The sum of all elements between k1'th and k2'th smallest elementsIntroduction Efficiently calculating the sum of items between the k1'th and k2'th smallest elements in an array is a key challenge in several fields, including data analysis, algorithm design, and statistics. In this, we'll look at three different approaches to tackling this challenge. We will look at the traditional sorting methodology, a selection algorithmbased method, and a segment treebased solution. Each technique has distinct tradeoffs in terms of time, space, and implementation complexity. For example, consider the array below: arr = [7, 10, 4, 3, 20, 15] We want to calculate the sum of all entries in this array that are between the second and fifth smallest. Solution: 1. Sorting the array in nondecreasing order yields: arr = [3, 4, 7, 10, 15, 20] 2. Retrieve the k1th and k2th smallest elements: Given that k1 = 2 and k2 = 5, the second smallest element (k1'th) is 4, and the fifth smallest element (k2'th) is 15. 3. Calculate the sum of components between the k1 and k2 smallest items:
So, in this case, the sum of all items between the second and fifth smallest entries in the array [7, 10, 4, 3, 20, 15] is 36. Method 1: SortingBased Approach The sortingbased method is a simple way to calculate the sum of items between an array's k1th and k2th smallest members. Let's look at each stage of this strategy and see how it works: 1. Sort the Array: The first step is to sort the array in nondecreasing order. It can be accomplished with efficient sorting algorithms like quick sort or merges ort. Sorting the array makes it easy to identify the k1'th and k2'th smallest items, which will be positioned at indices k11 and k21 after sorting. 2. Get the k1 and k2 smallest elements: After sorting the array, we may access the k1'th and k2'th smallest items from their corresponding places. These items reflect the lower and upper limits of the range for which we must compute the total. 3. Calculate the Sum: To calculate the sum, identify the k1'th and k2'th smallest items in the sorted array. Then, sum all elements between these limits, inclusive. It entails iterating through the sorted array and adding elements between [k11, k21]. Pseudocode Time Complexity:
Implementation Output: Explanation:
Method 2: Selection AlgorithmBased Approach The selection algorithmbased technique is an effective way to locate the k1 and k2 smallest items in an array without fully sorting it. Let's break down the phases of this technique to see how it works. 1. Implement a selection algorithm.
2. Traverse the array to calculate the sum:
3. Complexity analysis This method is faster than the sortingbased strategy, particularly for bigger datasets, because Quickselect has an anticipated temporal complexity of O(n). However, the recursive nature of the selection method necessitates a more complex implementation than the sortingbased technique. Pseudocode Implementation Output: Explanation
Method 3: Segment TreeBased Approach The segment treebased approach is an effective method for effectively resolving range queries on an array. In this approach, we preprocess the array to create a segment tree, which is a binary tree with each node containing information on a specified range of elements from the original array. Each node in the segment tree represents a range of indices and keeps aggregated information about the items in that range, such as the sum, minimum, and maximum. Here's how to use the segment treebased technique to compute the sum of items between the k1'th and k2'th smallest elements in an array: 1. Build a Segment Tree:
2. Query the segment tree:
3. Complexity analysis The segment treebased technique provides efficient range query capabilities with a time complexity of O(log n) per query after initial preprocessing. Although creating the segment tree requires O(n log n) time, subsequent searches may be executed quickly, making this approach acceptable for cases requiring repeated range queries. Pseudocode Implementation Output: Explanation 1. Building the Segment Tree: During the startup phase, we create a segment tree from the provided array. The segment tree is created recursively. At each node of the tree, we record the sum of elements inside the appropriate indices. 2. Querying the segment tree:
3. Example usage:
