Special Binary String


Problem Classification
The problem is a classic example of a string manipulation problem in the domain of computer science and algorithms. The problem includes elements of sorting (to achieve the lexicographically largest string) and the concept of substrings.
The “what” components of the problem are:
Input: A string
s
is given as input. This string represents a special binary string, which means it contains an equal number of ‘0’s and ‘1’s, and every prefix of the string has at least as many ‘1’s as ‘0’s.Operations: The operation defined in the problem is swapping two consecutive, nonempty, special substrings of
s
.Output: The objective is to find the lexicographically largest resulting string possible after applying the mentioned operations on the string.
This problem can be classified as a combination of string manipulation and sorting problem. The main task is to apply a sequence of swaps in such a way that the final result is the largest possible string in lexicographic order.
To solve this problem, understand how lexicographic order works, how substrings can be found and manipulated, and how to perform swaps in a string. String manipulation and understanding of sorting are fundamental skills needed in many areas of computer science, making this problem an interesting challenge that combines several key concepts.
Thought Process
To solve this problem, a major observation is that in a special binary string, each ‘1’ is always paired with the ‘0’ that occurs at the first time when the count of ‘0’s and ‘1’s are the same in a prefix of the string. This is because a special binary string always has more ‘1’s than ‘0’s in the prefix, and only when a ‘0’ comes to balance out a ‘1’, they form a pair, and all pairs are independent, thus forming a special binary substring.
Once we find out all these special binary substrings, we sort them in decreasing order because we want to form the lexicographically largest resulting string. After that, we again make a recursive call to solve the same problem for all substrings excluding the outermost ‘1’ and ‘0’ since they are already in their correct place.
Now, let’s move on to the coding part:


Here’s a linebyline explanation:
count = i = 0
,special_substrings = []
: Initializecount
andi
to zero, andspecial_substrings
as an empty list. Thecount
keeps track of the number of ‘1’s subtracted by the number of ‘0’s, whilei
is the start index of a special binary substring.for j, c in enumerate(s):
: Traverse through the string, withj
as the index andc
as the character.count = count + 1 if c == '1' else count  1
: Incrementcount
by 1 ifc
is ‘1’, else decrementcount
by 1.if count == 0:
: Ifcount
becomes 0, it means we’ve found a special binary substring.special_substrings.append('1' + self.makeLargestSpecial(s[i + 1:j]) + '0')
: Recursively find the special binary substrings for the current special binary substring excluding the outermost ‘1’ and ‘0’, and add it tospecial_substrings
.i = j + 1
: Updatei
to the start index of the next special binary substring.''.join(sorted(special_substrings, reverse=True))
: Finally, sortspecial_substrings
in descending order and join them to form a string, which is the lexicographically largest resulting string.
Language Agnostic Coding Drills
 Dissecting the code:
The distinct concepts the code contains are:
 Variable assignment: This concept involves assigning values to variables. Here, `count` and `i` are initialized to 0, and `special_substrings` is initialized as an empty list.
 Enumeration: The `enumerate` function is used to get both the index and the value from an iterable. Here, it is used to get each character `c` in the string `s` along with its index `j`.
 Conditional logic: This concept involves the use of `if` statements to execute different code blocks based on different conditions. Here, it is used to update `count` based on whether `c` is '1' or '0', and to check if `count` is 0.
 String concatenation: This concept involves joining strings together. Here, it is used to construct special binary substrings.
 Recursion: This concept involves a function calling itself. Here, the function `makeLargestSpecial` is called recursively to find the special binary substrings for the current special binary substring excluding the outermost '1' and '0'.
 Sorting and joining of list: This concept involves sorting a list and joining the elements to form a string. Here, it is used to form the lexicographically largest resulting string.
Listing out the concepts:
Variable assignment (Level: Easy): It is the most basic concept in programming which involves assigning values to variables. Here, it is used to initialize
count
andi
to zero, andspecial_substrings
as an empty list.Enumeration (Level: Easy): It is a common technique used to traverse through an iterable with access to both index and value. Here, it is used to get each character
c
in the strings
along with its indexj
.Conditional logic (Level: Easy): It allows the program to make decisions based on conditions. Here, it is used to increment or decrement
count
, and to check ifcount
is 0.String concatenation (Level: Easy): It is a simple concept that involves joining strings together. Here, it is used to construct special binary substrings.
Sorting and joining of list (Level: Medium): Sorting is a common operation in programming which puts elements in a certain order. Joining combines all elements in a list into a string. Here, it is used to form the lexicographically largest resulting string.
Recursion (Level: Hard): Recursion is a more advanced concept where a function calls itself. It can make the code cleaner but also more difficult to debug. Here, it is used to solve the subproblems.
Problemsolving approach:
The problem is solved using a combination of enumeration, conditional logic, string concatenation, and recursion. The first step involves traversing the string using enumeration and conditional logic to find special binary substrings. Each time a special binary substring is found, the function is called recursively to find its special binary substrings, and the result is added to the list special_substrings
. Finally, the list is sorted in descending order and joined to form the lexicographically largest resulting string. Each concept contributes to the overall solution in its unique way, and they all work together to solve the problem.
Targeted Drills in Python
Pythonbased coding drills for each identified concept:
 Variable assignment:
1 2
a = 5 b = "Hello, World!"
 Enumeration:
1 2 3
fruits = ["apple", "banana", "cherry"] for i, fruit in enumerate(fruits): print(f"index: {i}, value: {fruit}")
 Conditional logic:
1 2 3 4 5
a = 20 if a > 10: print("a is greater than 10.") else: print("a is not greater than 10.")
 String concatenation:
1 2 3 4
greeting = "Hello" name = "World" message = greeting + ", " + name + "!" print(message)
 Sorting and joining of list:
1 2 3 4
words = ["apple", "banana", "cherry"] words.sort(reverse=True) # Sort in descending order sentence = " ".join(words) print(sentence)
 Recursion:
1 2 3 4 5 6
def factorial(n): if n == 1: return 1 else: return n * factorial(n  1) print(factorial(5))
Problemspecific concept:
 Special Binary String Identification: In our problem, we have to identify special binary strings in the given string. A special binary string in this context is defined as a string where the number of ‘1’s is equal to the number of ‘0’s and every prefix of the binary string has at least as many 1’s as 0’s. A drill could be to create a function that checks if a given string meets these conditions. This concept is crucial as it forms the basis of our problem  manipulating these special binary strings to get the lexicographically largest string.
Integration of drills:
After understanding each of the individual concepts, we can start to combine them together to form the solution.
We start by initializing variables (count
, i
, special_substrings
). Then we iterate over the string s
using enumeration and conditional logic to find special binary substrings. Whenever we find a special binary substring, we call our function recursively on the substring excluding the outermost ‘1’ and ‘0’. We add the result to special_substrings
. Once we have iterated over the entire string, we sort special_substrings
in reverse order and join them together using string concatenation to form the lexicographically largest string. We return this as our final result.