Problem Description

The problem provides us with an array of 4 digits. Our task is to use these digits to form the latest possible valid 24-hour format time. The 24-hour time format is represented as "HH:MM", where "HH" is the hour part ranging from 00 to 23 and "MM" is the minute part ranging from 00 to 59. We must use each given digit exactly once. If it's possible to form a valid time, we should return it as a string in the "HH:MM" format; otherwise, if no valid combination exists to make a proper time, we should return an empty string.

The key considerations here include ensuring that the hours are within the valid range (00 to 23) and the minutes fall within the right range (00 to 59).

Intuition

To solve this problem, the solution adopts a brute-force strategy; it iterates over all possible permutations of the given four digits and checks which of these permutations can form a valid time. The idea here is that there are only 24 distinct permutations for 4 digits, which is a small and manageable number to examine exhaustively.

Here's the breakdown of the solution approach:

• The solution uses three nested loops to iterate over all possible combinations of the four digits without repetition. Each loop variable (i, j, k) represents an index for a digit in the array arr.
• To ensure no digit is repeated, the conditions i != j, i != k, and j != k are enforced.
• Out of these three chosen indices, the hour can be formed by combining the digits at index i and j. The minute is then formed by combining the digits at index k and the remaining digit (which is not used so far). The remaining index is calculated as 6 - i - j - k, leveraging the fact that the indices are 0, 1, 2, 3 for a 4-element array and their sum is 6.
• The hours (h) are valid if they are less than 24, and the minutes (m) are valid if they are less than 60.
• If a combination forms a valid time, it's converted to total minutes (h * 60 + m) to compare easily with other valid times found so far. The maximum value is maintained as ans.
• If no valid time is found, ans remains -1, and an empty string is returned.
• If a valid time is found, it is formatted correctly by converting the total minutes back into hours and minutes and then constructing a string with a proper "HH:MM" format using string formatting with padding zeroes where needed.

Thus, the solution explores all permutations to find the maximum possible valid time strictly using the 24-hour time constraints and efficiently formats the final answer.

Solution Approach

The implementation uses a simple backtracking algorithm to generate all permutations of the input array and validates each one as a potential 24-hour time.

To begin, a loop runs four times, from 0 to 3, corresponding to the first digit of the hour. Inside it, two nested loops run again from 0 to 3 to account for the second digit of the hour and the first digit of the minutes, respectively. The criteria for each of these loops to continue iterating are that their indices must not be equal (i != j != k) to avoid using the same digit more than once.

The digits for the hour are created by the expression arr[i] * 10 + arr[j], effectively placing arr[i] as the tens digit and arr[j] as the ones digit. Similarly, the minutes are created by the expression arr[k] * 10 + arr[6 - i - j - k], where 6 - i - j - k gives us the one unused index to complete the pair needed for the minutes.

Once h (hours) and m (minutes) are calculated, they're validated to check if they represent a valid time (i.e., h < 24 and m < 60). If the time is valid, we calculate the total minutes since 00:00 using h * 60 + m and update ans with the maximum of itself and the computed total minutes. This effectively stores the latest valid time seen so far.

After all permutations are examined, the solution checks if ans has been updated from its initial value of -1. If it hasn't, this means no valid time was found, and an empty string is returned. If ans has been updated, the total minutes are split back into hours and minutes. The formatting for the output string is done using string interpolation where ans // 60 gives the hours and ans % 60 gives the minutes. The :02 formatter is used to pad the output with leading zeroes if needed, ensuring the correct "HH:MM" format is maintained.

Though not explicitly stated, this implementation assumes that mathematical operations and comparisons are constant-time operations, and since there are only 4! (factorial of 4 i.e., 24) possible permutations, the overall time complexity of the algorithm is O(1) - although it's technically a brute-force solution, its complexity is bound by a small constant because the input size is fixed at 4.

In summary, the implementation utilizes:

• A brute-force approach to generate all possible permutations using nested loops.
• Mathematical operations and checks to construct valid 24-hour times from the permutations.
• A running maximum (ans) to determine the latest valid time.
• String formatting and modularity to convert the total minutes back into the required "HH:MM" output format.

Example Walkthrough

Let's say the input array of 4 digits is [1, 8, 3, 2]. We are tasked with finding the latest valid time that can be formed using these digits. Using the solution approach described above, we would proceed as follows:

1. Start by initializing the ans variable to -1 to track the latest valid time in minutes.

2. Begin iterating over all possible permutations of these four digits using three nested loops.

3. For the first permutation, we might start with i=0, j=1, k=2. Here, i, j, and k correspond to indices in the digits' array, so the digits would be arr[0] = 1, arr[1] = 8, arr[2] = 3.

4. Since we need two digits for the hour (HH) and two for the minutes (MM), we choose the first two for the hour and the last two for the minute. Here, i and j are for the hours, and k and the unused index for the minutes.

5. The hours would be arr[i] * 10 + arr[j], which is 1 * 10 + 8 = 18. The minutes would be arr[k] * 10 + arr[6 - i - j - k], which is 3 * 10 + arr[6 - 0 - 1 - 2] = 30 + arr[3] = 30 + 2 = 32. The time so far is 18:32.

6. The time 18:32 is a valid 24-hour format time since 18 < 24 and 32 < 60. Therefore, we calculate the total minutes as 18 * 60 + 32 = 1152 and update ans with this value since it's the first and thus the maximum found so far.

7. We continue this process, generating and checking each permutation. Some permutations would be invalid, for instance, if the chosen indices led to hours >= 24 or minutes >= 60; these permutations are ignored.

8. Assume that after checking all permutations, the largest value for ans remains 1152. This would be the latest valid time in terms of total minutes from 00:00.

9. We convert 1152 back into hours and minutes. 1152 // 60 gives us 19 hours and 1152 % 60 gives us 12 minutes.

10. We format these back into the string representation "19:12", padding the hours or minutes with a leading zero if necessary to adhere to the "HH:MM" format. However, in this case, no padding is needed.

Finally, the latest valid time we can form with the input array [1, 8, 3, 2] is "19:12", so this is the string we would return. If we had found no valid permutations, we would return an empty string.

Solution Implementation

Time and Space Complexity

The given code snippet is designed to find the largest possible time that can be displayed on a 24-hour clock, using digits from a given array of four integers.

Time Complexity:

The time complexity of the code is O(1) because the number of iterations is fixed and does not depend on the input size, which is always 4 elements in the arr list. Specifically, the number of iterations would be a fixed number since we are dealing with three nested loops each running for four possible options but skipping the indexes that have already been used (i != j and i != k and j != k). Therefore, the maximum number of iterations is bounded by 4 * 3 * 2 = 24 for each of the three loops (since one number is used as the first element of the hour, one as the first element of the minute, and one as the second element of the minute, while the remaining is the second element of the hour). Even though some iterations are skipped, 24 is the upper limit, which means the time complexity does not depend on the input size but is rather constant.

Space Complexity:

The space complexity of the algorithm is O(1). The extra space used by the algorithm does not grow with the input size. The variables i, j, k, h, m, and ans use a constant amount of space, and there are no additional data structures that grow with input size.

Thus, regardless of the input, the space used by the program remains constant.

Learn more about how to find time and space complexity quickly using problem constraints.