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Sort Integers by The Power Value

Number: 1488

Difficulty: Medium

Paid? No

Companies: Bloomberg, Adobe, Google

Problem Description

Given three integers lo, hi, and k, the task is to sort all integers in the interval [lo, hi] based on their "power value" in ascending order. The power value of an integer x is defined as the number of steps required to transform x to 1 using the following rules:

  1. If x is even, then x becomes x / 2.
  2. If x is odd, then x becomes 3 * x + 1.

If two numbers have the same power value, they are sorted by their numerical value in ascending order. The goal is to return the kth integer in this sorted list.

Key Insights

  • The computation of the power value for any number x involves simulating the process until x becomes 1.
  • Memoization can be used to optimize the repeated computations of power values.
  • After computing the power value for all numbers in the interval, sorting the numbers based on a tuple of (power value, integer) achieves the desired order.

Space and Time Complexity

Time Complexity: O(n * cost) where n = hi - lo + 1 and cost is the number of steps per number (which is usually small due to memoization). Space Complexity: O(n + m) where m is the additional space used for memoization of computed power values.

Solution

The solution uses a helper function to compute the power value of a given number using recursion with memoization. For each number in the interval [lo, hi]:

  1. Compute the power value.
  2. Store the number and its corresponding power value.
  3. Sort the list by (power value, number) using a stable sort.
  4. Retrieve the kth integer from the sorted list based on the problem's requirements.

The main data structures used are:

  • A dictionary (or map) for memoization.
  • A list (or vector/array) to store the pairs (number, power value) before sorting.

The algorithmic approach leverages recursion and memoization to efficiently compute the power values and then a sort to get the final order.

Code Solutions

# Python solution for "Sort Integers by The Power Value"

# Memoization dictionary to store computed power values
memo = {1: 0}

def get_power(x):
    # If the power of x is already computed, return it from memo
    if x in memo:
        return memo[x]
    # Recursively compute the next step based on whether x is even or odd
    if x % 2 == 0:
        memo[x] = 1 + get_power(x // 2)
    else:
        memo[x] = 1 + get_power(3 * x + 1)
    return memo[x]

def getKth(lo, hi, k):
    # Build list of tuples (power, number)
    power_list = [(get_power(x), x) for x in range(lo, hi + 1)]
    # Sort by power value first, then by the number itself if powers are equal
    power_list.sort()
    # Since k is 1-indexed, return the kth element's number
    return power_list[k - 1][1]

# Example usage:
print(getKth(12, 15, 2))  # Expected output: 13
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