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Sort Integers by The Power Value

The power of an integer x is defined as the number of steps needed to transform x into 1 using the following steps:

  • if x is even then x = x / 2
  • if x is odd then x = 3 * x + 1

For example, the power of x = 3 is 7 because 3 needs 7 steps to become 1 (3 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1).

Given three integers lo, hi and k. The task is to sort all integers in the interval [lo, hi] by the power value in ascending order, if two or more integers have the same power value sort them by ascending order.

Return the kth integer in the range [lo, hi] sorted by the power value.

Notice that for any integer x (lo <= x <= hi) it is guaranteed that x will transform into 1 using these steps and that the power of x is will fit in a 32-bit signed integer.

 

Example 1:

Input: lo = 12, hi = 15, k = 2
Output: 13
Explanation: The power of 12 is 9 (12 --> 6 --> 3 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1)
The power of 13 is 9
The power of 14 is 17
The power of 15 is 17
The interval sorted by the power value [12,13,14,15]. For k = 2 answer is the second element which is 13.
Notice that 12 and 13 have the same power value and we sorted them in ascending order. Same for 14 and 15.

Example 2:

Input: lo = 7, hi = 11, k = 4
Output: 7
Explanation: The power array corresponding to the interval [7, 8, 9, 10, 11] is [16, 3, 19, 6, 14].
The interval sorted by power is [8, 10, 11, 7, 9].
The fourth number in the sorted array is 7.

 

Constraints:

  • 1 <= lo <= hi <= 1000
  • 1 <= k <= hi - lo + 1

Solution Explanation for LeetCode 1387: Sort Integers by The Power Value

This problem requires sorting integers within a given range based on their "power value." The power value of an integer x is the number of steps it takes to reduce x to 1 using the following rules:

  • If x is even, x = x / 2.
  • If x is odd, x = 3 * x + 1.

The solution involves these steps:

  1. Power Value Calculation: A function f(x) is defined to compute the power value of any integer x. This function iteratively applies the rules until x becomes 1, counting the steps. Memoization (using @cache in Python or equivalent techniques in other languages) is crucial for efficiency, as many numbers will share common subsequences in their transformation to 1.

  2. Sorting: The integers in the range [lo, hi] are sorted. The sorting is done based on a custom comparison function. This comparison function prioritizes the power value; if two numbers have the same power value, it then sorts them in ascending order. This ensures that the kth element is found correctly.

  3. Result: The kth element of the sorted array is returned.

Time and Space Complexity Analysis

  • Time Complexity: The dominant factor is the sorting process. A standard sorting algorithm (like merge sort or quicksort used implicitly by Python's sorted or Java's Arrays.sort) has a time complexity of O(N log N), where N is the number of integers in the range [lo, hi]. Calculating the power value for each number is O(M) in the worst case, where M is the maximum number of steps required to reach 1 from any number in the range. The memoization significantly improves the overall time complexity by avoiding redundant calculations. While the exact maximum value of M is not easily determined mathematically and depends on the specific sequence of numbers, in practice M is quite small relative to N (e.g., below 200 for the input constraints) and doesn't significantly alter the dominant O(N log N). Therefore, the overall time complexity is effectively O(N log N).

  • Space Complexity: The space complexity is determined by the storage needed for the array of integers (O(N)) and, importantly, the space used by the memoization cache in f(x). In the worst case, the memoization cache could store the power values of all integers in the range [lo, hi], but again, because of the nature of the power value calculation and memoization, the overall space usage will likely remain linear with respect to N. Therefore, the space complexity is O(N).

Code Examples (with detailed explanations)

The code examples provided earlier demonstrate the solution in multiple programming languages. The key elements in each are:

  • f(x) function: This function is responsible for calculating the power value efficiently using memoization (or dynamic programming).
  • Sorting: The sorted function (Python), Arrays.sort (Java), sort (C++, Go), nums.sort (TypeScript), or nums.sort_by (Rust) is used with a custom comparator to handle the dual sorting criteria (power value then numerical value).

The memoization aspect of the f(x) function (using @cache in Python or similar methods in other languages) is essential for efficiency. Without it, the time complexity would be significantly worse, potentially reaching O(N * M), where M could be large for some sequences. However, due to memoization, repeated calculations are avoided, giving us the near-optimal O(N log N) performance.