Grammar-Based Compression of Unranked Trees

Abstract

We introduce forest straight-line programs (FSLPs) as a compressed representation of unranked ordered node-labelled trees. FSLPs are based on the operations of forest algebra and generalize tree straight-line programs. We compare the succinctness of FSLPs with two other compression schemes for unranked trees: top dags and tree straight-line programs of firstchild/next sibling encodings. Efficient translations between these formalisms are provided. Finally, we show that equality of unranked trees in the setting where certain symbols are associative or commutative can be tested in polynomial time. This generalizes previous results for testing isomorphism of compressed unordered ranked trees.

Balancing Straight-Line Programs

“…In [17] it was shown that from a top dag G one can compute in linear time an equivalent FSLP of size O(|G|). Vice versa, from an FSLP H for a tree t ∈ C a (for some a ∈ Σ) one can compute in time O(|Σ| · |H|) an equivalent top dag of size O(|Σ| · |H|).…”

“…Vice versa, from an FSLP H for a tree t ∈ C a (for some a ∈ Σ) one can compute in time O(|Σ| · |H|) an equivalent top dag of size O(|Σ| · |H|). The additional factor |Σ| in the transformation from FSLPs to top dags is unavoidable; see [17] for an example. Proof.…”

“…It can be seen as a regular tree grammar that produces only t. A drawback of DAG-compression is that the size of the DAG is lower-bounded by the height of the tree t. Hence, for deep narrow trees (like for instance caterpillar trees), the DAG-representation cannot achieve good compression. 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.…”

“…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]. Both types of algebras turn out to have finite subsumption bases.…”

“…Note that the definition of a top dag in[7] refers to the outcome of a particular top dag construction. In other words: for every tree t a very specific SLP over the cluster algebra is constructed and this SLP is called the top dag of t. Here, as in[17], we call any SLP over the cluster algebra a top dag.…”