Minimum Operations to Form Subsequence With Target Sum

Minimum Operations to Form Subsequence With Target Sum

1. What is this problem about?

You are given an array of integers that are all powers of 2 (1, 2, 4, 8, ...) and a target sum. You want to form a subsequence whose elements sum up to target. If you can't, you can perform an operation: take a number n in the array (where n > 1), remove it, and add two numbers n/2 to the array. You want to find the minimum number of these "splits" required.

2. Why is this asked in interviews?

This "Minimum Operations to Form Subsequence With Target Sum coding problem" is a high-level challenge from Media.net. It tests deep understanding of binary representation and greedy strategies. Since the numbers are all powers of 2, the problem is essentially about satisfying each bit of the target's binary representation using the available powers of 2.

3. Algorithmic pattern used

The pattern is Greedy with Bit Manipulation.

1. Represent the target in binary.
2. Count the frequency of each power of 2 available in the array.
3. Iterate through bits of the target from smallest to largest.
4. If a bit is set, try to use the available powers of 2 to satisfy it. If you don't have enough, you must find the next largest power of 2 available and "split" it down to the current bit.

4. Example explanation

• Target: 12 (binary 1100), Array: [8, 2]
• Bit 2 (value 4) is needed. We have 8 and 2.
• 2 is too small. We must split 8 into 4 and 4. (1 split)
• Now we have [4, 4, 2]. We use one 4.
• Bit 3 (value 8) is needed. We only have [4, 2] left.
• We need 4 more. We can't split 4, but we can combine things? No, the greedy logic says if we can't form the target, we keep splitting. Actually, in this case, 8 + 2 = 10 < 12, so it's impossible. But if the array was [16, 2], we would split 16 to 8, then split 8 to 4 to satisfy the bits.

5. Common mistakes candidates make

A major error is not realizing that smaller powers can be combined to satisfy a larger bit. For example, two 4s can satisfy the "8" bit. Another mistake is spliting a number more than necessary or not splitting the smallest possible large number, which would minimize operations.

6. Interview preparation tip

When dealing with powers of 2, always think in binary. Each power of 2 corresponds to a single bit. If you have a shortage at one bit, look at higher bits to split or lower bits to combine. This "bit-by-bit" satisfaction is a powerful technique for power-of-2 problems.