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Minimum Operations to Form Subsequence With Target Sum

Difficulty: Hard

Problem Description

You are given a 0-indexed array nums consisting of non-negative powers of 2, and an integer target. In one operation, you must apply the following changes to the array: choose an element nums[i] such that nums[i] > 1, remove nums[i] from the array, and add two occurrences of nums[i] / 2 to the end of nums. Return the minimum number of operations you need to perform so that nums contains a subsequence whose elements sum to target. If it is impossible to obtain such a subsequence, return -1.

Key Insights

  • The elements of nums are powers of 2, which allows for straightforward manipulation through halving.
  • To reach the target, we need to consider both the initial elements and those created through operations.
  • A greedy approach can be used to maximize the contribution of larger powers of 2 first, as they can be halved into smaller powers.
  • A careful counting of the number of operations is needed to determine the minimum required.

Space and Time Complexity

Time Complexity: O(n + log(target)), where n is the number of elements in nums. We iterate through nums and potentially through the powers of 2 up to target. Space Complexity: O(1), as we are using a fixed amount of additional space regardless of input size.

Solution

To solve this problem, we can use a greedy approach combined with a priority queue (or a simple list sorted in descending order). The idea is to always try to use the largest available element in nums to achieve the target. If the largest element is too large, we can keep halving it until we reach a usable size.

  1. Count occurrences of each power of 2 in nums.
  2. Starting from the largest power of 2, try to form the target by subtracting from it the largest available elements.
  3. If we reach a point where we cannot form the target, we perform operations to halve larger elements and add them back.
  4. Keep track of the number of operations performed and return it if we can achieve the target, otherwise return -1.

Code Solutions

def minOperations(nums, target):
    from collections import Counter
    count = Counter(nums)
    operations = 0
    
    # Convert to a sorted list of unique powers of 2
    powers = sorted(count.keys(), reverse=True)
    
    for power in powers:
        while count[power] > 0 and target >= power:
            target -= power
            count[power] -= 1
            
        while target < power and power > 1:
            count[power] -= 1
            operations += 1
            half_power = power // 2
            count[half_power] += 2
            
    return -1 if target > 0 else operations
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