Sort a nearly sorted (or K sorted) array
Last Updated :
11 Feb, 2025
Given an array arr[] and a number k . The array is sorted in a way that every element is at max k distance away from it sorted position. It means if we completely sort the array, then the index of the element can go from i – k to i + k where i is index in the given array. Our task is to completely sort the array.
Examples:
Input: arr= [6, 5, 3, 2, 8, 10, 9], k = 3
Output: [2, 3, 5, 6, 8, 9, 10]
Input: arr[]= [1, 4, 5, 2, 3, 6, 7, 8, 9, 10], k = 2
Output: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[Naive Approach] By sorting – O(n*log n) Time and O(1) Space
We can simply use any sorting algorithm of O(n*log n) time complexity to achieve sorted array.
[Expected Approach] Using Heap – O(n*log k) Time and O(k) Space
The main idea is to sort the array efficiently from left to right by considering elements from the right side. For each position i, we find the minimum element in the range [i, i + k] with help of min-heap. we don’t need to check numbers from left side as they are already at their correct positions.
Step by Step implementation:
- Push the first k elements of the array into the min-heap. These elements form the initial search range for sorting the first few indices.
- Iterate through the remaining elements of the array (from index k to n-1). For each element:
- Push it into the min-heap.
- The heap now contains k + 1 elements, so remove the smallest element (top of the heap) and place it at its correct position in the array (arr[i-k]).
- After processing all elements in the array, the heap still contains the last k elements. Extract these elements one by one and place them in the remaining positions of the array.
C++
// c++ program to sort k sorted array
#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;
// function to sorts a nearly sorted array
// where every element is at most
// k positions away from its target position.
void nearlySorted(vector<int> &arr, int k) {
// length of array
int n = arr.size();
// creating a min heap
priority_queue<int, vector<int>, greater<int>> pq;
// pushing first k elements in pq
for (int i = 0; i < k; i++)
pq.push(arr[i]);
int i;
for (i = k; i < n; i++) {
pq.push(arr[i]);
// size becomes k+1 so pop it
// and add minimum element in (i-k) index
arr[i - k] = pq.top();
pq.pop();
}
// puting remaining elements in array
while (!pq.empty()) {
arr[i - k] = pq.top();
pq.pop();
i++;
}
}
int main() {
vector<int> arr = {6, 5, 3, 2, 8, 10, 9};
int k = 3;
nearlySorted(arr, k);
for (int x : arr)
cout << x << ' ';
return 0;
}
// Java program to sort a nearly sorted array
import java.util.*;
class GfG {
// Function to sort a nearly sorted array
// where every element is at most
// k positions away from its target position.
static void nearlySorted(int[] arr, int k) {
// Length of array
int n = arr.length;
// Creating a min heap
PriorityQueue<Integer> pq = new PriorityQueue<>();
// Pushing first k elements in pq
for (int i = 0; i < k; i++)
pq.add(arr[i]);
int i;
for (i = k; i < n; i++) {
pq.add(arr[i]);
// Size becomes k+1 so pop it
// and add minimum element in (i-k) index
arr[i - k] = pq.poll();
}
// Putting remaining elements in array
while (!pq.isEmpty()) {
arr[i - k] = pq.poll();
i++;
}
}
public static void main(String[] args) {
int k = 3;
int[] arr = {6, 5, 3, 2, 8, 10, 9};
nearlySorted(arr, k);
for (int x : arr) {
System.out.print(x + " ");
}
}
}
# A Python program to sort a
# nearly sorted array.
import heapq
# Function to sort a nearly sorted array
# where every element is at most
# k positions away from its target position.
def nearlySorted(arr, k):
# Length of array
n = len(arr)
# Creating a min heap
pq = []
# Pushing first k elements in pq
for i in range(k):
heapq.heappush(pq, arr[i])
i = k
index = 0
while i < n:
heapq.heappush(pq, arr[i])
# Size becomes k+1 so pop it
# and add minimum element in (index) position
arr[index] = heapq.heappop(pq)
i += 1
index += 1
# Putting remaining elements in array
while pq:
arr[index] = heapq.heappop(pq)
index += 1
if __name__ == "__main__":
k = 3
arr = [6, 5, 3, 2, 8, 10, 9]
nearlySorted(arr, k)
print(" ".join(map(str, arr)))
// javascript program to sort k sorted array
class MinHeap {
constructor() {
this.heap = [];
}
push(val) {
this.heap.push(val);
this.heapifyUp();
}
pop() {
if (this.heap.length === 1) return this.heap.pop();
const min = this.heap[0];
this.heap[0] = this.heap.pop();
this.heapifyDown();
return min;
}
heapifyUp() {
let index = this.heap.length - 1;
while (index > 0) {
let parentIndex = Math.floor((index - 1) / 2);
if (this.heap[parentIndex] <= this.heap[index]) break;
[this.heap[parentIndex], this.heap[index]] = [this.heap[index], this.heap[parentIndex]];
index = parentIndex;
}
}
heapifyDown() {
let index = 0;
while (true) {
let left = 2 * index + 1;
let right = 2 * index + 2;
let smallest = index;
if (left < this.heap.length && this.heap[left] < this.heap[smallest]) {
smallest = left;
}
if (right < this.heap.length && this.heap[right] < this.heap[smallest]) {
smallest = right;
}
if (smallest === index) break;
[this.heap[index], this.heap[smallest]] = [this.heap[smallest], this.heap[index]];
index = smallest;
}
}
isEmpty() {
return this.heap.length === 0;
}
}
// Function to sort a nearly sorted array
// where every element is at most k positions away from its target position.
function nearlySorted(arr, k) {
let n = arr.length;
let pq = new MinHeap();
// Pushing first k elements in pq
for (let i = 0; i < k; i++) {
pq.push(arr[i]);
}
let index = 0;
for (let i = k; i < n; i++) {
pq.push(arr[i]);
// Size becomes k+1 so pop it
// and add minimum element in (index) position
arr[index] = pq.pop();
index++;
}
// Putting remaining elements in array
while (!pq.isEmpty()) {
arr[index] = pq.pop();
index++;
}
}
// Driver code
let arr = [6, 5, 3, 2, 8, 10, 9];
let k = 3;
nearlySorted(arr, k);
console.log(arr.join(" "));
Note: We can also use a Balanced Binary Search Tree instead of a Heap to store k+1 elements. The insert and delete operations on Balanced BST also take O(log k) time. So Balanced BST-based method will also take O(n log k) time, but the Heap based method seems to be more efficient as the minimum element will always be at the root. Also, Heap doesn’t need extra space for left and right pointers.
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