# Sort an Array in Wave Form

Sorting arrays is a common task in computer science and programming. Often, the requirements are to simply sort the array in ascending or descending order. However, sometimes more complex arrangements are needed. One such arrangement is sorting the array elements in a wave pattern - alternating between large and small values. This wave sort presents an interesting problem, and the algorithm requires careful swapping after sorting to achieve the wave effect.

In this article, we will look at the wave sort technique to rearrange an unsorted array of integers in a wave fashion. We will go through the algorithm step-by-step. Code examples in Python.

## What is Wave Form?

Waveform sorting refers to arranging the elements of an array in a sequence such that elements alternatively change from large to small and then small to large. Visualizing the final sorted array in a graph will look like a wave with crests and troughs.

For example, consider the array [3, 6, 5, 10, 7, 20]. If we sort this array in wave form, we should get [5, 3, 10, 6, 20, 7]. We can observe the following properties:

• The elements are arranged in an alternating crests and troughs sequence - 5 > 3 < 10 > 6 < 20 > 7.
• It is not fully sorted in either ascending or descending order.
• Every odd-positioned element is greater than its adjacent left element. For example, 10 > 6 at 3rd position.
• Every even-positioned element is smaller than its adjacent left element. For example, 3 < 5 at 2nd position.

Therefore, waveform sorting aims to rearrange elements in this fashion rather than in simple ascending or descending order.

The algorithm involves sorting the array normally and then swapping adjacent elements pairwise. This results in small and large values exchanging positions in each pair, creating the wave pattern.

The wave-sorted array has applications where generating data in wave patterns is required. For example, while testing functions that process alternating large and small values. It is also sometimes used in mathematics for specific sequences.

Overall, waveform sorting creates an unusual arrangement of data that switches between crests and troughs in a wave-like visual pattern.

## Advantages of using Wave Form in Sorting

• Alternating pattern: The wave form sorted array has an alternating pattern of large and small values. This can be useful for generating test data or finding mathematical sequences.
• Different from full sort: Wave form is not just fully sorted in ascending or descending order. It provides a unique arrangement.
• Stability: The original order of equal elements is maintained after wave form sorting. This property of stability can be useful in many cases.
• In-place sorting: Wave form sorting is done in place by swapping elements. No extra space is needed.
• Easy to implement: The algorithm is simple, just sorting and swapping. It can be coded easily in any programming language.
• O(nLogn) time complexity: For an array of size n, wave form sorting takes O(nLogn) time since the main operation is sorting.
• O(1) space complexity: No extra memory is needed for the algorithm, so space complexity is constant O(1).
• Can be parallelized: The initial sorting step can be parallelized for optimization in large arrays.
• Applicable to other data: The waveform ordering concept can be applied to other data structures like linked lists, trees, etc.

So, in summary, the alternating pattern, stability, in-place operation, easy coding, optimal time/space complexity and parallelism make wave form sorting useful over conventional sorting techniques in certain cases. The wave pattern has its unique applications.

## Applications and Uses of Wave Form

• Alternating test data: Wave-sorted arrays can generate alternating large-small test cases for testing functions. This data acts better than random or fully sorted data.
• Signal processing: In signal processing, a wave pattern occurs naturally when alternating components are present. Wave form sorting is applicable when processing such wave signals.
• Image dithering: In graphics, alternating pixel intensities create dithering effects. Wave arranged pixel values can produce these visual effects.
• Load balancing: In computing, wave form data can help distribute load better by alternating between heavy and light tasks.
• Waveform analysis: In statistics, waveform analysis uses wave patterns to study cyclical changes over time. Sorting data in wave form aids this analysis.
• Mathematical sequences: Some mathematical formulas and equations require inputs in a wave pattern to generate specific sequences.
• Waveform visualization: The crests and troughs of the wave formed sorted array can be visualized when plotted on a graph.
• Alternating games: A wave sorted sequence of easy and hard levels can provide an alternating gaming challenge.
• Testing edge cases: The unusual order tests boundary conditions well compared to standard sorted data.
• Finding wave segments: Wave form sorting can identify sub-arrays with a wave property in a larger array.

So, in summary, generating alternating test data, processing wave signals, creating patterns and sequences, analyzing cyclical data, balancing loads, testing edge cases etc., are some applications where wave form sorting provides benefits over other sorting techniques. The alternating wave pattern has unique use cases.

## Python Implementation of Wave Form

Algorithm

1. Sort the input array arr[] in ascending order using the built-in sort() function. This puts smaller elements before larger elements.
2. Initialize a variable 'i' to 0 to use as an index into the array arr[].
3. Start a loop to iterate from 0 to n-1, stepping by 2 in each iteration. This allows us to access alternate elements of the array (odd and even indexed elements).
4. Swap the element arr[i] with element arr[i+1] inside the loop. As 'i' increments by 2 each time, adjacent element pairs are swapped.
5. After the loop, the array is sorted in wave form. Print the array elements to verify.

Approach

• The calling function first rearranges elements in ascending order.
• The loop iterates by a step of 2 using 'i', allowing access to odd and even indexed pairs.
• Swapping adjacent pairs inverts the large-small relationship between them.
• As 'i' increments alternately, each odd-even pair gets swapped.
• Finally, a wavy crests and troughs pattern is achieved.

Example

Input: arr[] = [10, 90, 49, 2, 1, 5, 23]

Sort: [1, 2, 5, 10, 23, 49, 90]

After swap: [2, 1, 10, 5, 90, 23, 49] (wave form)

Time Complexity - O(nLogn) for sort() + O(n) for swap loop => O(nLogn)

Space Complexity - O(1) as it's done in place without any extra space.

So, in summary, initializing 'i' for accessing alternates, swapping adjacent pairs, and performing it in-place results in an efficient wave sort algorithm with optimal time and space complexity.

Output:

Explanation:

• First, the given arr is sorted using the built-in sort() function.
• A loop is run from 0 to n-1, stepping by 2 using a variable i.
• The loop swaps adjacent pairs - arr[i] with arr[i+1].
• This results in a final wave sorted array, which is printed.
• The adjacent pairs (10, 5), (6, 3), (20, 100) etc. get swapped resulting in wave form sorting.
• The time complexity is O(nLogn) and space complexity is O(1), making it efficient. This program demonstrates the implementation of wave sort algorithm in Python.

## Conclusion:

In this article, we explored the technique of sorting an unsorted array of integers in a wave pattern. This wave sorting rearranges the elements into alternating crests and troughs instead of just ascending or descending order.

We looked at a simple and efficient algorithm that first sorts the array normally, then swaps adjacent elements pairwise to achieve the wave effect. The algorithm has an optimal time complexity of O(nLogn) and constant O(1) space complexity.

Code examples were provided in Python to demonstrate implementation. The Python program leveraged built-in sort and in-place swapping to implement the wave sort efficiently.

Wave form sorting has some interesting applications in computing and mathematics. Some use cases include generating alternating test data, processing wave signals, creating dithering effects, balancing loads and analyzing waveforms. It provides a different data arrangement compared to typical sorting.

In summary, the wave sort algorithm provides an unusual reordering of array elements from small to large to small and so on in a wave pattern. With its unique applications, this technique can be useful to standard sorting methods in certain scenarios.