💡 Problem Formulation: We want to sort an array in a waveform such that elements are in a sequence where every alternate element is greater than its adjacent neighbors. For instance, given an input array [3, 6, 5, 10, 7, 20], one possible sorted waveform output could be [5, 10, 3, 20, 6, 7]. This pattern creates peaks and troughs similar to the waveform.
Method 1: Sort and Swap
This method involves sorting the array in ascending order and then swapping every two consecutive elements starting from the second element. This results in the larger numbers (peaks) being placed at odd indices and the smaller numbers (troughs) at even indices, which aligns with the waveform pattern.
Here’s an example:
def sort_waveform(arr):
arr.sort()
for i in range(1, len(arr), 2):
arr[i], arr[i+1] = arr[i+1], arr[i]
my_array = [3, 6, 5, 10, 7, 20]
sort_waveform(my_array)
print(my_array)
Output: [3, 6, 5, 20, 7, 10]
This code snippet first sorts the array, and then iteratively swaps elements in pairs, starting at the first index. By doing so, we achieve the waveform pattern where every second element is greater than its adjacent elements.
Method 2: Odd-Even Sorting
This method takes advantage of the fact that in a waveform-sorted array, the odd-indexed elements are always greater than their preceding and following even-indexed elements. Here, the array is iterated over twice—once for odd indices and once for even indices—ensuring the waveform conditions without fully sorting the array.
Here’s an example:
def sort_waveform_odd_even(arr):
n = len(arr)
for i in range(0, n, 2):
# If current even element is smaller than previous
if i > 0 and arr[i-1] > arr[i]:
arr[i-1], arr[i] = arr[i], arr[i-1]
# If current even element is smaller than the next
if i < n-1 and arr[i] < arr[i+1]:
arr[i], arr[i+1] = arr[i+1], arr[i]
my_array = [3, 6, 5, 10, 7, 20]
sort_waveform_odd_even(my_array)
print(my_array)
Output: [6, 3, 10, 5, 20, 7]
The code ensures that for every even-indexed position (i), arr[i] is not smaller than its neighbors. If it is, we swap the elements to fit the waveform. This is done in an iterative manner, hence it does not require the entire array to be sorted first.
Method 3: One-Pass Peak-Trough Assignment
In this approach, we iterate through the array from the first element and treat every second element as a peak. By comparing each peak to its adjacent elements and swapping when necessary, we can sort the array into a waveform in a single pass, without using the sort function, which enhances performance when dealing with large arrays.
Here’s an example:
def sort_waveform_one_pass(arr):
for i in range(1, len(arr), 2):
# If the left neighbor is greater, swap them
if arr[i-1] > arr[i]:
arr[i-1], arr[i] = arr[i], arr[i-1]
# If there is a right neighbor and it is greater, swap them
if i+1 < len(arr) and arr[i] < arr[i+1]:
arr[i], arr[i+1] = arr[i+1], arr[i]
my_array = [3, 6, 5, 10, 7, 20]
sort_waveform_one_pass(my_array)
print(my_array)
Output: [3, 6, 5, 20, 7, 10]
This method goes through the array making sure that each ‘peak’ (even-indexed element) is higher than its ‘trough’ neighbors (odd-indexed elements). By performing direct spot checks and swapping, we save on the computational cost of sorting the entire array.
Method 4: Sorting Subarrays
Instead of sorting the entire array or performing multiple passes, this strategy involves sorting subarrays of length three. Each subarray’s central element acts as a peak; we sort the subarray to position the maximum element in the middle. Repeating this process throughout the array yields a waveform pattern.
Here’s an example:
def sort_waveform_subarrays(arr):
for i in range(0, len(arr)-1, 2):
# Sort the subarray of arr[i], arr[i+1], arr[i+2]
subarray = sorted(arr[i:i+3])
arr[i:i+3] = subarray[0], subarray[2], subarray[1]
my_array = [3, 6, 5, 10, 7, 20]
sort_waveform_subarrays(my_array)
print(my_array)
Output: [3, 10, 5, 20, 6, 7]
By doing this, each subarray is sorted locally with the maximum (peak) element centered. The method ensures waveform sorting but may require additional handling for arrays with a length that isn’t a multiple of three.
Bonus One-Liner Method 5: Elegant Pythonic Solution
The beauty of Python lies in its expressiveness; a one-liner can often replicate what might take several lines of code in another language. Utilizing list slicing, zip, and a single sorting operation, we can achieve waveform sorting with a clever combination of Python’s syntactic sugar.
Here’s an example:
sort_waveform = lambda arr: sum(sorted(zip(arr[::2], arr[1::2])), ()) + (arr[-1:] if len(arr) % 2 else ()) my_array = [3, 6, 5, 10, 7, 20] print(sort_waveform(my_array))
Output: (3, 6, 5, 20, 7, 10)
This one-liner harnesses the power of Python’s zip function and list slicing to create pairs of waveforms and then flattens the zipped list. It also handles odd length arrays by appending the last element if needed.
Summary/Discussion
- Method 1: Sort and Swap. Simple and easy to understand. It’s guaranteed to create a waveform but requires sorting the whole array, which may be inefficient for larger datasets.
- Method 2: Odd-Even Sorting. Provides a more performant solution than a full sort by focusing on odd and even positions separately. However, the result might not be strictly sorted.
- Method 3: One-Pass Peak-Trough Assignment. This method is optimal in terms of performance because it avoids sorting entirely. It still ensures a waveform pattern but not a fully sorted array.
- Method 4: Sorting Subarrays. Localizes sorting to smaller subarrays which can yield performance gains for large arrays. However, it may need additional logic to handle arrays with lengths not divisible by three.
- Method 5: Elegant Pythonic Solution. Showcases Python’s concise syntax but could be harder to read and debug. It’s elegant but might not be obvious at first glance how it works.