Gravity-Driven String Extraction: Find the Lexicographically Largest

Imagine a specialized container, represented as a 2D grid, holding various items denoted by characters. Due to a peculiar gravity mechanism, when an item is extracted from the top of a column, the item directly below it conceptually "falls" to take its place. Your task is to extract items in a specific sequence over multiple rounds to form strings, and your ultimate goal is to identify the lexicographically largest string among all strings formed. This challenge tests your ability to model sequential processing, manage dynamic data structures, and perform lexicographical comparisons, skills vital in text processing, data interpretation, and algorithmic competitions.

Problem Description

You are given a 2D character array, box, representing a grid of m rows and n columns. Your objective is to extract characters from this box over m distinct rounds and determine the lexicographically largest string formed in any of these rounds.

The process of character extraction and string formation unfolds as follows:

1. Initialization:

   • For each column j (from 0 to n-1), prepare a conceptual "stack" of characters. This stack should be populated by taking characters box[i][j] for i from 0 to m-1, in that precise order. That is, box[0][j] is considered the topmost character in the stack, box[1][j] is next, and so on, down to box[m-1][j] at the bottom of the stack.
   • Initialize a string variable, lexicographically_largest_string, to an empty string or the smallest possible string value to store the maximum string found so far.

2. Extraction Rounds (Repeat m times):

   • In each round, you will form a temporary string, current_round_string, of length n.
   • For each column j (from 0 to n-1), perform the following:
     • Remove (pop) the topmost character from the stack corresponding to column j.
     • Append this popped character to current_round_string.
   • After collecting n characters (one from each column), you will have formed a complete current_round_string. Compare current_round_string with lexicographically_largest_string. If current_round_string is lexicographically larger than lexicographically_largest_string, update lexicographically_largest_string to current_round_string.

3. Final Result:

   • After completing all m extraction rounds, return the final value of lexicographically_largest_string.

Examples

Example 1:

Input:
box = [["#",".","#"],
       ["*","@","*"]]
Output: "*@*"
Explanation:
The box has m=2 rows and n=3 columns.

1. Initial column stacks (topmost character first):
   Col 0: ['#', '*']
   Col 1: ['.', '@']
   Col 2: ['#', '*']

2. Round 1:
   - Pop from Col 0: '#'. `current_round_string` is "#"
   - Pop from Col 1: '.'. `current_round_string` is "#."
   - Pop from Col 2: '#'. `current_round_string` is "#.#"
   String for Round 1: "#.#"
   `lexicographically_largest_string` becomes "#.#" (initial empty string is smaller).
   Stacks after Round 1:
   Col 0: ['*']
   Col 1: ['@']
   Col 2: ['*']

3. Round 2:
   - Pop from Col 0: '*'. `current_round_string` is "*"
   - Pop from Col 1: '@'. `current_round_string` is "*@"
   - Pop from Col 2: '*'. `current_round_string` is "*@*"
   String for Round 2: "*@*"
   Comparing "#.#" and "*@*", "*@*" is lexicographically larger (since '*' > '#').
   `lexicographically_largest_string` becomes "*@*"

After 2 rounds, the final answer is "*@*".

Example 2:

Input:
box = [["a","b","c"]]
Output: "abc"
Explanation:
The box has m=1 row and n=3 columns.

1. Initial column stacks:
   Col 0: ['a']
   Col 1: ['b']
   Col 2: ['c']

2. Round 1:
   - Pop from Col 0: 'a'. `current_round_string` is "a"
   - Pop from Col 1: 'b'. `current_round_string` is "ab"
   - Pop from Col 2: 'c'. `current_round_string` is "abc"
   String for Round 1: "abc"
   `lexicographically_largest_string` becomes "abc"

After 1 round, the final answer is "abc".

Example 3:

Input:
box = [["z","y"],
       ["x","w"],
       ["v","u"]]
Output: "zy"
Explanation:
The box has m=3 rows and n=2 columns.

1. Initial column stacks:
   Col 0: ['z', 'x', 'v']
   Col 1: ['y', 'w', 'u']

2. Round 1:
   - Pop from Col 0: 'z'. `current_round_string` is "z"
   - Pop from Col 1: 'y'. `current_round_string` is "zy"
   String for Round 1: "zy"
   `lexicographically_largest_string` becomes "zy"
   Stacks after Round 1:
   Col 0: ['x', 'v']
   Col 1: ['w', 'u']

3. Round 2:
   - Pop from Col 0: 'x'. `current_round_string` is "x"
   - Pop from Col 1: 'w'. `current_round_string` is "xw"
   String for Round 2: "xw"
   Comparing "zy" and "xw", "zy" is lexicographically larger (since 'z' > 'x').
   `lexicographically_largest_string` remains "zy"
   Stacks after Round 2:
   Col 0: ['v']
   Col 1: ['u']

4. Round 3:
   - Pop from Col 0: 'v'. `current_round_string` is "v"
   - Pop from Col 1: 'u'. `current_round_string` is "vu"
   String for Round 3: "vu"
   Comparing "zy" and "vu", "zy" is lexicographically larger (since 'z' > 'v').
   `lexicographically_largest_string` remains "zy"

After 3 rounds, the final answer is "zy".

Constraints

• m (number of rows) is between 1 and 100.
• n (number of columns) is between 1 and 100.
• box[i][j] consists of lowercase English letters ('a'-'z'), uppercase English letters ('A'-'Z'), digits ('0'-'9'), and common symbols like . (period), # (hash), * (asterisk), @ (at-sign).
• The total number of characters (m * n) will not exceed 10^4.
• The solution should be efficient enough to handle the given constraints, aiming for optimal time and space complexity.

Notes

• Remember that lexicographical comparison is based on the character's ordinal (ASCII/Unicode) values. For example, 'Z' is lexicographically smaller than 'a', and digits are generally smaller than letters.
• Consider using an appropriate data structure for each column's "stack" that allows efficient popping from the top (e.g., a list acting as a queue, or a deque/double-ended queue).
• The problem asks for the single lexicographically largest string found across all m rounds, not a concatenation or combination of all the round strings.