From c85312da89dcc5bb1ad397feffc0e055dc576e85 Mon Sep 17 00:00:00 2001 From: Dharni0607 <30770547+Dharni0607@users.noreply.github.com> Date: Tue, 9 Jul 2019 20:50:43 +0530 Subject: [PATCH] updates in closest pair of points algorithm (#979) * updated closest pair of points (n*(logn)^2) to (n*logn) --- divide_and_conquer/closest_pair_of_points.py | 49 +++++++++++--------- 1 file changed, 28 insertions(+), 21 deletions(-) diff --git a/divide_and_conquer/closest_pair_of_points.py b/divide_and_conquer/closest_pair_of_points.py index cc5be428d..ee06d2706 100644 --- a/divide_and_conquer/closest_pair_of_points.py +++ b/divide_and_conquer/closest_pair_of_points.py @@ -1,27 +1,27 @@ """ -The algorithm finds distance btw closest pair of points in the given n 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 -& by applying divide and conquer approach, +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 lie on different sides of partition +>> 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. +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 * (logn)^2) +Time complexity: O(n * log n) """ -import math - - def euclidean_distance_sqr(point1, point2): - return pow(point1[0] - point2[0], 2) + pow(point1[1] - point2[1], 2) + return (point1[0] - point2[0]) ** 2 + (point1[1] - point2[1]) ** 2 def column_based_sort(array, column = 0): @@ -66,7 +66,7 @@ def dis_between_closest_in_strip(points, points_counts, min_dis = float("inf")): return min_dis -def closest_pair_of_points_sqr(points, points_counts): +def closest_pair_of_points_sqr(points_sorted_on_x, points_sorted_on_y, points_counts): """ divide and conquer approach Parameters : @@ -79,12 +79,16 @@ def closest_pair_of_points_sqr(points, points_counts): # base case if points_counts <= 3: - return dis_between_closest_pair(points, points_counts) + return dis_between_closest_pair(points_sorted_on_x, points_counts) # recursion mid = points_counts//2 - closest_in_left = closest_pair_of_points(points[:mid], mid) - closest_in_right = closest_pair_of_points(points[mid:], points_counts - 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:], + points_counts - mid) closest_pair_dis = min(closest_in_left, closest_in_right) """ cross_strip contains the points, whose Xcoords are at a @@ -92,22 +96,25 @@ def closest_pair_of_points_sqr(points, points_counts): """ cross_strip = [] - for point in points: - if abs(point[0] - points[mid][0]) < closest_pair_dis: + for point in points_sorted_on_x: + if abs(point[0] - points_sorted_on_x[mid][0]) < closest_pair_dis: cross_strip.append(point) - cross_strip = column_based_sort(cross_strip, 1) 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): - return math.sqrt(closest_pair_of_points_sqr(points, points_counts)) + 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 -points = [(2, 3), (12, 30), (40, 50), (5, 1), (12, 10), (0, 2), (5, 6), (1, 2)] -points = column_based_sort(points) -print("Distance:", closest_pair_of_points(points, len(points))) +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)))