Sort an Increasing-Decreasing Array

Given a K-increasing-decreasing array arr[], the task is to sort the given array. An array is said to be K-increasing-decreasing if elements repeatedly increase upto a certain index after which they decrease, then again increase, a total of K times. The diagram below shows a 4-increasing-decreasing array.

Example:



Input: arr[] = {57, 131, 493, 294, 221, 339, 418, 458, 442, 190}
Output: 57 131 190 221 294 339 418 442 458 493

Input: arr[] = {1, 2, 3, 4, 3, 2, 1}
Output: 1 1 2 2 3 3 4

Approach: The brute-force approach is to sort the array without taking advantage of the k-increasing-decreasing property. The time complexity of this approach is O(n logn) where n is the length of the array.

If k is significantly smaller than n, a better approach can be found with less time complexity. For example, if k=2, the input array consists of two subarrays, one increasing, the other decreasing. Reversing the second subarray yields two sorted arrays and the result is then merged which can be done in O(n) time. Generalizing, we could first reverse the order of each of the decreasing subarrays. For example, in the above figure, the array could be decomposed into four sorted arrays as {57, 131, 493}, {221, 294}, {339, 418, 458} and {190, 442}. Now the min-heap technique can be used to merge these sorted arrays.

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// A pair of pairs, first element is going to
// store value, second element index of array
// and third element index in the array
typedef pair<int, pair<int, int> > ppi;
  
// This function takes an array of arrays as an
// argument and all arrays are assumed to be
// sorted
// It merges them together and returns the
// final sorted output
vector<int> mergeKArrays(vector<vector<int> > arr)
{
    vector<int> output;
  
    // Create a min heap with k heap nodes
    // Every heap node has first element of an array
    priority_queue<ppi, vector<ppi>, greater<ppi> > pq;
  
    for (int i = 0; i < arr.size(); i++)
        pq.push({ arr[i][0], { i, 0 } });
  
    // Now one by one get the minimum element
    // from min heap and replace it with next
    // element of its array
    while (pq.empty() == false) {
        ppi curr = pq.top();
        pq.pop();
  
        // i ==> Array Number
        // j ==> Index in the array number
        int i = curr.second.first;
        int j = curr.second.second;
  
        output.push_back(curr.first);
  
        // The next element belongs to same array as
        // current
        if (j + 1 < arr[i].size())
            pq.push({ arr[i][j + 1], { i, j + 1 } });
    }
  
    return output;
}
  
// Function to sort the alternating
// increasing-decreasing array
vector<int> SortKIncDec(const vector<int>& A)
{
    // Decompose the array into a
    // set of sorted arrays
    vector<vector<int> > sorted_subarrays;
    typedef enum { INCREASING,
                   DECREASING } SubarrayType;
    SubarrayType subarray_type = INCREASING;
    int start_idx = 0;
    for (int i = 0; i <= A.size(); i++) {
  
        // If the current subarrays ends here
        if (i == A.size()
            || (A[i - 1] < A[i]
                && subarray_type == DECREASING)
            || (A[i - 1] >= A[i]
                && subarray_type == INCREASING)) {
  
            // If the subarray is increasing
            // then add from the start
            if (subarray_type == INCREASING) {
                sorted_subarrays.emplace_back(A.cbegin() + start_idx,
                                              A.cbegin() + i);
            }
  
            // If the subarray is decreasing
            // then add from the end
            else {
                sorted_subarrays.emplace_back(A.crbegin()
                                                  + A.size() - i,
                                              A.crbegin()
                                                  + A.size()
                                                  - start_idx);
            }
            start_idx = i;
            subarray_type = (subarray_type == INCREASING
                                 ? DECREASING
                                 : INCREASING);
        }
    }
  
    // Merge the k sorted arrays`
    return mergeKArrays(sorted_subarrays);
}
  
// Driver code
int main()
{
    vector<int> arr = { 57, 131, 493, 294, 221,
                        339, 418, 458, 442, 190 };
  
    // Get the sorted array
    vector<int> ans = SortKIncDec(arr);
  
    // Print the sorted array
    for (int i = 0; i < ans.size(); i++)
        cout << ans[i] << " ";
  
    return 0;
}

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Output:

57 131 190 221 294 339 418 442 458 493

Time Complexity: O(n*logk), where n is the length of array.



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