Number Of Subsequences That Satisfy The Given Sum Condition [Leetcode]

In this blog post, we will delve into LeetCode problem 1498, Number Of Subsequences That Satisfy The Given Sum Condition This problem falls under the category of "Two Pointers" and is classified as a medium difficulty challenge.

It is worth noting that this problem has been encountered by candidates in interviews with companies such as Amazon.

Problem Overview

The problem statement is as follows: You are given an array of integers nums and an integer target.

Your task is to return the number of non-empty subsequences of nums such that the sum of the minimum and maximum element in the subsequence is less than or equal to target.

Since the answer may be a large number, you are required to return it modulo 10^9 + 7.

Example 1:

Let's understand this problem through an example:

Input:

nums = [3, 5, 6, 7]
target = 9

Output:

4


Explanation: In this example, there are 4 subsequences that satisfy the condition:
1. [3] -> Min value + max value <= target (3 + 3 <= 9)
2. [3, 5] -> (3 + 5 <= 9)
3. [3, 5, 6] -> (3 + 6 <= 9)
4. [3, 6] -> (3 + 6 <= 9)

Example 2:

Input:

nums = [3, 3, 6, 8]
target = 10

Output:

6


Explanation: In this example, there are 6 subsequences that satisfy the condition.

Notice that the input array nums can contain repeated numbers.

Example 3:

Input:

nums = [2, 3, 3, 4, 6, 7]
target = 12

Output:

61


Explanation: In this example, there are 63 non-empty subsequences, but two of them do not satisfy the condition ([6, 7], [7]).

So, the number of valid subsequences is 61.

Understanding Constraints

Before we dive into the solution, let's take a closer look at the constraints of this problem:

• 1 <= nums.length <= 10^5
• 1 <= nums[i] <= 10^6
• 1 <= target <= 10^6

These constraints inform us that the input array nums can be quite large and that the values within it can range from 1 to 10^6. Therefore, we need to design an efficient algorithm to handle such inputs.

Efficient Approach with Two Pointers

Now, let's explore an efficient approach to solving this problem.

def numSubseq(nums, target):
    nums.sort()  # Sort the input array

    res, mod = 0, (10**9 + 7)

    left, right = 0, len(nums) - 1
    while left <= right:
        if (nums[left] + nums[right]) > target:
            right -= 1
        else:
            res += 1 << (right - left)
            left += 1
    return res % mod

In this approach, we start by sorting the input array nums.

Sorting is crucial because it allows us to easily determine the minimum and maximum elements within a subsequence.

We initialize two pointers, left and right, pointing to the beginning and end of the sorted array, respectively.

These pointers will help us create subsequences efficiently.

The main idea is to find pairs of elements (one from the left side of the array and one from the right side) such that their sum is less than or equal to the target.

By counting the number of valid pairs, we can calculate the total number of subsequences that meet the condition.

The loop continues as long as the left pointer is less than or equal to the right pointer.

We compare the sum of the elements pointed to by left and right with the target.

If the sum is greater than the target, we decrement the right pointer to find a smaller element from the right side.

If the sum is less than or equal to the target, it means we can form valid subsequences.

The key insight is that for each element at the left pointer, we can include multiple elements from the right side to create valid subsequences.

The number of such elements is determined by right - left.

So, for each left pointer position, we add 2^(right – left) to the result.

Finally, we return the result modulo 10^9 + 7 to handle the large number of subsequences.

Python Code Solution

Now that we have discussed the efficient approach, let's provide the Python code solution for the problem:

def numSubseq(nums, target):
    nums.sort()  # Sort the input array

    res, mod = 0, (10**9 + 7)

    left, right = 0, len(nums) - 1
    while left <= right:
        if (nums[left] + nums[right]) > target:
            right -= 1
        else:
            res += 1 << (right - left)
            left += 1
    return res % mod

This Python function, numSubseq, takes two arguments: nums, the input array of integers, and target, the target value for the condition.

It returns the number of valid subsequences satisfying the given condition.

Time and Space Complexity

Let's analyze the time and space complexity of our solution:

• Time Complexity: O(n log n)
  • The dominant factor in the time complexity is the sorting of the input array, which takes O(n log n) time, where n is the length of the array.
  • The two-pointer approach after sorting runs in linear time, as we traverse the array once.
• Space Complexity: O(1)
  • We use a constant amount of extra space for variables like res, mod, left, and right.

Therefore, the space complexity is O(1).

Reasoning Behind Our Approach

Our approach is based on the observation that sorting the input array allows us to easily find pairs of elements that meet the condition.

By using two pointers and counting the number of subsequences for each minimum element, we can efficiently compute the total number of valid subsequences.

Sorting the array is a key step because it allows us to determine the minimum and maximum elements within a subsequence efficiently.

This simplifies the process of finding pairs that meet the condition.

We also use a two-pointer approach to traverse the sorted array, which helps us find the valid subsequences without redundantly considering pairs that cannot satisfy the condition.

The use of bit manipulation (the << operator) allows us to calculate the number of subsequences for each minimum element without explicitly iterating through all possibilities.

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Conclusion

In this blog post, we've explored LeetCode problem 1498, Number Of Subsequences That Satisfy The Given Sum Condition We've provided a Python solution that

efficiently calculates the number of non-empty subsequences meeting the specified condition.

By sorting the input array and using a two-pointer approach, we can handle large inputs with ease.

If you found this explanation helpful or have any questions or suggestions, please feel free to comment and share your thoughts.

Solving coding challenges like this one can be a rewarding experience for both beginners and experienced programmers, and it's an essential skill for technical interviews.

You can access the original problem on LeetCode at the following link: Number of Subsequences That Satisfy The Given Sum Condition.

Happy coding!