Minimum Time to Activate String

The problem asks you to find the minimum time to activate a string by replacing characters in a specific order. At each time step, a character at the index in the order array is replaced by ''. A valid substring contains at least one ' '. The goal is to determine the minimum time required to achieve at least 'k' valid substrings.

Problem Statement

You are given a string s of length n and an integer array order, where order is a permutation of the numbers in the range [0, n - 1]. Starting from time t = 0, replace the character at index order[t] in s with * at each time step. The string s becomes active when at least one * is present in every valid substring of length greater than 0.

A substring is considered valid if it contains at least one '*'. You are asked to find the minimum time t when at least k valid substrings exist, or return -1 if it's impossible. The input guarantees that the string and order array are valid, but you must carefully account for the constraints when designing your solution.

Examples

Example 1

Input: s = "abc", order = [1,0,2], k = 2

Output: 0

The string s becomes active at t = 0 . Thus, the answer is 0.

Example 2

Input: s = "cat", order = [0,2,1], k = 6

Output: 2

The string s becomes active at t = 2 . Thus, the answer is 2.

Example 3

Input: s = "xy", order = [0,1], k = 4

Output: -1

Even after all replacements, it is impossible to obtain k = 4 valid substrings. Thus, the answer is -1.

Constraints

• 1 <= n == s.length <= 105
• order.length == n
• 0 <= order[i] <= n - 1
• s consists of lowercase English letters.
• order is a permutation of integers from 0 to n - 1.
• 1 <= k <= 109

Solution Approach

Binary Search on Time

The problem can be solved efficiently using binary search on the time t value. For each potential time t, we check whether replacing characters at the first t+1 positions in order leads to at least k valid substrings.

Count Valid Substrings

At each binary search step, calculate the number of valid substrings formed by the first t+1 replacements. This can be done by marking positions in s and calculating the number of substrings containing at least one *.

Final Calculation and Edge Cases

After binary search, if the time t results in fewer than k valid substrings, return -1. If we find the required t, then that's the minimum time to activate the string with the desired number of valid substrings.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is dominated by the binary search, which runs in O(log n). For each binary search step, we calculate the valid substrings, which takes O(n). Hence, the overall time complexity is O(n log n). The space complexity is O(n) due to the need to store the string and the order array.

What Interviewers Usually Probe

• The candidate demonstrates understanding of binary search and efficient substring counting.
• Look for an explanation of how to manage the time complexity when dealing with large values of n and k.
• Candidates who struggle with edge cases like k being too large or impossible to reach may need further guidance.

Common Pitfalls or Variants

Common pitfalls

• Overlooking the fact that not all t values will result in valid substrings; carefully managing how many valid substrings are generated is crucial.
• Focusing on brute-force approaches without considering binary search may lead to inefficient solutions for larger inputs.
• Not considering cases where it's impossible to reach k valid substrings, leading to a failure in returning -1.

Follow-up variants

• Changing the order array to a randomized sequence might add difficulty by altering the index replacement order.
• Increasing k beyond feasible limits could require optimizations for large values.
• Extending the problem to handle non-alphabetic characters or mixed-case strings could introduce new complexities.

FAQ

What is the best approach to solve the Minimum Time to Activate String problem?

The most efficient approach is to use binary search on time, checking for valid substrings at each step.

How does binary search apply to this problem?

Binary search is used to find the minimum time t that generates at least k valid substrings after replacing characters in the order array.

What happens if the number of valid substrings is less than k?

If it's impossible to achieve k valid substrings, return -1.

How do I efficiently count valid substrings?

You can count valid substrings by marking positions where '' appears and calculating the number of substrings containing at least one ' ' in a single pass.

What is the time complexity of the solution?

The time complexity is O(n log n), where n is the length of the string.