divide and conquer and brute force algorithms for array-inversions counting

Author: maxwell-aladago

divide_and_conquer/inversions.py

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+from __future__ import print_function, absolute_import, division
+
+"""
+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`
+
+    """
+
+    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'
+
+    """
+    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`
+
+    """
+
+    R = []
+    i = j = num_inversion = 0
+    while i < len(P) and j < len(Q):
+        if P[i] > Q[j]:
+            # if P[1] is > Q[j], then P[k] for all k > i, P[K] > Q[j]
+            # and therefore are inversions. This 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()
+
+