Number of Atoms - Solution & Explanation

This approach uses a recursive parsing strategy combined with a stack to handle the nested nature of formulas within parenthesis. When encountering a parenthesis, we call the function recursively to process the sub-formula, using the stack to apply multipliers correctly once recursion unwinds back to the outer level.

We create a recursive function parse_formula that tracks the current index in the formula. We loop through characters, processing each segment as either an atom, number, opening parenthesis, or closing parenthesis. For each atom or sub-formula, we determine its count and add it to a dictionary, which accumulates the counts of each atom in the current context. When encountering a closing parenthesis, we return from the recursion with the accumulated counts for that segment to be multiplied accordingly. Finally, the main function uses a sorted list of atoms for output formatting.

This approach leverages an iterative stack-based strategy to process the formula without recursion. By using an explicit stack, the nested structure is managed more explicitly by keeping track of counts at each level of parentheses, applying multipliers when unwinding back.

This C++ solution uses an iterative approach with a stack to manage ongoing calculations and resolve nested parentheses structures. A map is maintained for the current level of parsing, storing atom counts and using a stack to save the state when entering new parentheses. On encountering a closing parenthesis, values are multiplied and merged back into the previous level. The std::map is employed for automatic alphabetical ordering of keys.