You are given a string s of even length consisting of digits from 0 to 9, and two integers a and b.
You can apply either of the following two operations any number of times and in any order on s:
a to all odd indices of s (0-indexed). Digits post 9 are cycled back to 0. For example, if s = "3456" and a = 5, s becomes "3951".s to the right by b positions. For example, if s = "3456" and b = 1, s becomes "6345".Return the lexicographically smallest string you can obtain by applying the above operations any number of times on s.
A string a is lexicographically smaller than a string b (of the same length) if in the first position where a and b differ, string a has a letter that appears earlier in the alphabet than the corresponding letter in b. For example, "0158" is lexicographically smaller than "0190" because the first position they differ is at the third letter, and '5' comes before '9'.
Example 1:
Input: s = "5525", a = 9, b = 2 Output: "2050" Explanation: We can apply the following operations: Start: "5525" Rotate: "2555" Add: "2454" Add: "2353" Rotate: "5323" Add: "5222" Add: "5121" Rotate: "2151" Add: "2050" There is no way to obtain a string that is lexicographically smaller than "2050".
Example 2:
Input: s = "74", a = 5, b = 1 Output: "24" Explanation: We can apply the following operations: Start: "74" Rotate: "47" Add: "42" Rotate: "24" There is no way to obtain a string that is lexicographically smaller than "24".
Example 3:
Input: s = "0011", a = 4, b = 2 Output: "0011" Explanation: There are no sequence of operations that will give us a lexicographically smaller string than "0011".
Constraints:
2 <= s.length <= 100s.length is even.s consists of digits from 0 to 9 only.1 <= a <= 91 <= b <= s.length - 1This approach leverages a breadth-first search (BFS) to explore all possible string configurations by applying the given operations. We keep track of already-visited states to avoid processing them again, storing the lexicographically smallest string found in the process.
The key point is to simulate the effect of each operation (addition and rotation) and use a queue to handle the states and explore until no new configurations can be found.
The Python solution uses BFS to explore each possible configuration of the string obtained by performing the operations. Starting from the initial string, it performs an addition operation and a rotation operation, pushing these new strings to the queue to be evaluated later, ensuring that each configuration is only visited once by using a set.
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Time Complexity: O(n * 10 * n), where n is the length of the string, as the cycle for each index via addition is at most 10, and there are n possible configurations for rotations.
Space Complexity: O(n * n) for storing possible configurations in the 'visited' set.
This approach reduces complexity by detecting cycles in possible configurations without exhausting all possibilities. It leverages mathematical principles governing rotations and additions, where applying operationsform sequences and constraints that can terminate the search early once a smallest string is identified.
By identifying recurring patterns in transformation sequences, computational resources can be conserved, and optimizations applied, particularly in identifying redundant operations that do not lead to smaller strings.
This Python code reduces operations by recognizing cycles in resulting transformations the operations can achieve. It systematically applies a limited number of additions and explores all rotations, using combinations to ensure minimal complexity is faced.
Time Complexity: O(10 * n^2), reduced due to early detection of cycles and constraints on possibilities given operation properties.
Space Complexity: O(n).
| Approach | Complexity |
|---|---|
| Brute Force with BFS to Check All Combinations | Time Complexity: O(n * 10 * n), where n is the length of the string, as the cycle for each index via addition is at most 10, and there are n possible configurations for rotations. Space Complexity: O(n * n) for storing possible configurations in the 'visited' set. |
| Mathematical Cycle Detection to Minimize Operations | Time Complexity: O(10 * n^2), reduced due to early detection of cycles and constraints on possibilities given operation properties. Space Complexity: O(n). |
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