[Solved] You are given an array of positive integers and a target sum K.... Which of the following approaches guarantees an optimal solution with polynomial time complexity? | Subsets & Combinations

Subsets & Combinations - Count of Subsets With Sum K

You are given an array of positive integers and a target sum K. You need to find how many subsets of the array sum exactly to K. Which of the following approaches guarantees an optimal solution with polynomial time complexity?

AGreedy algorithm that picks the largest elements first until the sum reaches or exceeds K

BPure recursion that tries all subsets without memoization

CDynamic Programming using a bottom-up tabulation approach that counts subsets for all sums up to K

DSorting the array and using two pointers to find pairs that sum to K

Step-by-Step Solution

Solution:

Step 1: Understand problem constraints
The problem requires counting all subsets summing to K, which involves exploring combinations, not just pairs or greedy picks.
Step 2: Evaluate approaches
Greedy and two-pointer methods fail because they do not consider all subsets. Pure recursion is correct but exponential. Bottom-up DP efficiently counts subsets for all sums up to K, ensuring polynomial time.
Final Answer:
Option C -> Option C
Quick Check:
DP tabulation counts subsets for all sums -> polynomial time [OK]

Quick Trick: Counting subsets with sum K requires DP, not greedy or two-pointer [OK]

Common Mistakes:

MISTAKES

Assuming greedy or two-pointer works for subset sums
Confusing counting subsets with finding pairs

Trap Explanation:

PITFALL

Greedy and two-pointer approaches look simpler but fail to consider all subset combinations, making them incorrect for counting subsets.

Interviewer Note:

CONTEXT

Tests if candidate can identify the correct algorithmic pattern beyond brute force or greedy.

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Which of the following approaches guarantees an optimal solution with polynomial time complexity?

Step 1: Understand problem constraints

Step 2: Evaluate approaches

Final Answer:

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