Minimum Operations to Form Subsequence With Target Sum

C++
Java

Problem Description

So, you want to form a subsequence with a target sum? Well, welcome to the club! It’s like trying to find the last piece of a jigsaw puzzle that your cat decided to play with. The problem is simple yet infuriating: given an array of integers, you need to determine the minimum number of operations required to form a subsequence that sums up to a given target.

Imagine you’re at a party, and everyone is trying to form a conga line (the subsequence) to reach the snack table (the target sum). But wait! Some party-goers (the numbers in the array) are too busy dancing and can’t join the line. Your job is to figure out how to get the maximum number of people in line with the least amount of effort.

Code Solution


import collections
import math

class Solution:
    def minOperations(self, nums: list[int], target: int) -> int:
        kNoMissingBit = 31
        maxBit = 31
        ans = 0
        minMissingBit = kNoMissingBit
        count = collections.Counter(int(math.log2(num)) for num in nums)

        for bit in range(maxBit):
            if target >> bit & 1:
                if count[bit] > 0:
                    count[bit] -= 1
                else:
                    minMissingBit = min(minMissingBit, bit)
            if minMissingBit != kNoMissingBit and count[bit] > 0:
                count[bit] -= 1
                ans += bit - minMissingBit
                minMissingBit = kNoMissingBit
            count[bit + 1] += count[bit] // 2

        return ans if minMissingBit == maxBit else -1

Approach Explanation

The code uses a bit manipulation approach to determine how many operations are needed to form the target sum. It counts the occurrences of each power of two in the input numbers and checks if the bits required for the target sum can be formed using these numbers. If a required bit is missing, it looks for the smallest available bit to “break” and use it instead, counting the operations needed along the way.

Time and Space Complexity

Complexity Type	Complexity
Time Complexity	O(n)
Space Complexity	O(1)

Real-World Example

Imagine you’re trying to save money for a vacation (the target sum). You have a piggy bank (the array of integers) filled with coins of different denominations. Each time you want to buy a souvenir, you need to break down your coins (operations) to reach the exact amount needed. The challenge is to do this with the least hassle, just like forming that perfect subsequence!