diff --git a/divide_and_conquer/closest_pair_of_points.py b/divide_and_conquer/closest_pair_of_points.py index b6f633964..11dac7e0a 100644 --- a/divide_and_conquer/closest_pair_of_points.py +++ b/divide_and_conquer/closest_pair_of_points.py @@ -1,55 +1,54 @@ """ -The algorithm finds distance between closest pair of points +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 +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, +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 +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 +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) -""" -""" - doctests - >>> euclidean_distance_sqr([1,2],[2,4]) - 5 - >>> dis_between_closest_pair([[1,2],[2,4],[5,7],[8,9],[11,0]],5) - 5 - >>> dis_between_closest_in_strip([[1,2],[2,4],[5,7],[8,9],[11,0]],5) - 85 - >>> points = [(2, 3), (12, 30), (40, 50), (5, 1), (12, 10), (3, 4)] - >>> print("Distance:", closest_pair_of_points(points, len(points))) - "Distance: 1.4142135623730951" +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 + """ + brute force approach to find distance between closest pair points - Parameters : - points, points_count, min_dis (list(tuple(int, int)), int, int) - - Returns : + 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): @@ -61,14 +60,17 @@ def dis_between_closest_pair(points, points_counts, min_dis = float("inf")): def dis_between_closest_in_strip(points, points_counts, min_dis = float("inf")): - """ closest pair of points in strip + """ + closest pair of points in strip - Parameters : - points, points_count, min_dis (list(tuple(int, int)), int, int) - - Returns : + 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): @@ -82,29 +84,32 @@ def dis_between_closest_in_strip(points, points_counts, min_dis = float("inf")): 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 + 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], + 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:], + 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 + + """ + cross_strip contains the points, whose Xcoords are at a distance(< closest_pair_dis) from mid's Xcoord """ @@ -113,21 +118,23 @@ def closest_pair_of_points_sqr(points_sorted_on_x, points_sorted_on_y, points_co 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, + 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, + 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)] + points = [(2, 3), (12, 30), (40, 50), (5, 1), (12, 10), (3, 4)] print("Distance:", closest_pair_of_points(points, len(points))) - -