Can CYK Parsing algorithm generate the parsing tree in O(n^3)?

Question

I found this question What is the usage of CYK algorithm in the real world considering we have algorithms with a much better Time complexity? saying CYK Parsing algorithm can compute any Context Free Grammar in $O(n^3)$.

But this does not make sense because I cannot see how NonDeterministic Context Free grammar can be parsed in $O(n^3)$. Then, while reading Hopcroft and Ullman (2006) pages 304 I found:

In $O(n^3)$ time the algorithm constructs a table that tells whether w is in L $\in$ Note that when computing this running time the grammar itself is considered fixed and its size contributes only a constant factor to the running time which is measured in terms of the length of the string w whose membership in L is being tested.

So, the correct affirmation is: There is no way a NonDeterministic Context Free Grammar parsing tree can be built in $O(n^3)$ time, only its word membership can be determined in $O(n^3)$ time by the CYK algorithm. NonDeterministic Context-Free Grammars Parsing Trees can only be built in Exponential Time (worst case)?

Related:

1. Chomsky Hierarchy and P vs NP
2. What is the big-O (worst-case upper bound) for time and space requirement of the different Chomsky classes?

Answer 1

It depends what you mean by build a parse tree.

You can build a parse forest in $O(n^3)$ time and space. The forest represents all parse trees, even an infinite number of parse trees, because it is a graph, not a tree. From a parse forest, it is possible to produce a single parse tree in time linear to the size of the forest, and it is possible to iterate through all possible parses.

So if you either consider that "build a parse tree" means "find one of the possible parses" or if you consider that it means "build a datastructure which represents all possible parses", then you can certainly do that with a CYK grammar in $O(n^3)$.

If, on the other hand, you intend "build a parse tree" to mean "build all parse trees for a sentence", then it is not limited to exponential time; it is easy to write a grammar with an infinite number of possible parses for certain sentences.