Question

Please answer by mathematical language:

An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location.

Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time Ω(n log k), You may assume that n is a multiple of k.

Answer 1

We can sort such arrays more efficiently with the help of Heap data structure. Following is the detailed process that uses Heap.
1) Create a Min Heap of size k+1 with first k+1 elements. This will take O(k) time (See this GFact)
2) One by one remove min element from heap, put it in result array, and add a new element to heap from remaining elements.

Removing an element and adding a new element to min heap will take Logk time. So overall complexity will be O(k) + O((n-k)*logK).

**Program in c++

#include <bits/stdc++.h>

using namespace std;

   

// Given an array of size n, where every element

// is k away from its target position, sorts the

// array in O(nLogk) time.

int sortK(int arr[], int n, int k)

{

    // Insert first k+1 items in a priority queue (or min heap)

    //(A O(k) operation). We assume, k < n.

    priority_queue<int, vector<int>, greater<int> > pq(arr, arr + k + 1);

   

    // i is index for remaining elements in arr[] and index

    // is target index of for current minimum element in

    // Min Heapm 'hp'.

    int index = 0;

    for (int i = k + 1; i < n; i++) {

        arr[index++] = pq.top();

        pq.pop();

        pq.push(arr[i]);

    }

   

    while (pq.empty() == false) {

        arr[index++] = pq.top();

        pq.pop();

    }

}

   

// A utility function to print array elements

void printArray(int arr[], int size)

{

    for (int i = 0; i < size; i++)

        cout << arr[i] << " ";

    cout << endl;

}

   

// Driver program to test above functions

int main()

{

    int k = 3;

    int arr[] = { 2, 6, 3, 12, 56, 8 };

    int n = sizeof(arr) / sizeof(arr[0]);

    sortK(arr, n, k);

   

    cout << "Following is sorted arrayn";

    printArray(arr, n);

   

    return 0;

}

Output:

Following is sorted array
2 3 6 8 12 56

The Min Heap based method takes O(nLogk) time and uses O(k) auxiliary space.