Kth Largest Element in an Array

Problem Description

Find the kth largest element in an integer array. Note that it is the kth largest element in the sorted order, not the kth distinct element. The goal is to achieve a solution without fully sorting the array.

Key Insights

• kth largest element can be converted to finding the element at index n-k if the array were sorted.
• A min-heap of size k is a practical method to maintain the k largest elements seen so far.
• The Quickselect algorithm can find the kth largest element in average O(n) time with in-place partitioning.
• Choose an appropriate method based on performance needs and worst-case scenarios.

Space and Time Complexity

Time Complexity:

• Min-Heap: O(n log k) by maintaining a heap of size k.
• Quickselect: Average O(n) but worst-case O(n^2) with poor pivot selection. Space Complexity:
• Min-Heap: O(k)
• Quickselect: O(1) additional space with in-place partitioning.

Solution

We can solve the problem using either a min-heap or the Quickselect algorithm.

1. Min-Heap Approach:

   • Initialize a min-heap with the first k elements.
   • Iterate through the remaining elements; if an element is greater than the heap’s root (the smallest in the heap), replace the root.
   • After processing all elements, the root of the min-heap is the kth largest element.

2. Quickselect Approach:

   • Choose a pivot and partition the array so that elements larger than the pivot come before it.
   • Determine the position of the pivot; if it matches the index corresponding to the kth largest element, return it.
   • Recurse into the appropriate part of the array.

Key points to consider include maintaining heap size for the first approach and proper pivot selection for the Quickselect approach.

Code Solutions