Python/divide_and_conquer/closest_pair_of_points.py

"""
The algorithm finds distance between closest pair of points
in the given n points.
Approach used -> Divide and conquer
The points are sorted based on Xco-ords and
then based on Yco-ords separately.
And by applying divide and conquer approach,
minimum distance is obtained recursively.

>> Closest points can lie on different sides of partition.
This case handled by forming a strip of points
whose Xco-ords distance is less than closest_pair_dis
from mid-point's Xco-ords. Points sorted based on Yco-ords
are used in this step to reduce sorting time.
Closest pair distance is found in the strip of points. (closest_in_strip)

min(closest_pair_dis, closest_in_strip) would be the final answer.

Time complexity: O(n * log n)
"""


def euclidean_distance_sqr(point1, point2):
    """
    >>> euclidean_distance_sqr([1,2],[2,4])
    5
    """
    return (point1[0] - point2[0]) ** 2 + (point1[1] - point2[1]) ** 2


def column_based_sort(array, column = 0):
    """
    >>> column_based_sort([(5, 1), (4, 2), (3, 0)], 1)
    [(3, 0), (5, 1), (4, 2)]
    """
    return sorted(array, key = lambda x: x[column])


def dis_between_closest_pair(points, points_counts, min_dis = float("inf")):
    """
    brute force approach to find distance between closest pair points

    Parameters :
    points, points_count, min_dis (list(tuple(int, int)), int, int)

    Returns :
    min_dis (float):  distance between closest pair of points

    >>> dis_between_closest_pair([[1,2],[2,4],[5,7],[8,9],[11,0]],5)
    5

    """

    for i in range(points_counts - 1):
        for j in range(i+1, points_counts):
            current_dis = euclidean_distance_sqr(points[i], points[j])
            if current_dis < min_dis:
                min_dis = current_dis
    return min_dis


def dis_between_closest_in_strip(points, points_counts, min_dis = float("inf")):
    """
    closest pair of points in strip

    Parameters :
    points, points_count, min_dis (list(tuple(int, int)), int, int)

    Returns :
    min_dis (float):  distance btw closest pair of points in the strip (< min_dis)

    >>> dis_between_closest_in_strip([[1,2],[2,4],[5,7],[8,9],[11,0]],5)
    85
    """

    for i in range(min(6, points_counts - 1), points_counts):
        for j in range(max(0, i-6), i):
            current_dis = euclidean_distance_sqr(points[i], points[j])
            if current_dis < min_dis:
                min_dis = current_dis
    return min_dis


def closest_pair_of_points_sqr(points_sorted_on_x, points_sorted_on_y, points_counts):
    """ divide and conquer approach

    Parameters :
    points, points_count (list(tuple(int, int)), int)

    Returns :
    (float):  distance btw closest pair of points

    >>> closest_pair_of_points_sqr([(1, 2), (3, 4)], [(5, 6), (7, 8)], 2)
    8
    """

    # base case
    if points_counts <= 3:
        return dis_between_closest_pair(points_sorted_on_x, points_counts)

    # recursion
    mid = points_counts//2
    closest_in_left = closest_pair_of_points_sqr(points_sorted_on_x,
                                                 points_sorted_on_y[:mid],
                                                 mid)
    closest_in_right = closest_pair_of_points_sqr(points_sorted_on_y,
                                                  points_sorted_on_y[mid:],
                                                  points_counts - mid)
    closest_pair_dis = min(closest_in_left, closest_in_right)

    """
    cross_strip contains the points, whose Xcoords are at a
    distance(< closest_pair_dis) from mid's Xcoord
    """

    cross_strip = []
    for point in points_sorted_on_x:
        if abs(point[0] - points_sorted_on_x[mid][0]) < closest_pair_dis:
            cross_strip.append(point)

    closest_in_strip = dis_between_closest_in_strip(cross_strip,
                     len(cross_strip), closest_pair_dis)
    return min(closest_pair_dis, closest_in_strip)


def closest_pair_of_points(points, points_counts):
    """
    >>> closest_pair_of_points([(2, 3), (12, 30)], len([(2, 3), (12, 30)]))
    28.792360097775937
    """
    points_sorted_on_x = column_based_sort(points, column = 0)
    points_sorted_on_y = column_based_sort(points, column = 1)
    return (closest_pair_of_points_sqr(points_sorted_on_x,
                                       points_sorted_on_y,
                                       points_counts)) ** 0.5


if __name__ == "__main__":
    points = [(2, 3), (12, 30), (40, 50), (5, 1), (12, 10), (3, 4)]
    print("Distance:", closest_pair_of_points(points, len(points)))