2389. Longest Subsequence With Limited Sum

Problem Description

The problem gives us two integer arrays: nums which is of length n, and queries which is of length m. Our task is to find, for each query, the maximum size of a subsequence that can be extracted from nums such that the sum of its elements does not exceed the value given by the corresponding element in queries.

It's important to recognize what a subsequence is. A subsequence is a new sequence generated from the original array where some (or no) elements are deleted, but the order of the remaining elements is retained. It's also worth noticing that there might be more than one subsequence that meets the condition for a query, and among those, we are looking to report the length of the largest one.

For example, suppose nums = [4, 5, 2], and we have a single-element queries = [6]. The subsequences of nums that have sums less than or equal to 6 are [4], [5], [2], and [4, 2]. The longest of these is [4, 2], which has a length of 2.

Intuition

To solve this problem, we first need to sort the nums array. Sorting will help us easily find subsequences that maximize size while minimizing their sum—since we'll have guaranteed that we're always adding the smallest possible numbers (from the sorted array) to reach the sum required in the queries.

Next, we calculate the prefix sum of the nums array after sorting it. This prefix sum array (s) represents the sum of elements of nums up to and including the ith element. So the first element of the prefix sum will be equal to the first element of nums, the second will be the sum of the first two elements of nums, and so on.

Once we have the prefix sum array, we can easily determine, for each query, how many elements can be taken from the nums array without exceeding the query sum. We do this by finding the rightmost index in the prefix sum array that is less than or equal to the query value. This index tells us the number of elements that can be included in the subsequence for a particular query.

The bisect_right function from Python's bisect module is a binary search function which returns the index where an element should be inserted to maintain the order. In our case, for each query, it finds the first element in the sorted prefix sum array s that is greater than the query value—thus, the index we get is actually the length of the subsequence, because array indices are 0-based in Python.

Our final answer is an array of lengths, each corresponding to the maximum subsequence lengths that can be picked for each query without exceeding their sums.

The beauty of this approach is that by utilizing the sorted array and prefix sums, the time-consuming task of checking each possible subsequence is avoided, leading to a much more efficient solution.

Learn more about Greedy, Binary Search, Prefix Sum and Sorting patterns.

Solution Approach

The implementation of the solution can be broken down into the following steps:

1. Sort the Input Array: We start by sorting nums in ascending order. This arrangement guarantees that when we choose subsequences, we are choosing the smallest available elements first, which helps us maximize the number of elements in the subsequence for any given sum.

   1nums.sort()

2. Calculate Prefix Sums: We then compute a prefix sum array s from the sorted array. The prefix sum is a common technique used in algorithm design for quickly finding the sum of elements in a subarray. In Python, the accumulate function from the itertools module can compute the prefix sums efficiently.

   1s = list(accumulate(nums))

3. Search with Binary Search: For each query, we use the bisect_right function to perform a binary search over the prefix sum array. This binary search will find the point where the query value (the sum we are not supposed to exceed) would be inserted to keep the array s sorted. Since arrays in Python are 0-indexed, the insertion index found by bisect_right directly corresponds to the number of elements we can sum up from the sorted nums array to not exceed the query value. This is equivalent to the length of the required subsequence.

   1[bisect_right(s, q) for q in queries]

The implementation uses the following concepts:

• Sorting: An essential part of this approach is to sort the nums array. Sorting helps us with a greedy-like approach to keep adding the smallest elements to reach the subsequence sum.

• Prefix Sums: The prefix sum array is constructed to have quick access to the sum of numbers from the start of the nums array up to any given point.

• Binary Search: The bisect_right function helps us to quickly find the rightmost location where a given query value would fit in the prefix sum array, which translates directly to the answer of how many elements can be used.

• Greedy Approach: The sorted array in combination with the prefix sums represents a greedy choice to choose the smallest elements to form subsequences.

The overall complexity of the algorithm is dominated by the sort operation, which is (O(n \log n)), where n is the length of the nums array. The prefix sum and binary searches for each query are (O(n)) and (O(\log n)) respectively, which are eclipsed by the sort operation when analyzing overall algorithm performance.

Example Walkthrough

Let's take small example arrays to walk through the solution approach described.

Suppose we have nums = [3, 7, 5, 6] and queries = [7, 12, 5].

1. Sort the Input Array: We sort nums in ascending order.

   1nums.sort()  # nums becomes [3, 5, 6, 7]

2. Calculate Prefix Sums: Next, we calculate the prefix sums of the sorted nums array.

   1s = list(accumulate(nums))  # s becomes [3, 8, 14, 21]

3. Search with Binary Search: We then perform binary searches on s by using bisect_right for each query.

   For the query value 7:

   1bisect_right(s, 7)  # returns 2 because '7' could be inserted at index 2 to keep 's' sorted

   The subsequence [3, 5] sums up to 8, which is greater than 7. Thus, we only take the first element [3], resulting in a max subsequence length of 1.

   For the query value 12:

   1bisect_right(s, 12)  # returns 3 because '12' could be inserted at index 3 to keep 's' sorted

   The subsequence [3, 5, 6] sums up to 14, exceeding 12. We exclude the last element and get [3, 5], resulting in a max subsequence length of 2.

   For the query value 5:

   1bisect_right(s, 5)  # returns 1 because '5' could be inserted at index 1 to keep 's' sorted

   Here, only the first element [3] is less than or equal to 5, giving us a max subsequence length of 1.

The final answer for this example would be an array [1, 2, 1] corresponding to the max subsequence lengths for each query.

Solution Implementation

Time and Space Complexity

The given code snippet defines a function answerQueries which takes a list of numbers nums and a list of queries queries, then returns a list of integers. Let's analyze the time and space complexity of this function:

Time Complexity

1. nums.sort(): Sorting the nums list has a time complexity of O(n log n), where n is the length of nums.
2. list(accumulate(nums)): The accumulate function computes the prefix sums of the sorted nums list. Since it goes through each element once, it has a time complexity of O(n).
3. [bisect_right(s, q) for q in queries]: For each query q in queries, the bisect_right function performs a binary search on the list s of prefix sums. Each binary search operation has a time complexity of O(log n), and there are m queries, so the overall time complexity for this step is O(m log n), where m is the length of queries.

Combining these, the overall time complexity is O(n log n) + O(n) + O(m log n). Since O(n log n) is the dominant term, the time complexity simplifies to O(n log n + m log n).

Space Complexity

1. Sorting the list nums in-place doesn't require additional space, so its space complexity is O(1).
2. The list s of prefix sums would require additional space proportional to the length of nums. Hence, the space complexity for this step is O(n).
3. The list comprehension for bisect_right does not use additional space apart from the output list, which contains m elements. So, the space complexity for the output list is O(m).

Since O(n) and O(m) are not part of the same operation, we consider both for the overall space complexity. Thus, the overall space complexity of the function is O(n + m).

Learn more about how to find time and space complexity quickly using problem constraints.