Combination Sum Iv Leetcode Problem 377 [Python Solution]

Welcome back!

Today, we're going to dive into the fascinating world of dynamic programming by solving the LeetCode problem, Combination Sum Iv This problem falls under the category of 1-D Dynamic Programming and is classified as a medium-level challenge.

By the end of this guide, you'll have a solid understanding of the problem, its constraints, a brute force approach, and an efficient dynamic programming solution in Python.

Problem Overview

Question: Given an array of distinct integers nums and a target integer target, your task is to determine the number of possible combinations that add up to the given target.

The catch here is that different sequences of the same values that sum up to the target are counted as different combinations.

Let's illustrate this with an example.

Example 1:

Suppose you have the following input:

• nums = [1, 2, 3]
• target = 4

Your goal is to find the number of possible combinations that add up to the target.

Here's what the combinations look like:

1. (1, 1, 1, 1)
2. (1, 1, 2)
3. (1, 2, 1)
4. (1, 3)
5. (2, 1, 1)
6. (2, 2)
7. (3, 1)

In this case, different sequences, such as (1, 1, 2) and (2, 1, 1), are counted as distinct combinations.

Example 2:

Let's consider another example:

• nums = [9]
• target = 3

In this scenario, it's impossible to form any combination that adds up to the target.

Therefore, the expected output is 0.

Constraints:

Before we delve into the solution, it's essential to understand the problem's constraints:

• 1 <= nums.length <= 200
• 1 <= nums[i] <= 1000
• All elements of nums are unique.
• 1 <= target <= 1000

Efficient Python Code Solution

Now, let's get into the actual Python code solution for the Combination Sum Iv problem.

We'll be using dynamic programming to tackle this challenge.

def combinationSum4(nums, target):
    # Initialize a dictionary to store intermediate results
    dp = {0: 1}

    # Iterate through the range of 1 to target
    for total in range(1, target + 1):
        dp[total] = 0
        for n in nums:
            dp[total] += dp.get(total - n, 0)
    return dp[target]

This Python function efficiently computes the number of combinations by leveraging dynamic programming.

Here's a breakdown of the code:

1. We initialize a dictionary dp where keys represent the remaining target values we want to compute, and values represent the number of ways to reach that target value.

We start with 0, which means there's one way to reach a target of 0 (by not choosing any number).

1. We loop through each total from 1 to target (inclusive).

For each total, we initialize it as 0 because we haven't counted any combinations yet.

1. Now, the most crucial part of our dynamic programming approach is nested within this loop.

We iterate through the elements in nums to consider each number as a potential choice to form combinations.

For each number n in nums, we update dp[total] by adding the value of dp[total - n].

1. Here, dp.get(total - n, 0) is a critical part of the code.

It retrieves the number of ways to reach total - n from our dictionary dp.

If total - n is not in the dictionary (meaning we can't reach that value using our current numbers), we return 0.

1. Finally, we return the value of dp[target, which contains the total number of combinations to reach the target using the provided nums.

Time and Space Complexity

Understanding the time and space complexity of your code is essential.

In this solution, we're using dynamic programming, which results in the following complexities:

• Time Complexity: O(target * n), where target is the target value and n is the length of the nums array.

We have to consider each value in the nums array for every possible target value from 1 to target.

• Space Complexity: O(target), as our dynamic programming dictionary dp stores results for each possible target value.

Reasoning Behind Our Approach

In this dynamic programming approach, we break down the problem into smaller subproblems.

Instead of repeatedly calculating the same subproblems, we store the results in our dp dictionary, which allows us to avoid redundant calculations and significantly improve efficiency.

The basic idea is to calculate the number of combinations for each target value, starting from 1 and working our way up to the given target.

By considering each number in nums as a potential choice and utilizing the results of previous subproblems stored in dp, we can efficiently compute the final answer.

This approach optimizes the time complexity and makes the solution both elegant and efficient.

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Conclusion

In this guide, we've explored the Combination Sum Iv problem on LeetCode.

We've provided a clear problem overview, discussed constraints, and presented an efficient Python solution using dynamic programming.

Understanding this approach and its time and space complexity is crucial for tackling similar problems in the world of competitive programming and software development.

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Happy coding!

Question Link: Combination Sum IV on LeetCode