Solution

The correct answer is O(n²m).

Key Points

• The CYK algorithm is a parsing algorithm for context-free grammars, often used to determine whether a given string can be generated by a grammar.
• The algorithm uses a table, often called the "P table," which is a 2D array where P[i][j] stores information about the non-terminal symbols that can generate the substring from position i to j in the input string.
• For a sentence of length n and a grammar with m non-terminal symbols, the space required to store this table is proportional to O(n²m).
• This space complexity arises because:
  • There are approximately n² entries in the table, corresponding to all substrings of the input string of different lengths.
  • For each entry, we may need to store information for up to m non-terminal symbols.

Step-by-Step Solution / Explanation

• Step 1: Understanding the dimensions of the P table
  • The P table is a triangular 2D table used to store parsing information for substrings of the input string.
  • Given a string of length n, the table has approximately n² entries, as it stores information for all possible substrings of the string.
• Step 2: Accounting for the number of non-terminal symbols
  • For each table entry, the algorithm may need to store information about all the m non-terminal symbols in the grammar.
  • This means that the memory required for each entry is proportional to m.
• Step 3: Calculating the total space complexity
  • The total space complexity is the product of the number of entries in the table and the space required for each entry.
  • Total space complexity = O(n²) * O(m) = O(n²m).

Additional Information

• Time Complexity of CYK Algorithm:
  • The time complexity of the CYK algorithm is O(n³m), where n is the length of the input string and m is the number of non-terminal symbols in the grammar.
  • This arises because for each entry in the table, the algorithm may need to consider all possible splits of the substring and all combinations of non-terminal symbols.
• Applications of the CYK Algorithm:
  • The CYK algorithm is commonly used in natural language processing and computational linguistics to parse sentences according to a given grammar.
  • It is also used in formal language theory and compiler design for syntax analysis of programming languages.
• Limitations of the CYK Algorithm:
  • The algorithm requires the grammar to be in Chomsky Normal Form (CNF), which may require preprocessing to convert a given grammar into CNF.
  • The algorithm is not efficient for very large input strings or grammars with a large number of non-terminal symbols.