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Given a list of allowed triples consisting of lowercase letters 'A' to 'Z', we need to determine if it is possible to build a pyramid from the bottom row of letters to a single top letter by stacking the allowed letters. The bottom row is given as a string, and the allowed triples are formatted as follows: if (A, B) -> C, then in the bottom row, if there are letters A and B stacked consecutively, you can stack a letter C on top of them. The goal is to find if we can reduce the bottom row to a single letter on the top of the pyramid using these stacking rules.

Clarifying Questions

  1. Q: What is the maximum length of the bottom string and the number of allowed triples?
    • A: The length of the bottom string is up to 7, and the number of allowed triples is up to 15600.
  2. Q: Are there any specific constraints on the input?
    • A: The bottom row consists only of uppercase English letters. The allowed triples are also constrained to uppercase English letters spanning from ‘A’ to ‘Z’.

Strategy

  1. Represent Allowed Triples in a Dictionary: We will use a dictionary where the key is a pair of letters and the value is a list of possible letters that can be stacked on top of each pair.

  2. Backtracking to Build the Pyramid: We will use a recursive backtracking approach to try all possible constructions of the pyramid. If at any level we can’t match a pair to one of the allowed triples, we backtrack.

  3. Recursive Function: The recursive function will take the current level of the pyramid and generate the next level above it. It will continue this process until it either succeeds or exhausts all possibilities.

Code

from collections import defaultdict

def pyramidTransition(bottom, allowed):
    # Create a dictionary for possible transitions
    transitions = defaultdict(list)
    for triplet in allowed:
        transitions[(triplet[0], triplet[1])].append(triplet[2])
    
    def can_form_pyramid(current_level):
        if len(current_level) == 1:
            return True
        
        next_level = []
        return can_form_pyramid_recursive(current_level, 0, next_level)
    
    def can_form_pyramid_recursive(current_level, index, next_level):
        if index == len(current_level) - 1:
            return can_form_pyramid(next_level)
        
        key = (current_level[index], current_level[index + 1])
        if key in transitions:
            for letter in transitions[key]:
                next_level.append(letter)
                if can_form_pyramid_recursive(current_level, index + 1, next_level):
                    return True
                next_level.pop()        
        return False
    
    return can_form_pyramid(bottom)

# Example usage
bottom = "BCD"
allowed = ["BCG", "CDE", "GEA", "FFF"]
print(pyramidTransition(bottom, allowed)) # Should return True or False

Explanation

  1. Data Structure: A defaultdict of lists is used to store the allowed transitions.

  2. Recursive Function can_form_pyramid_recursive:
    • Takes current_level, index, and next_level as arguments.
    • It tries to build the next_level from the current_level by using allowed transitions.
    • When the recursion advances to the next level successfully, it continues until it reaches the top with only one letter.
  3. Base Condition: The function checks if the current_level is of length 1, meaning we have successfully formed the pyramid.

Time Complexity

Given the constraints, the maximum depth would be 7, leading to potential exponential branching with worst-case considerations handled by backtracking. Thus, the recursive approach is efficient for the given constraints and should work within reasonable time limits for competitive programming scenarios.

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