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"""
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)))
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