## Wave Array Difficulty: Easy

You are given an unsorted array of integers( ``` arr ``` ) of length ``` n ``` , write a program to sort it in wave form.

Problem Note:

• The array elements in the resultant array must be such that ``` arr >= arr <= arr >= arr <= arr ..... ```
• If there are multiple sorted orders in wave form, return the one which is lexicographically smallest.
• The array may contain duplicates.

Example 1

``````Input: arr = [5, 2, 9, 3, 2]
Output: [2, 2, 5, 3, 9]
Explanation: In the above example, you can see 2 >= 2 <= 5 >= 3 <= 9. Thus we get, arr = [2, 2, 5, 3, 9] as output which is sorted in wave form. ``````

Example 2

``````Input: arr = [3, 2, 9, 6, 4, 1]
Output: [2, 1, 4, 3, 9, 6]
Explanation: In the above example, you can see 2 >= 1 <= 4 >= 3 <= 9 >= 6. Thus we get, arr = [2, 1, 4, 3, 9, 6] as output which is sorted in wave form.``````

Example 3

``````Input: arr = [4, 2, 9, 1, 21, 43, 24]
Output: [2, 1, 9, 4, 24, 21, 43]
Explanation: In the above example, you can see 2 >= 1 <= 9 >= 4 <= 24 >= 21 <= 43. Thus we get, arr = [2, 1, 9, 4, 24, 21, 43] as output which is sorted in wave form. ``````

#### Solutions

For this problem, we will be discussing two possible solutions:-

1. Sorting and swapping: Sort the array and swap the adjacent values.
2. Comparing neighbours: Compare neighbouring elements and swap if the problem condition doesn’t satisfy.

You can try the problem here.

#### 1. Sorting and Swapping

The problem asks that the array elements in the resultant array must be such that ``` arr >= arr <= arr >= arr <= arr ..... ``` . So, one can find many different solutions for one input. However, the problem statement also requires to return the lexicographically smallest array. So, We can generalize the outputs in a simple way.

We can directly sort the input array so that ``` arr <= arr <= arr <= arr <= arr ..... ``` now, swapping the alternate elements i.e. ``` arr with arr ``` and ``` arr with arr ``` and ``` arr with arr ``` and …, we will have the resultant array satisfying the problem condition.

Solutions steps

• Sort the array in ascending order.

Pseudo Code

``````int[] waveArray(int arr[], int n) {
// Sort the input array
arr.sort()

i = 0
while(i < n-1){
swap(arr[i], arr[i+1])
i = i + 2
}
return arr
}``````

Complexity Analysis

Time Complexity: O(n log n) (How?)

Space Complexity: O(1)

Critical Ideas To Think

• Can we solve this problem by sorting the array in decreasing order and then swapping adjacent values?
• The problem requires the array to satisfy just the relationship between the neighbouring elements, so is sorting even required?
• Try to optimize the running time complexity.

#### 2. Comparing Neighbors

We can optimize the previous approach as we can only focus on maintaining the condition for neighbouring elements. However, this will not guarantee the lexicographically smallest array in return.

If we make sure that all even positioned (at index 0, 2, 4, ..) elements are greater than their adjacent odd positioned elements, we won’t have to check the oddly positioned element.

Solution Steps

Traverse all even positioned elements of the input array, and do the following  — >

• If the current element is smaller than the previous odd element, swap previous and current.
• If the current element is smaller than the next odd element, swap next and current.

Pseudo Code

``````int[] waveArray(int arr[], int n) {
i = 0
while (i < n) {
// If current even index element is smaller than previous
if (i > 0 and arr[i-1] > arr[i] )
swap(arr[i], arr[i-1])
// If current even index element is smaller than next
if (i < n-1 and arr[i] < arr[i+1] )
swap(arr[i], arr[i + 1])
i = i + 2
}
return arr
}``````

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Critical Ideas To Think

• Does the solution return the lexicographically smallest array? If No, Why?
• Do you think any solution is possible which guarantees lexicographically smallest array with linear run-time complexity?
• How did we make sure that only comparing the neighbouring elements and swapping elements as per the required condition will satisfy the problem condition?
• If the input array is already sorted, then the output array according to this approach will be a lexicographically smallest array. True or False?
• Looking at the pseudo-code, what are the previous and next elements for each current element?

#### Comparison of Different Solutions #### Suggested Problems To Solve

• Check if an array is a wave array
• Find if an integer p exists in the array such that the number of integers greater than p in the array equals to p .
• Sort array with squares
• Sort array of 0s, 1s, and 2s.

If you have any more approaches or you find an error/bug in the above solutions, please comment down below.

Happy Coding!

Enjoy Algorithms!