Explanation

- The **FM-index** is a data structure used for efficient full-text searching, especially for applications like searching within compressed texts. It is based on the Burrows-Wheeler Transform (BWT) and allows fast searching while using less space than traditional indexing methods.

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Steps

• Construct the Burrows-Wheeler Transform (BWT) of the text.
• Build the suffix array for the text.
• Preprocess the text to store information about the first and last columns of the BWT matrix.
• Use the FM-index to perform backward searches by repeatedly scanning the text from the last column to the first column.

Time Complexity

• Preprocessing time: O(n log n) for BWT and suffix array construction.
• Query time: O(m log n), where m is the length of the pattern being searched, and n is the length of the text.

class FMIndex:
    def __init__(self, text):
        self.text = text
        self.bwt = self.burrows_wheeler_transform(text)
        self.suffix_array = self.build_suffix_array(text)
        
    def burrows_wheeler_transform(self, text):
        # Generate the Burrows-Wheeler Transform (BWT) for the text
        pass
    
    def build_suffix_array(self, text):
        # Build the suffix array for the text
        pass
 
    def search(self, pattern):
        # Implement backward search with FM-index
        pass
 
# Example usage
fm_index = FMIndex("banana")
print(fm_index.search("ana"))  # Example search