Special binary strings are binary strings with the following two properties:
0's is equal to the number of 1's.1's as 0's.You are given a special binary string s.
A move consists of choosing two consecutive, non-empty, special substrings of s, and swapping them. Two strings are consecutive if the last character of the first string is exactly one index before the first character of the second string.
Return the lexicographically largest resulting string possible after applying the mentioned operations on the string.
Example 1:
Input: s = "11011000" Output: "11100100" Explanation: The strings "10" [occuring at s[1]] and "1100" [at s[3]] are swapped. This is the lexicographically largest string possible after some number of swaps.
Example 2:
Input: s = "10" Output: "10"
Constraints:
1 <= s.length <= 50s[i] is either '0' or '1'.s is a special binary string.The key observation in Special Binary String is that a valid special string behaves similarly to a balanced parentheses sequence: the number of 1s equals the number of 0s, and every prefix has at least as many 1s as 0s. This allows us to treat each valid segment as an independent unit.
Traverse the string and split it into the smallest special substrings. Whenever the balance between 1 and 0 returns to zero, we have identified one valid block. For each block, recursively apply the same logic to its inner portion to maximize its arrangement.
After processing each block recursively, sort the resulting special substrings in descending lexicographical order and concatenate them. This greedy-recursive strategy ensures the overall string becomes the largest possible special binary string.
The approach mainly relies on recursion and sorting of substrings. The typical complexity is around O(n^2 log n) in the worst case due to recursive substring processing and sorting, with O(n) auxiliary space from recursion and substring storage.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Recursive decomposition with greedy sorting of special substrings | O(n^2 log n) | O(n) |
NeetCode
Use these hints if you're stuck. Try solving on your own first.
Draw a line from (x, y) to (x+1, y+1) if we see a "1", else to (x+1, y-1). A special substring is just a line that starts and ends at the same y-coordinate, and that is the lowest y-coordinate reached. Call a mountain a special substring with no special prefixes - ie. only at the beginning and end is the lowest y-coordinate reached. If F is the answer function, and S has mountain decomposition M1,M2,M3,...,Mk, then the answer is: reverse_sorted(F(M1), F(M2), ..., F(Mk)). However, you'll also need to deal with the case that S is a mountain, such as 11011000 -> 11100100.
This approach leverages recursion to decompose the special binary string into smaller special substrings, sort these components, and then rebuild the string to achieve the lexicographically largest string. This works because by sorting the special substrings in descending order, larger lexicographical strings are formed.
Time Complexity: O(n^2), where n is the length of the string, due to recursive decomposition and sorting.
Space Complexity: O(n), for storing the results and recursion stack.
1var makeLargestSpecial = function(s) {
2 let count = 0, i = 0;
3 let res = [];
4 The JavaScript function segments the string using a similar logic as other languages, sorts the fragments in descending order to produce the largest attempt result.
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Yes, problems like Special Binary String appear in advanced coding interviews because they test recursion, string manipulation, and greedy thinking. Variants of this problem are commonly used to evaluate a candidate's ability to recognize structural patterns in strings.
The solution mainly relies on strings, recursion, and dynamic substring handling. A list or array is commonly used to store the extracted special substrings before sorting them in descending order.
The optimal strategy decomposes the string into the smallest valid special substrings and processes them recursively. After maximizing each substring internally, the substrings are sorted in descending lexicographical order and concatenated to form the largest possible result.
Each special substring contains smaller special substrings within it. Recursion naturally handles this nested structure, allowing the algorithm to optimize inner segments first before combining them for the final result.