inversions Algorithm
The inversions algorithm is a computational technique used to determine the number of inversions in a given sequence of integers. An inversion is a pair of elements within the sequence where the earlier element is larger than the later one. Inversion count is often used as a measure of the disorder or chaos in a sequence, as it helps in understanding how far the sequence is from being sorted. The algorithm utilizes divide-and-conquer strategy to efficiently compute the inversion count, making it well-suited for solving problems in sorting, data compression, and other applications where the relative order of elements is important.
One popular implementation of the inversions algorithm is the modified merge sort, which is an extension of the classic merge sort algorithm. The modified merge sort divides the input sequence into two equal halves and recursively computes the inversion count for each half. Once the counts for the two halves are obtained, it merges the sorted halves while simultaneously counting the inversions that occur across the halves. By combining the counts from the left half, right half, and the cross-half inversions, the algorithm obtains the total inversion count of the input sequence. The time complexity of the inversions algorithm, like the merge sort, is O(n log n), where n is the number of elements in the sequence, making it an efficient technique for counting inversions in large datasets.
"""
Given an array-like data structure A[1..n], how many pairs
(i, j) for all 1 <= i < j <= n such that A[i] > A[j]? These pairs are
called inversions. Counting the number of such inversions in an array-like
object is the important. Among other things, counting inversions can help
us determine how close a given array is to being sorted
In this implementation, I provide two algorithms, a divide-and-conquer
algorithm which runs in nlogn and the brute-force n^2 algorithm.
"""
def count_inversions_bf(arr):
"""
Counts the number of inversions using a a naive brute-force algorithm
Parameters
----------
arr: arr: array-like, the list containing the items for which the number
of inversions is desired. The elements of `arr` must be comparable.
Returns
-------
num_inversions: The total number of inversions in `arr`
Examples
---------
>>> count_inversions_bf([1, 4, 2, 4, 1])
4
>>> count_inversions_bf([1, 1, 2, 4, 4])
0
>>> count_inversions_bf([])
0
"""
num_inversions = 0
n = len(arr)
for i in range(n - 1):
for j in range(i + 1, n):
if arr[i] > arr[j]:
num_inversions += 1
return num_inversions
def count_inversions_recursive(arr):
"""
Counts the number of inversions using a divide-and-conquer algorithm
Parameters
-----------
arr: array-like, the list containing the items for which the number
of inversions is desired. The elements of `arr` must be comparable.
Returns
-------
C: a sorted copy of `arr`.
num_inversions: int, the total number of inversions in 'arr'
Examples
--------
>>> count_inversions_recursive([1, 4, 2, 4, 1])
([1, 1, 2, 4, 4], 4)
>>> count_inversions_recursive([1, 1, 2, 4, 4])
([1, 1, 2, 4, 4], 0)
>>> count_inversions_recursive([])
([], 0)
"""
if len(arr) <= 1:
return arr, 0
else:
mid = len(arr) // 2
P = arr[0:mid]
Q = arr[mid:]
A, inversion_p = count_inversions_recursive(P)
B, inversions_q = count_inversions_recursive(Q)
C, cross_inversions = _count_cross_inversions(A, B)
num_inversions = inversion_p + inversions_q + cross_inversions
return C, num_inversions
def _count_cross_inversions(P, Q):
"""
Counts the inversions across two sorted arrays.
And combine the two arrays into one sorted array
For all 1<= i<=len(P) and for all 1 <= j <= len(Q),
if P[i] > Q[j], then (i, j) is a cross inversion
Parameters
----------
P: array-like, sorted in non-decreasing order
Q: array-like, sorted in non-decreasing order
Returns
------
R: array-like, a sorted array of the elements of `P` and `Q`
num_inversion: int, the number of inversions across `P` and `Q`
Examples
--------
>>> _count_cross_inversions([1, 2, 3], [0, 2, 5])
([0, 1, 2, 2, 3, 5], 4)
>>> _count_cross_inversions([1, 2, 3], [3, 4, 5])
([1, 2, 3, 3, 4, 5], 0)
"""
R = []
i = j = num_inversion = 0
while i < len(P) and j < len(Q):
if P[i] > Q[j]:
# if P[1] > Q[j], then P[k] > Q[k] for all i < k <= len(P)
# These are all inversions. The claim emerges from the
# property that P is sorted.
num_inversion += len(P) - i
R.append(Q[j])
j += 1
else:
R.append(P[i])
i += 1
if i < len(P):
R.extend(P[i:])
else:
R.extend(Q[j:])
return R, num_inversion
def main():
arr_1 = [10, 2, 1, 5, 5, 2, 11]
# this arr has 8 inversions:
# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)
num_inversions_bf = count_inversions_bf(arr_1)
_, num_inversions_recursive = count_inversions_recursive(arr_1)
assert num_inversions_bf == num_inversions_recursive == 8
print("number of inversions = ", num_inversions_bf)
# testing an array with zero inversion (a sorted arr_1)
arr_1.sort()
num_inversions_bf = count_inversions_bf(arr_1)
_, num_inversions_recursive = count_inversions_recursive(arr_1)
assert num_inversions_bf == num_inversions_recursive == 0
print("number of inversions = ", num_inversions_bf)
# an empty list should also have zero inversions
arr_1 = []
num_inversions_bf = count_inversions_bf(arr_1)
_, num_inversions_recursive = count_inversions_recursive(arr_1)
assert num_inversions_bf == num_inversions_recursive == 0
print("number of inversions = ", num_inversions_bf)
if __name__ == "__main__":
main()
