Combination Sum I Explained in Depth with Intuition and Code Solution

When diving into algorithmic problems, the challenge of finding combinations of numbers that sum up to a specific target often arises. One such problem is known as Combination Sum I, a classic problem that tests our understanding of recursion and backtracking. In this post, we’ll explore the problem in-depth, break down its intuition, and provide a clear code solution.

Problem Statement

The Combination Sum I problem can be formally stated as follows:

Given an array of distinct integers candidates and a target integer target, return all unique combinations of candidates where the chosen numbers sum up to target. You may use an element from candidates as many times as needed. The solution set must not contain duplicate combinations.

Example

Let’s consider the following example for better understanding:

• Input: candidates = [2, 3, 6, 7], target = 7
• Output: [[2, 2, 3], [7]]

In this example, the combinations that sum to 7 are [2, 2, 3] and [7].

Intuition Behind the Solution

To tackle the Combination Sum I problem, we can utilize a backtracking approach. The intuition here is to explore all possible combinations of numbers that may lead us to the target sum. The backtracking algorithm will allow us to build combinations incrementally and backtrack whenever we exceed the target.

Steps Involved:

1. Start with an empty combination: We’ll begin with an empty combination and a target sum to reach.
2. Iterate through the candidates: For each candidate, we can either include it in our combination or skip it.
3. Recursive exploration: If we include a candidate, we will continue exploring with the same candidate (since we can reuse it) and a reduced target.
4. Check for completion: If our current combination sums to the target, we add it to our results.
5. Backtrack: If the current sum exceeds the target, we backtrack to explore other combinations.

Code Solution

Now, let’s translate our thought process into code. Below is a Python implementation using backtracking:

def combination_sum(candidates, target):
    def backtrack(start, path, target):
        # Base case: if target is achieved
        if target == 0:
            result.append(path)
            return
        # Iterate over the candidates
        for i in range(start, len(candidates)):
            # If the candidate exceeds the target, break the loop
            if candidates[i] > target:
                break
            # Include the candidate in the path and continue
            backtrack(i, path + [candidates[i]], target - candidates[i])
    
    candidates.sort()  # Sort candidates to facilitate early stopping
    result = []
    backtrack(0, [], target)
    return result

# Example usage
candidates = [2, 3, 6, 7]
target = 7
print(combination_sum(candidates, target))

Explanation of the Code:

• We define a helper function backtrack that takes the current starting index, the current path of numbers, and the remaining target.
• We check if the target is zero, indicating a valid combination, and store it in our results.
• We loop through the candidates starting from the current index, checking if adding a candidate would surpass the target.
• If it doesn’t, we recursively call backtrack with the updated path and reduced target.

Conclusion

The Combination Sum I problem beautifully demonstrates the power of backtracking in finding all possible combinations that meet a certain criterion. By carefully managing the state and employing recursion, we can explore a vast search space efficiently.

Feel free to test the code with different inputs and explore how the backtracking algorithm adapts to various scenarios. Happy coding!

Top Comments

1. User A: “This explanation of backtracking really helped clarify the approach! Thanks for breaking it down step by step.”
2. User B: “I love how you included the code snippet. It makes it so much easier to understand the implementation!”
3. User C: “The example provided was super helpful. I was confused at first, but now I feel more confident tackling similar problems!”
4. User D: “Great post! I appreciate the thought process behind the solution. Looking forward to more algorithm explanations!”

By sharing insights and code solutions like this, we can further our understanding of algorithmic challenges while helping others on their coding journey.

Related Posts

• LeetCode - Two Sum Solution with Code