the redundancy over previous proposals, approaching optimal space usage for certain operations. By implementing the range min-max tree, tree operations such as parent, depth, and subtree size computations are efficiently reduced to basic operations on smaller constructs, resolving within constant time. This approach contrasts against earlier methods, which often required complex auxiliary structures for each operation, accumulating larger overheads.

Dynamic Succinct Tree Representation

The dynamic framework is more intricate due to the additional requirements of node insertions and deletions. For the dynamic case, the authors achieve $2n + O(n \log\log n/\log n)$ bits of space, with operations executed in $O(\log n / \log \log n)$ time, which is optimal for many operations in a dynamic context. The use of dynamic min-max trees allows a seamless adjustment of the structure with tree modifications. New data structures facilitate $O(\log^{1+\epsilon} n)$ time for subtree detaching and attaching, showing flexibility in supporting complex modifications.

Applications and Implications

The proposed techniques have broader implications beyond ordinal trees. They enhance solutions to adjacent problems such as range minimum/maximum queries within bit vectors, where elements differ by ±1, achieving efficient space reductions. Additionally, the ability to maintain dynamic arrays of numbers that support operations like sum and search efficiently in $O(\log n / \log\log n)$ time is exemplified. This capability optimizes sequence manipulation, including dynamic compressed bitmaps and sequences, within zero-order entropy bounds. These enhancements provide an advanced basis for dynamically managing and querying text indices and collections, extending to more efficient construction of compressed text indices and full-text self-indexes.

Future Directions

The research opens several avenues for future exploration, including achieving lower bounds on redundancy for dynamic operations. It poses open questions on achieving $O(\log n/\log \log n)$ complexity for all dynamic operations. Additionally, exploring applications in broader contexts, such as more complex text index structures or high-dimensional data navigation, could extend this work's impact.

In conclusion, the paper presents significant advancements in both theoretical insights and practical implementations for efficiently handling both static and dynamic trees. The introduction of range min-max trees offers a promising approach to streamline operations, demonstrating both space and time efficiencies that surpass previous methods in the field of succinct data structures.