CYK Algorithm Space Complexity Analysis
The Cocke-Younger-Kasami (CYK) algorithm is used for parsing strings based on a given context-free grammar, typically one in Chomsky Normal Form (CNF). It uses dynamic programming to determine if a string can be generated by the grammar.
Understanding the P Table
The core data structure in CYK is a table, often called the 'P' table or a DP table. This table stores information about which non-terminal symbols can generate substrings of the input sentence.
- The table is typically indexed based on the start and end positions of substrings within the sentence.
- Let the input sentence have n words.
- The table effectively stores results for all possible contiguous substrings.
- A common representation uses dimensions related to the sentence length. For a sentence of length n, there are $O(n^2)$ possible substrings.
- Each cell in the table (or associated with a specific substring) stores a set of non-terminal symbols.
- Let m be the number of non-terminal symbols in the grammar. In the worst case, a single cell might need to store all m non-terminals.
Calculating Space Complexity
The space complexity is determined by the size of this P table.
- Number of Substrings: There are $O(n^2)$ possible contiguous substrings in a sentence of length n. For example, substrings of length 1, length 2, ..., length n.
- Information per Substring: For each substring, the algorithm stores the set of non-terminals that can derive it. The maximum size of this set is m (the total number of non-terminals).
- Total Space: The total space required is the product of the number of substrings and the maximum space needed per substring.
Space = (Number of Substrings) $\times$ (Max Symbols per Substring)
Space = $O(n^2) \times O(m)$
Space = $O(n^2m)$
Therefore, the space complexity of the CYK algorithm for the P table is $O(n^2m)$.
Important Questions from Context Free Grammars - Teaching
- Match the LIST-I with LIST-II
LIST-I Grammar LIST-II All productions are of the form A. Regular Grammar I. $A \to aX$, where $a \in T$ and $X \in V^*$ B. Unrestricted Grammar II. $A \to xB$, $A \to x$ or $A \to Bx$, $A \to x$ where $A, B \in V$ and $x \in T^*$ C. Chomsky Normal Form III. $x \to y$, where $x \in (V \cup T)^+$ and $y \in (V \cup T)^*$ D. Greibach Normal Form IV. $A \to BC$ or $A \to a$, where A, B, C are in V and a is in T.
