Input: nums = [5, 2, 4, 3, 1, 6], k = 3
Output: 4

In Plain English

New to this one? Start here. Below is the core idea in everyday language, plus a real-world analogy—then the sections that follow build on it with the formal reasoning, code, and step-by-step walkthroughs.

Intuition - Min-Heap

A straightforward solution to this problem is to sort the array in reverse order and return the number at the (k - 1)th index:

This solution takes $O(n\log(n))$ time, but as we only need the kth largest integer, sorting all the elements in the input array might not be necessary. Let’s explore some solutions which take this into account.

An interesting thing to realize is that if we know what the top k largest integers are, we’d know that the smallest of these would be the kth largest integer:

This leads to the idea that instead of sorting the entire array, we can keep track of the top k largest integers in the array. Is there a way to maintain the top k largest integers, while having access to the smallest of these integers?

A min-heap seems like it should work well for this purpose because it provides efficient access to the smallest element. Let’s explore how to use it to maintain the top k integers in the array.

Consider the below example with k = 3. Let’s begin by pushing the first k integers to the heap:

We shouldn’t immediately push the next element, 3, in the heap because the heap already contains k integers. This gives us a choice:

• Skip 3, or:
• Remove an integer from the heap and push 3 in.

The smallest integer in the heap is 2. Integer 3 is more likely to belong in the top k integers than 2 is, so let’s pop 2 from the heap and push 3 in:

The next integer is 1, which is smaller than the smallest integer in the heap. So, we know it’s not among the top k integers:

The final integer is 6. It’s larger than the smallest integer in the heap. So, let’s pop off the top of the heap and push in 6:

At this point, the k integers in the heap represent the top k largest integers from the array, with the smallest of these integers located at the top of the heap. This smallest value is the kth largest integer in the entire array, so we can just return this integer from the top of the heap:

Implementation - Min-Heap

from typing import List
import heapq
    
def kth_largest_integer_min_heap(nums: List[int], k: int) -> int:
    min_heap = []
    heapq.heapify(min_heap)
    for num in nums:
        # Ensure the heap has at least 'k' integers.
        if len(min_heap) < k:
            heapq.heappush(min_heap, num)
        # If 'num' is greater than the smallest integer in the heap, pop off this
        # smallest integer from the heap and push in 'num'.
        elif num > min_heap[0]:
            heapq.heappop(min_heap)
            heapq.heappush(min_heap, num)
    return min_heap[0]

int kthLargestIntegerMinHeap(vector<int> nums, int k) {
    // A min-heap: the smallest element is always on top.
    priority_queue<int, vector<int>, greater<int>> minHeap;
    for (int num : nums) {
        if ((int)minHeap.size() < k) {
            minHeap.push(num);                     // not full yet
        } else if (num > minHeap.top()) {          // beats the smallest kept
            minHeap.pop();
            minHeap.push(num);
        }
    }
    return minHeap.top();                          // smallest of the top k
}

int kthLargestIntegerMinHeap(int[] nums, int k) {
    // A min-heap: the smallest element is always on top.
    java.util.PriorityQueue<Integer> minHeap = new java.util.PriorityQueue<>();
    for (int num : nums) {
        if (minHeap.size() < k) {
            minHeap.offer(num);                    // not full yet
        } else if (num > minHeap.peek()) {         // beats the smallest kept
            minHeap.poll();
            minHeap.offer(num);
        }
    }
    return minHeap.peek();                         // smallest of the top k
}

Complexity Analysis

Time complexity: The time complexity of kth_largest_integer_min_heap is $O(n\log(k))$ because for each integer, we perform at most one push and pop operation on the min-heap, which has a size no larger than $k$. Each heap operation takes $O(\log(k))$ time.

Space complexity: The space complexity is $O(k)$ because the heap can grow to a size of $k$.

Intuition - Quickselect

The previous approach involved keeping track of the k largest integers to identify the kth largest integer. This effectively required us to keep these k values partially sorted through the use of a heap. But what if there’s a way to position only the kth largest value in its expected sorted position, without sorting the other numbers?

Quickselect can be used to achieve this.

Quickselect is an algorithm that leverages the partition step of quicksort, which positions a value in its sorted position. Quickselect is generally used to find the kth smallest element, whereas, in this problem, we're asked to find the kth largest element. To utilize quickselect for this purpose, we can instead find the (n - k)th smallest integer, which is equivalent to finding the kth largest element:

Now, let’s dive into how quickselect works. To understand quickselect, we recommend you study the quicksort solution provided in the Sort Array problem.

Quickselect works identically to quicksort in that we:

1. Choose a pivot.
2. Partition the array into two parts by moving elements around the pivot so that:

• Numbers less than the pivot are moved to its left.
• Numbers greater than the pivot are moved to its right.

The primary difference between these two algorithms lies in the recursion step:

• In quicksort, we recursively process both the left and right parts.
• In quickselect, we only need to recursively process one of these parts.

Let’s explore why only one part is chosen.

Deciding which part of a partition to process
Suppose we pick a random number as a pivot. After performing a partition, this pivot will be positioned correctly. From here, there are three possibilities for the position of the (n - k)th smallest integer:

• If the pivot is positioned before index n - k, it means the (n - k)th smallest integer must be somewhere to the right of the pivot. So, perform quickselect on the right part:

  

• If the pivot is positioned after index n - k, perform quickselect on the left part:

  

• If the pivot is positioned exactly at index n - k, the pivot itself is the (n - k)th smallest integer. So, we just return the pivot:

  

Implementation - Quickselect

Note that the partition function in this implementation is identical to the partition function in Sort Array. Also, this implementation selects a random pivot, as per the optimization discussed in Sort Array.

import random
from typing import List
    
def kth_largest_integer_quickselect(nums: List[int], k: int) -> int:
    return quickselect(nums, 0, len(nums) - 1, k)
    
def quickselect(nums: List[int], left: int, right: int, k: int) -> None:
    n = len(nums)
    if left >= right:
        return nums[left]
    random_index = random.randint(left, right)
    nums[random_index], nums[right] = nums[right], nums[random_index]
    pivot_index = partition(nums, left, right)
    # If the pivot comes before 'n - k', the ('n - k')th smallest integer is somewhere
    # to its right. Perform quickselect on the right part.
    if pivot_index < n - k:
        return quickselect(nums, pivot_index + 1, right, k)
    # If the pivot comes after 'n - k', the ('n - k')th smallest integer is somewhere
    # to its left. Perform quickselect on the left part.
    elif pivot_index > n - k:
        return quickselect(nums, left, pivot_index - 1, k)
    # If the pivot is at index 'n - k', it's the ('n - k')th smallest integer.
    else:
        return nums[pivot_index]
    
def partition(nums: List[int], left: int, right: int) -> int:
    pivot = nums[right]
    lo = left
    for i in range(left, right):
        if nums[i] < pivot:
            nums[lo], nums[i] = nums[i], nums[lo]
            lo += 1
    nums[lo], nums[right] = nums[right], nums[lo]
    return lo

int partitionArray(vector<int>& nums, int left, int right) {
    int pivot = nums[right];
    int lo = left;
    for (int i = left; i < right; i++) {
        if (nums[i] < pivot) {
            int t = nums[lo]; nums[lo] = nums[i]; nums[i] = t;
            lo++;
        }
    }
    int t = nums[lo]; nums[lo] = nums[right]; nums[right] = t;
    return lo;
}

int quickselect(vector<int>& nums, int left, int right, int target) {
    if (left == right) {
        return nums[left];
    }
    int r = left + rand() % (right - left + 1);
    int t = nums[r]; nums[r] = nums[right]; nums[right] = t;
    int p = partitionArray(nums, left, right);
    if (p == target) {
        return nums[p];
    } else if (p < target) {
        return quickselect(nums, p + 1, right, target);
    } else {
        return quickselect(nums, left, p - 1, target);
    }
}

int kthLargestIntegerQuickselect(vector<int> nums, int k) {
    int target = (int)nums.size() - k;   // (n-k)-th smallest
    return quickselect(nums, 0, (int)nums.size() - 1, target);
}

int kthLargestIntegerQuickselect(int[] nums, int k) {
    int target = nums.length - k;        // (n-k)-th smallest
    return quickselect(nums, 0, nums.length - 1, target);
}

int quickselect(int[] nums, int left, int right, int target) {
    if (left == right) {
        return nums[left];
    }
    int r = left + (int)(Math.random() * (right - left + 1));
    int t = nums[r]; nums[r] = nums[right]; nums[right] = t;
    int p = partitionArray(nums, left, right);
    if (p == target) {
        return nums[p];
    } else if (p < target) {
        return quickselect(nums, p + 1, right, target);
    } else {
        return quickselect(nums, left, p - 1, target);
    }
}

int partitionArray(int[] nums, int left, int right) {
    int pivot = nums[right];
    int lo = left;
    for (int i = left; i < right; i++) {
        if (nums[i] < pivot) {
            int t = nums[lo]; nums[lo] = nums[i]; nums[i] = t;
            lo++;
        }
    }
    int t = nums[lo]; nums[lo] = nums[right]; nums[right] = t;
    return lo;
}

Complexity Analysis

Time complexity: The time complexity of kth_largest_integer_quickselect can be analyzed in terms of the average and worst cases:

• Average case: $O(n)$. In the average case, quickselect partitions the array and reduces the problem size by approximately half each time by performing a recursive call on only one part of each partition. A linear partition is performed during each of these recursive calls. This results in a total time complexity of $O(n) + O(n/2) + O(n/4) + O(n/8) + \dots$, which is a geometric series that sums to $O(n)$.

• Worst case: $O(n^2)$. The worst-case scenario occurs when the pivot selection consistently results in extremely unbalanced partitions. This can result in the problem size only being reduced by one element after each partition, leading to a total time complexity of $O(n)+O(n-1)+O(n-2)+…$, which can be simplified to an $O(n^2)$ time complexity.

Space complexity: The space complexity can also be analyzed in terms of the average and worst cases:

• Average Case: $O(\log(n))$. In the average case, the depth of the recursive call stack is approximately $\log_2(n)$.

• Worst Case: $O(n)$. In the worst case, the depth of the recursive call stack can be as deep as $n$.

Dry Runs

Two step-by-step walkthroughs of the algorithm below: a valid case (the expected happy path) and an invalid / edge case that exercises the tricky parts. Each step shows the program state as the code runs—the same steps apply to Python, C++ and Java.