Solution Approach
Solution Overview
There are 3 main approaches to solve this problem:
- Sorting Approach - Complex approach
- Min Heap Approach - Complex approach
- QuickSelect Approach - Complex approach
Approach 1: Sorting Approach
Let's start by understanding the problem: we need to find the kth largest element in an unsorted array.
Thinking Process: The most straightforward approach is to sort the array in descending order and return the element at index k-1. This is a simple and intuitive solution that directly follows the problem statement.
Intuition: If we sort the array in descending order, the kth largest element will be at index k-1. This approach is easy to implement but may not be the most efficient for large arrays or when k is small compared to the array size.
Algorithm:
- Sort the array in descending order.
- Return the element at index k-1.
Time Complexity:
Where n is the length of the array. Sorting the array takes O(n log n) time.
Space Complexity:
We only use a constant amount of extra space regardless of the input size (assuming the sort is in-place).
Approach 2: Min Heap Approach
We can optimize the previous approach by using a min heap of size k to find the kth largest element.
Thinking Process: Instead of sorting the entire array, we can maintain a min heap of size k. We iterate through the array and add each element to the heap. If the heap size exceeds k, we remove the smallest element. After processing all elements, the top of the heap will be the kth largest element.
Intuition: A min heap of size k allows us to efficiently maintain the k largest elements in the array. The smallest element in the heap (the root) will be the kth largest element in the array. This approach is more efficient than sorting when k is much smaller than the array size.
Algorithm:
- Create a min heap of size k.
- Iterate through the array:
- a. Add each element to the heap.
- b. If the heap size exceeds k, remove the smallest element.
- Return the top of the heap (the kth largest element).
Time Complexity:
Where n is the length of the array and k is the value of k. Adding each element to the heap takes O(log k) time, and we do this for n elements, resulting in O(n log k) time.
Space Complexity:
We need O(k) space to store the heap of size k.
Approach 3: QuickSelect Approach
We can further optimize the solution using the QuickSelect algorithm to find the kth largest element in O(n) average time.
Thinking Process: The QuickSelect algorithm is a selection algorithm to find the kth smallest element in an unordered list. We can adapt it to find the kth largest element by either finding the (n-k+1)th smallest element or by modifying the partitioning step to work with descending order.
The algorithm works by selecting a pivot element and partitioning the array such that all elements less than the pivot are on the left, and all elements greater than the pivot are on the right. The pivot is then in its final sorted position. If the pivot's position is the kth position, we've found our answer. Otherwise, we recursively apply the algorithm to the appropriate partition.
Intuition: The QuickSelect algorithm is more efficient than sorting when we only need to find the kth element. It has an average time complexity of O(n), which is better than the O(n log n) time complexity of sorting.
Algorithm:
- Define a partition function that selects a pivot and partitions the array such that all elements less than the pivot are on the left, and all elements greater than the pivot are on the right.
- Apply the QuickSelect algorithm:
- a. Choose a pivot element.
- b. Partition the array around the pivot.
- c. If the pivot's position is the kth position, return the pivot.
- d. If the pivot's position is less than the kth position, recursively apply the algorithm to the right partition.
- e. If the pivot's position is greater than the kth position, recursively apply the algorithm to the left partition.
Time Complexity:
Where n is the length of the array. The QuickSelect algorithm has an average time complexity of O(n), although its worst-case time complexity is O(n²).
Space Complexity:
We only use a constant amount of extra space regardless of the input size (assuming the partition is in-place).
Pseudocode
123function findKthLargest(nums, k): sort nums in descending order return nums[k-1]String Problems
Learning Objective
Understand different approaches to solve the element rank finder problem and analyze their efficiency.
Solution Approaches
Approach 1: Sorting Approach
Let's start by understanding the problem: we need to find the kth largest element in an unsorted array.
Thinking Process: The most straightforward approach is to sort the array in descending order and return the element at index k-1. This is a simple and intuitive solution that directly follows the problem statement.
Intuition: If we sort the array in descending order, the kth largest element will be at index k-1. This approach is easy to implement but may not be the most efficient for large arrays or when k is small compared to the array size.
Algorithm:
- Sort the array in descending order.
- Return the element at index k-1.
Approach 2: Min Heap Approach
We can optimize the previous approach by using a min heap of size k to find the kth largest element.
Thinking Process: Instead of sorting the entire array, we can maintain a min heap of size k. We iterate through the array and add each element to the heap. If the heap size exceeds k, we remove the smallest element. After processing all elements, the top of the heap will be the kth largest element.
Intuition: A min heap of size k allows us to efficiently maintain the k largest elements in the array. The smallest element in the heap (the root) will be the kth largest element in the array. This approach is more efficient than sorting when k is much smaller than the array size.
Algorithm:
- Create a min heap of size k.
- Iterate through the array:
- a. Add each element to the heap.
- b. If the heap size exceeds k, remove the smallest element.
- Return the top of the heap (the kth largest element).
Approach 3: QuickSelect Approach
We can further optimize the solution using the QuickSelect algorithm to find the kth largest element in O(n) average time.
Thinking Process: The QuickSelect algorithm is a selection algorithm to find the kth smallest element in an unordered list. We can adapt it to find the kth largest element by either finding the (n-k+1)th smallest element or by modifying the partitioning step to work with descending order.
The algorithm works by selecting a pivot element and partitioning the array such that all elements less than the pivot are on the left, and all elements greater than the pivot are on the right. The pivot is then in its final sorted position. If the pivot's position is the kth position, we've found our answer. Otherwise, we recursively apply the algorithm to the appropriate partition.
Intuition: The QuickSelect algorithm is more efficient than sorting when we only need to find the kth element. It has an average time complexity of O(n), which is better than the O(n log n) time complexity of sorting.
Algorithm:
- Define a partition function that selects a pivot and partitions the array such that all elements less than the pivot are on the left, and all elements greater than the pivot are on the right.
- Apply the QuickSelect algorithm:
- a. Choose a pivot element.
- b. Partition the array around the pivot.
- c. If the pivot's position is the kth position, return the pivot.
- d. If the pivot's position is less than the kth position, recursively apply the algorithm to the right partition.
- e. If the pivot's position is greater than the kth position, recursively apply the algorithm to the left partition.
Complexity Analysis
Sorting Approach Approach
Time Complexity
Where n is the length of the array. Sorting the array takes O(n log n) time.
Space Complexity
We only use a constant amount of extra space regardless of the input size (assuming the sort is in-place).
Min Heap Approach Approach
Time Complexity
Where n is the length of the array and k is the value of k. Adding each element to the heap takes O(log k) time, and we do this for n elements, resulting in O(n log k) time.
Space Complexity
We need O(k) space to store the heap of size k.
QuickSelect Approach Approach
Time Complexity
Where n is the length of the array. The QuickSelect algorithm has an average time complexity of O(n), although its worst-case time complexity is O(n²).
Space Complexity
We only use a constant amount of extra space regardless of the input size (assuming the partition is in-place).
Pseudocode
Sorting Approach Approach
123function findKthLargest(nums, k): sort nums in descending order return nums[k-1]Min Heap Approach Approach
1234567function findKthLargest(nums, k): create a min heap for each num in nums: add num to the heap if heap size > k: remove the smallest element from the heap return the top of the heapQuickSelect Approach Approach
123456789101112131415161718192021222324252627function findKthLargest(nums, k): return quickSelect(nums, 0, nums.length - 1, k - 1) function quickSelect(nums, left, right, k): if left == right: return nums[left] pivotIndex = partition(nums, left, right) if pivotIndex == k: return nums[pivotIndex] else if pivotIndex < k: return quickSelect(nums, pivotIndex + 1, right, k) else: return quickSelect(nums, left, pivotIndex - 1, k) function partition(nums, left, right): pivot = nums[right] i = left for j from left to right - 1: if nums[j] >= pivot: swap nums[i] and nums[j] i++ swap nums[i] and nums[right] return i