Tree compression with top trees

Abstract

We introduce a new compression scheme for labeled trees based on top trees [3]. Our compression scheme is the first to simultaneously take advantage of internal repeats in the tree (as opposed to the classical DAG compression that only exploits rooted subtree repeats) while also supporting fast navigational queries directly on the compressed representation. We show that the new compression scheme achieves close to optimal worst-case compression, can compress exponentially better than DAG compression, is never much worse than DAG compression, and supports navigational queries in logarithmic time.

Tree Compression Using String Grammars

“…Here, the word RAM model is used, where memory cells can store numbers with log N bits and arithmetic operations on log N -bit numbers can be carried out in constant time. An analogous result was shown in [7,21] for top dags. Here, we show the same result for SLPs that produce (preorder traversals of) ranked trees.…”

“…In [1] so called elementary ordered tree grammars are used, and a polynomial time compressor with an approximation ratio of O(n 5/6 ) is presented. Also the top dags from [7] can be seen as a variation of TSLPs for unranked trees. Recently, in [21] it was shown that for every tree of size n with σ many node labels, the top dag has size O( n·log log σ n log σ n ), which improved the bound from [7].…”

“…Also the top dags from [7] can be seen as a variation of TSLPs for unranked trees. Recently, in [21] it was shown that for every tree of size n with σ many node labels, the top dag has size O( n·log log σ n log σ n ), which improved the bound from [7]. An extension of TSLPs to higher order tree grammars was proposed in [27].…”

Balancing Straight-Line Programs

“…With Theorem 9.2 it follows that from a forest straight-line program (resp., top dag) of size m that defines a tree of size n, one can compute in linear time an equivalent forest straight-line program (resp., top dag) of size O(m) and depth O(log n). This solves an open problem from [7], where the authors proved that from a tree t of size n, whose minimal DAG has size m (measured in number of edges in the DAG), one can construct in linear time a top dag for t of size O(m · log n) and depth O(log n). It remained open whether one can get rid of the additional factor log n in the size bound.…”

“…This can be overcome by representing a tree t by a linear context-free tree grammar that produces only t. Such grammars are also known as tree straight-line programs in the case of ranked trees [10,29,30] and forest straight-line programs in the case of unranked trees [17]. The latter are tightly related to top dags [7,4,13,21], which are another tree compression formalism, also akin to grammars. Forest straight-line programs and top dags can be defined as circuits over certain algebras, called forest algebras [9,17] and cluster algebras [17].…”