Summarize this content to 400 words in Persian Lang <\/p>\n
\u0645\u0642\u062f\u0645\u0647 \u0627\u06cc \u0628\u0631 \u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc<\/p>\n
\u0628\u0631\u0631\u0633\u06cc \u0627\u062c\u0645\u0627\u0644\u06cc \u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc:<\/p>\n
\u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc \u0634\u0627\u062e\u0647 \u0627\u06cc \u0627\u0632 \u0639\u0644\u0648\u0645 \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631 \u0648 \u0631\u06cc\u0627\u0636\u06cc\u0627\u062a \u0627\u0633\u062a \u06a9\u0647 \u0628\u0647 \u0645\u0637\u0627\u0644\u0639\u0647 \u0627\u062c\u0633\u0627\u0645 \u0647\u0646\u062f\u0633\u06cc \u0648 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u062f\u06cc\u0631\u06cc\u062a \u0622\u0646\u0647\u0627 \u0645\u06cc \u067e\u0631\u062f\u0627\u0632\u062f. \u0627\u06cc\u0646 \u0634\u0627\u0645\u0644 \u062d\u0644 \u0645\u0633\u0627\u0626\u0644 \u0647\u0646\u062f\u0633\u06cc \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u062a\u06a9\u0646\u06cc\u06a9 \u0647\u0627\u06cc \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc \u0627\u0633\u062a \u0648 \u0628\u0631\u0627\u06cc \u06a9\u0627\u0631\u0628\u0631\u062f\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u062f\u0631 \u06af\u0631\u0627\u0641\u06cc\u06a9 \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631\u06cc\u060c \u0637\u0631\u0627\u062d\u06cc \u0628\u0647 \u06a9\u0645\u06a9 \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631 (CAD)\u060c \u0631\u0648\u0628\u0627\u062a\u06cc\u06a9 \u0648 \u0633\u06cc\u0633\u062a\u0645 \u0647\u0627\u06cc \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u062c\u063a\u0631\u0627\u0641\u06cc\u0627\u06cc\u06cc (GIS) \u0627\u0633\u0627\u0633\u06cc \u0627\u0633\u062a.<\/p>\n
\u06a9\u0627\u0631\u0628\u0631\u062f\u0647\u0627\u06cc \u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc:<\/p>\n
\u06af\u0631\u0627\u0641\u06cc\u06a9 \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631\u06cc: \u0631\u0646\u062f\u0631\u06cc\u0646\u06af \u0648 \u062f\u0633\u062a\u06a9\u0627\u0631\u06cc \u0627\u0634\u06a9\u0627\u0644 \u0648 \u0633\u0637\u0648\u062d.<\/p>\n
\u0637\u0631\u0627\u062d\u06cc \u0628\u0647 \u06a9\u0645\u06a9 \u06a9\u0627\u0645\u067e\u06cc\u0648\u062a\u0631 (CAD): \u0637\u0631\u0627\u062d\u06cc \u0648 \u062a\u062d\u0644\u06cc\u0644 \u0642\u0637\u0639\u0627\u062a \u0648 \u0633\u0627\u0632\u0647 \u0647\u0627\u06cc \u0645\u06a9\u0627\u0646\u06cc\u06a9\u06cc.<\/p>\n
\u0631\u0628\u0627\u062a\u06cc\u06a9: \u0628\u0631\u0646\u0627\u0645\u0647 \u0631\u06cc\u0632\u06cc \u0645\u0633\u06cc\u0631 \u0648 \u062a\u062d\u0644\u06cc\u0644 \u062d\u0631\u06a9\u062a<\/p>\n
\u0633\u06cc\u0633\u062a\u0645 \u0647\u0627\u06cc \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u062c\u063a\u0631\u0627\u0641\u06cc\u0627\u06cc\u06cc (GIS): \u062a\u062c\u0632\u06cc\u0647 \u0648 \u062a\u062d\u0644\u06cc\u0644 \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u0645\u06a9\u0627\u0646\u06cc \u0648 \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u062c\u063a\u0631\u0627\u0641\u06cc\u0627\u06cc\u06cc.<\/p>\n
\u0627\u0644\u06af\u0648 \u0634\u0646\u0627\u0633\u06cc: \u0634\u0646\u0627\u0633\u0627\u06cc\u06cc \u0627\u0644\u06af\u0648\u0647\u0627 \u0648 \u0627\u0634\u06a9\u0627\u0644 \u062f\u0631 \u067e\u0631\u062f\u0627\u0632\u0634 \u062a\u0635\u0648\u06cc\u0631.<\/p>\n
\u0627\u062c\u0633\u0627\u0645 \u0648 \u0639\u0645\u0644\u06cc\u0627\u062a \u0647\u0646\u062f\u0633\u06cc \u067e\u0627\u06cc\u0647<\/p>\n
\u0646\u0642\u0627\u0637\u060c \u062e\u0637\u0648\u0637\u060c \u0642\u0637\u0639\u0627\u062a \u062e\u0637 \u0648 \u0633\u0637\u0648\u062d:<\/p>\n
\u0646\u06a9\u062a\u0647 \u0647\u0627: \u0648\u0627\u062d\u062f\u0647\u0627\u06cc \u0627\u0633\u0627\u0633\u06cc \u062f\u0631 \u0647\u0646\u062f\u0633\u0647 \u06a9\u0647 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u06cc\u06a9 \u0645\u06a9\u0627\u0646 \u062f\u0631 \u0641\u0636\u0627 \u0647\u0633\u062a\u0646\u062f.<\/p>\n
\u062e\u0637\u0648\u0637: \u062f\u0631 \u0647\u0631 \u062f\u0648 \u062c\u0647\u062a \u0628\u06cc \u0646\u0647\u0627\u06cc\u062a \u06af\u0633\u062a\u0631\u0634 \u06cc\u0627\u0641\u062a\u0647 \u0648 \u0628\u0627 \u062f\u0648 \u0646\u0642\u0637\u0647 \u0645\u0634\u062e\u0635 \u0645\u06cc \u0634\u0648\u0646\u062f.<\/p>\n
\u0628\u062e\u0634 \u0647\u0627\u06cc \u062e\u0637: \u0628\u062e\u0634\u06cc \u0627\u0632 \u06cc\u06a9 \u062e\u0637 \u06a9\u0647 \u062a\u0648\u0633\u0637 \u062f\u0648 \u0646\u0642\u0637\u0647 \u067e\u0627\u06cc\u0627\u0646\u06cc \u062a\u0639\u0631\u06cc\u0641 \u0634\u062f\u0647 \u0627\u0633\u062a.<\/p>\n
\u0647\u0648\u0627\u067e\u06cc\u0645\u0627\u0647\u0627: \u0633\u0637\u0648\u062d \u0645\u0633\u0637\u062d \u06a9\u0647 \u0628\u06cc \u0646\u0647\u0627\u06cc\u062a \u062f\u0631 \u062f\u0648 \u0628\u0639\u062f \u06af\u0633\u062a\u0631\u0634 \u0645\u06cc \u06cc\u0627\u0628\u0646\u062f.<\/p>\n
\u0628\u0631\u062f\u0627\u0631\u0647\u0627 \u0648 \u0639\u0645\u0644\u06cc\u0627\u062a \u0628\u0631\u062f\u0627\u0631:<\/p>\n
\u0628\u0631\u062f\u0627\u0631\u0647\u0627: \u06a9\u0645\u06cc\u062a \u0647\u0627 \u0631\u0627 \u0628\u0627 \u0642\u062f\u0631 \u0648 \u062c\u0647\u062a \u0646\u0645\u0627\u06cc\u0634 \u062f\u0647\u06cc\u062f. \u0628\u0631\u0627\u06cc \u062a\u0648\u0635\u06cc\u0641 \u062a\u0631\u062c\u0645\u0647 \u0647\u0627 \u0648 \u062c\u0647\u062a \u0647\u0627 \u062f\u0631 \u0641\u0636\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n
\u0639\u0645\u0644\u06cc\u0627\u062a \u0628\u0631\u062f\u0627\u0631: \u0634\u0627\u0645\u0644 \u062c\u0645\u0639\u060c \u062a\u0641\u0631\u06cc\u0642\u060c \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0646\u0642\u0637\u0647 \u0627\u06cc\u060c \u0636\u0631\u0628\u062f\u0631\u06cc\u060c \u0648 \u0645\u0642\u06cc\u0627\u0633 \u0628\u0646\u062f\u06cc. \u0627\u06cc\u0646 \u0639\u0645\u0644\u06cc\u0627\u062a \u0628\u0631\u0627\u06cc \u0627\u0646\u062c\u0627\u0645 \u062a\u0628\u062f\u06cc\u0644\u0627\u062a \u0647\u0646\u062f\u0633\u06cc \u0648 \u0645\u062d\u0627\u0633\u0628\u0627\u062a \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n
\u062a\u0628\u062f\u06cc\u0644\u200c\u0647\u0627\u06cc \u0647\u0646\u062f\u0633\u06cc \u067e\u0627\u06cc\u0647 (\u062a\u0631\u062c\u0645\u0647\u060c \u0686\u0631\u062e\u0634\u060c \u0645\u0642\u06cc\u0627\u0633\u200c\u0628\u0646\u062f\u06cc):<\/p>\n
\u062a\u0631\u062c\u0645\u0647: \u062d\u0631\u06a9\u062a \u06cc\u06a9 \u0634\u06cc \u0647\u0646\u062f\u0633\u06cc \u0627\u0632 \u06cc\u06a9 \u0645\u06a9\u0627\u0646 \u0628\u0647 \u0645\u06a9\u0627\u0646 \u062f\u06cc\u06af\u0631 \u0628\u062f\u0648\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0634\u06a9\u0644 \u06cc\u0627 \u0627\u0646\u062f\u0627\u0632\u0647 \u0622\u0646.<\/p>\n
\u0686\u0631\u062e\u0634: \u0686\u0631\u062e\u0627\u0646\u062f\u0646 \u06cc\u06a9 \u062c\u0633\u0645 \u0628\u0647 \u062f\u0648\u0631 \u06cc\u06a9 \u0646\u0642\u0637\u0647 \u062b\u0627\u0628\u062a (\u06cc\u0627 \u0645\u0628\u062f\u0623) \u0628\u0627 \u0632\u0627\u0648\u06cc\u0647 \u0645\u0639\u06cc\u0646.<\/p>\n
\u0645\u0642\u06cc\u0627\u0633 \u0628\u0646\u062f\u06cc: \u062a\u063a\u06cc\u06cc\u0631 \u0627\u0646\u062f\u0627\u0632\u0647 \u06cc\u06a9 \u062c\u0633\u0645 \u0628\u0627 \u062d\u0641\u0638 \u0634\u06a9\u0644 \u0622\u0646\u060c \u0628\u0627 \u0628\u0632\u0631\u06af \u06a9\u0631\u062f\u0646 \u06cc\u0627 \u06a9\u0648\u0686\u06a9 \u06a9\u0631\u062f\u0646 \u0622\u0646.<\/p>\n
\u0645\u062a\u0631\u06cc\u06a9 \u0648 \u0627\u0646\u062f\u0627\u0632\u0647 \u06af\u06cc\u0631\u06cc \u0641\u0627\u0635\u0644\u0647:<\/p>\n
\u0641\u0627\u0635\u0644\u0647 \u0627\u0642\u0644\u06cc\u062f\u0633\u06cc: \u0641\u0627\u0635\u0644\u0647 \u062e\u0637 \u0645\u0633\u062a\u0642\u06cc\u0645 \u0628\u06cc\u0646 \u062f\u0648 \u0646\u0642\u0637\u0647 \u062f\u0631 \u0641\u0636\u0627\u06cc \u0627\u0642\u0644\u06cc\u062f\u0633\u06cc.<\/p>\n
\u0641\u0627\u0635\u0644\u0647 \u0645\u0646\u0647\u062a\u0646: \u0641\u0627\u0635\u0644\u0647 \u0628\u06cc\u0646 \u062f\u0648 \u0646\u0642\u0637\u0647 \u062f\u0631 \u0627\u0645\u062a\u062f\u0627\u062f \u0645\u062d\u0648\u0631\u0647\u0627 \u062f\u0631 \u0632\u0627\u0648\u06cc\u0647 \u0642\u0627\u0626\u0645 \u0627\u0646\u062f\u0627\u0632\u0647 \u06af\u06cc\u0631\u06cc \u0645\u06cc \u0634\u0648\u062f.<\/p>\n
\u0633\u0627\u06cc\u0631 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627: \u0634\u0627\u0645\u0644 \u0641\u0627\u0635\u0644\u0647 \u0686\u0628\u06cc\u0634\u0641 \u0648 \u0641\u0627\u0635\u0644\u0647 \u0645\u06cc\u0646\u06a9\u0648\u0641\u0633\u06a9\u06cc \u0627\u0633\u062a \u06a9\u0647 \u062f\u0631 \u0632\u0645\u06cc\u0646\u0647 \u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0647\u0646\u062f\u0633\u06cc \u0648 \u06a9\u0627\u0631\u0628\u0631\u062f\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n
\u0646\u0645\u0648\u0646\u0647 \u0647\u0627\u06cc\u06cc \u0627\u0632 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc<\/p>\n
\u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc \u0634\u0627\u0645\u0644 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641\u06cc \u0628\u0631\u0627\u06cc \u062d\u0644 \u0645\u0648\u062b\u0631 \u0645\u0633\u0627\u0626\u0644 \u0647\u0646\u062f\u0633\u06cc \u0627\u0633\u062a. \u062f\u0631 \u0627\u06cc\u0646\u062c\u0627 \u0646\u0645\u0648\u0646\u0647\u200c\u0647\u0627\u06cc\u06cc \u0627\u0632 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627\u06cc \u0631\u0627\u06cc\u062c \u067e\u06cc\u0627\u062f\u0647\u200c\u0633\u0627\u0632\u06cc \u0634\u062f\u0647 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u067e\u0627\u0631\u0627\u062f\u0627\u06cc\u0645 \u062a\u0642\u0633\u06cc\u0645 \u06a9\u0646:<\/p>\n
\u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u0627\u0645\u062a\u06cc\u0627\u0632 (1D)<\/p>\n
\u062f\u0631 \u06cc\u06a9 \u0628\u0639\u062f\u06cc\u060c \u06cc\u0627\u0641\u062a\u0646 \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u0646\u0642\u0627\u0637 \u0631\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0631\u0648\u06cc\u06a9\u0631\u062f \u062a\u0642\u0633\u06cc\u0645 \u0648 \u063a\u0644\u0628\u0647 \u0628\u0647 \u0637\u0648\u0631 \u0645\u0648\u062b\u0631 \u0627\u0646\u062c\u0627\u0645 \u062f\u0627\u062f. \u0627\u06cc\u062f\u0647 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0646\u0642\u0627\u0637 \u0631\u0627 \u0628\u0647 \u0635\u0648\u0631\u062a \u0628\u0627\u0632\u06af\u0634\u062a\u06cc \u0628\u0647 \u06af\u0631\u0648\u0647 \u0647\u0627\u06cc \u06a9\u0648\u0686\u06a9\u062a\u0631 \u062a\u0642\u0633\u06cc\u0645 \u06a9\u0646\u06cc\u062f \u0648 \u0646\u0632\u062f\u06cc\u06a9 \u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u0631\u0627 \u062f\u0631 \u0647\u0631 \u06af\u0631\u0648\u0647 \u067e\u06cc\u062f\u0627 \u06a9\u0646\u06cc\u062f.<\/p>\n
\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646:<\/p>\n
def closest_pair_1d(points):
\n def find_closest_pair(sorted_points):
\n # Base case: If there are only two points, return them
\n if len(sorted_points) == 2:
\n return sorted_points[0], sorted_points[1]\n # Base case: If there are three points, return the closest pair
\n if len(sorted_points) == 3:
\n return min(
\n ((sorted_points[0], sorted_points[1]),
\n (sorted_points[1], sorted_points[2]),
\n (sorted_points[0], sorted_points[2])),
\n key=lambda pair: abs(pair[1] – pair[0])
\n )
\n # Divide the points into two halves
\n mid = len(sorted_points) \/\/ 2
\n left_half = sorted_points[:mid]\n right_half = sorted_points[mid:]\n # Find the closest pair in each half
\n closest_pair_left = find_closest_pair(left_half)
\n closest_pair_right = find_closest_pair(right_half)
\n # Find the closest pair across the split
\n closest_pair_split = (left_half[-1], right_half[0])
\n # Return the overall closest pair
\n return min(closest_pair_left, closest_pair_right, closest_pair_split, key=lambda pair: abs(pair[1] – pair[0]))<\/p>\n
if len(points) < 2:
\n raise ValueError(“At least two points are required”)
\n # Sort the points
\n sorted_points = sorted(points)
\n # Find the closest pair using divide and conquer
\n return find_closest_pair(sorted_points)<\/p>\n
# Example usage
\npoints = [0, 1, 3, 5, 7, 9, 11]\nclosest_points = closest_pair_1d(points)
\nprint(f”The closest pair of points is: {closest_points}”)<\/p>\n
\u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u0627\u0645\u062a\u06cc\u0627\u0632 (2 \u0628\u0639\u062f\u06cc)<\/p>\n
\u062f\u0631 \u062f\u0648 \u0628\u0639\u062f\u06cc\u060c \u0645\u0633\u0626\u0644\u0647 \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u0646\u0642\u0637\u0647 \u067e\u06cc\u0686\u06cc\u062f\u0647 \u062a\u0631 \u0627\u0633\u062a. \u0627\u06cc\u0646 \u0634\u0627\u0645\u0644 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0646\u0642\u0627\u0637 \u0648 \u06cc\u0627\u0641\u062a\u0646 \u0628\u0627\u0632\u06af\u0634\u062a\u06cc \u0646\u0632\u062f\u06cc\u06a9\u200c\u062a\u0631\u06cc\u0646 \u062c\u0641\u062a \u062f\u0631 \u0628\u062e\u0634\u200c\u0647\u0627\u06cc \u062a\u0642\u0633\u06cc\u0645\u200c\u0634\u062f\u0647\u060c \u0633\u067e\u0633 \u0628\u0631\u0631\u0633\u06cc \u062f\u0631 \u0633\u0631\u0627\u0633\u0631 \u0645\u0631\u0632 \u062a\u0642\u0633\u06cc\u0645 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/p>\n
\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646:<\/p>\n
import math<\/p>\n
def distance(point1, point2):
\n return math.sqrt((point1[0] – point2[0]) ** 2 + (point1[1] – point2[1]) ** 2)<\/p>\n
def closest_pair_2d(points):
\n def closest_pair_recursive(points_sorted_by_x, points_sorted_by_y):
\n n = len(points_sorted_by_x)<\/p>\n
if n <= 3:
\n return brute_force_closest_pair(points_sorted_by_x)<\/p>\n
mid = n \/\/ 2
\n left_half_x = points_sorted_by_x[:mid]\n right_half_x = points_sorted_by_x[mid:]\n
midpoint = points_sorted_by_x[mid][0]\n left_half_y = list(filter(lambda x: x[0] <= midpoint, points_sorted_by_y))
\n right_half_y = list(filter(lambda x: x[0] > midpoint, points_sorted_by_y))<\/p>\n
(p1, q1, d1) = closest_pair_recursive(left_half_x, left_half_y)
\n (p2, q2, d2) = closest_pair_recursive(right_half_x, right_half_y)<\/p>\n
if d1 < d2:
\n d = d1
\n min_pair = (p1, q1)
\n else:
\n d = d2
\n min_pair = (p2, q2)<\/p>\n
(p3, q3, d3) = closest_split_pair(points_sorted_by_x, points_sorted_by_y, d, min_pair)<\/p>\n
if d3 < d:
\n return (p3, q3, d3)
\n else:
\n return min_pair[0], min_pair[1], d<\/p>\n
def brute_force_closest_pair(points):
\n min_dist = float(‘inf’)
\n p1 = None
\n p2 = None
\n for i in range(len(points)):
\n for j in range(i + 1, len(points)):
\n d = distance(points[i], points[j])
\n if d < min_dist:
\n min_dist = d
\n p1, p2 = points[i], points[j]\n return p1, p2, min_dist<\/p>\n
def closest_split_pair(points_sorted_by_x, points_sorted_by_y, delta, best_pair):
\n n = len(points_sorted_by_x)
\n mid_x = points_sorted_by_x[n \/\/ 2][0]\n
Sy = [point for point in points_sorted_by_y if mid_x – delta <= point[0] <= mid_x + delta]\n
best = delta
\n len_Sy = len(Sy)
\n for i in range(len_Sy – 1):
\n for j in range(i + 1, min(i + 7, len_Sy)):
\n p, q = Sy[i], Sy[j]\n dst = distance(p, q)
\n if dst < best:
\n best = dst
\n best_pair = (p, q)
\n return best_pair[0], best_pair[1], best<\/p>\n
points_sorted_by_x = sorted(points, key=lambda x: x[0])
\n points_sorted_by_y = sorted(points, key=lambda x: x[1])<\/p>\n
p1, p2, min_dist = closest_pair_recursive(points_sorted_by_x, points_sorted_by_y)
\n return p1, p2, min_dist<\/p>\n
# Example usage
\npoints = [(2, 3), (2, 4), (12, 30), (40, 50), (5, 1), (12, 10), (3, 4)]\nclosest_points = closest_pair_2d(points)
\nprint(f”The closest pair of points is: {closest_points[0]} and {closest_points[1]} with a distance of {closest_points[2]}”)<\/p>\n
\u0627\u0633\u06a9\u0646 \u06af\u0631\u0627\u0647\u0627\u0645 (\u0647\u0627\u0644 \u0645\u062d\u062f\u0628)<\/p>\n
\u06af\u0631\u0627\u0647\u0627\u0645 \u0627\u0633\u06a9\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u06cc \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u0628\u062f\u0646\u0647 \u0645\u062d\u062f\u0628 \u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u06cc \u0627\u0632 \u0646\u0642\u0627\u0637 \u0628\u0647 \u0635\u0648\u0631\u062a \u062f\u0648 \u0628\u0639\u062f\u06cc \u0627\u0633\u062a. \u0646\u0642\u0627\u0637 \u0631\u0627 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0632\u0627\u0648\u06cc\u0647 \u0642\u0637\u0628\u06cc \u0646\u0633\u0628\u062a \u0628\u0647 \u06cc\u06a9 \u0645\u062d\u0648\u0631 \u0645\u0631\u062a\u0628 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0633\u067e\u0633 \u0628\u062f\u0646\u0647 \u0645\u062d\u062f\u0628 \u0631\u0627 \u0645\u06cc \u0633\u0627\u0632\u062f.<\/p>\n
\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646:<\/p>\n
import math<\/p>\n
def graham_scan(points):
\n # Find the point with the lowest y-coordinate, break ties by x-coordinate
\n pivot = min(points, key=lambda p: (p[1], p[0]))
\n points.remove(pivot)<\/p>\n
# Sort the points based on polar angle with the pivot
\n points.sort(key=lambda p: (math.atan2(p[1] – pivot[1], p[0] – pivot[0]), p[1], p[0]))<\/p>\n
# Add the pivot back to the beginning
\n points.insert(0, pivot)<\/p>\n
# Initialize the hull with the first two points
\n hull = [points[0], points[1]]\n
for p in points[2:]:
\n while len(hull) > 1 and orientation(hull[-2], hull[-1], p) != 2:
\n hull.pop()
\n hull.append(p)<\/p>\n
return hull<\/p>\n
def orientation(p, q, r):
\n # Function to find the orientation of the triplet (p, q, r)
\n # 0 -> p, q, and r are collinear
\n # 1 -> Clockwise
\n # 2 -> Counterclockwise
\n val = (q[1] – p[1]) * (r[0] – q[0]) – (q[0] – p[0]) * (r[1] – q[1])
\n if val == 0:
\n return 0
\n elif val > 0:
\n return 1
\n else:
\n return 2<\/p>\n
# Sample points
\npoints = [(0, 0), (1, 1), (2, 2), (2, 0), (0, 2), (0.5, 2.5), (2.5, 2)]\n
# Call the function
\nconvex_hull = graham_scan(points)<\/p>\n
# Print the result
\nprint(“Convex Hull:”, convex_hull)<\/p>\n
\u062c\u0627\u0631\u0648\u06cc\u0633 \u0645\u0627\u0631\u0686 (\u06a9\u0627\u062f\u0648\u067e\u06cc\u0686) (\u06a9\u0627\u0633\u0647 \u0645\u062d\u062f\u0628)<\/p>\n
Jarvis March (Gift Wrapping) \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u062f\u06cc\u06af\u0631\u06cc \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u0628\u062f\u0646\u0647 \u0645\u062d\u062f\u0628 \u0627\u0633\u062a. \u0627\u0632 \u0633\u0645\u062a \u0686\u067e \u062a\u0631\u06cc\u0646 \u0646\u0642\u0637\u0647 \u0634\u0631\u0648\u0639 \u0645\u06cc \u0634\u0648\u062f \u0648 \u062f\u0648\u0631 \u0646\u0642\u0627\u0637 \u0645\u06cc \u067e\u06cc\u0686\u062f \u062a\u0627 \u0628\u062f\u0646\u0647 \u0631\u0627 \u067e\u06cc\u062f\u0627 \u06a9\u0646\u062f.<\/p>\n
\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u067e\u0627\u06cc\u062a\u0648\u0646:<\/p>\n
def gift_wrapping(points):
\n hull = []\n start = min(points, key=lambda p: p[0]) # Find the leftmost point
\n on_hull = start
\n while True:
\n hull.append(on_hull)
\n next_point = points[0]\n for p in points:
\n if (next_point == on_hull) or (orientation(on_hull, next_point, p) == 2):
\n next_point = p
\n on_hull = next_point
\n if on_hull == start:
\n break
\n return hull<\/p>\n
def orientation(p, q, r):
\n # Function to find the orientation of the triplet (p, q, r)
\n # 0 -> p<\/p>\n
, q, and r are collinear
\n # 1 -> Clockwise
\n # 2 -> Counterclockwise
\n val = (q[1] – p[1]) * (r[0] – q[0]) – (q[0] – p[0]) * (r[1] – q[1])
\n if val == 0:
\n return 0
\n elif val > 0:
\n return 1
\n else:
\n return 2<\/p>\n
# Sample points
\npoints = [(0, 0), (1, 1), (2, 2), (2, 0), (0, 2), (0.5, 2.5), (2.5, 2)]\n
# Call the function
\nconvex_hull = gift_wrapping(points)<\/p>\n
# Print the result
\nprint(“Convex Hull:”, convex_hull)<\/p>\n
\u0627\u06cc\u0646 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0647\u0627 \u06cc\u06a9 \u0646\u0642\u0637\u0647 \u0634\u0631\u0648\u0639 \u0639\u0645\u0644\u06cc \u0628\u0631\u0627\u06cc \u06a9\u0627\u0648\u0634 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0647\u0646\u062f\u0633\u0647 \u0645\u062d\u0627\u0633\u0628\u0627\u062a\u06cc \u0641\u0631\u0627\u0647\u0645 \u0645\u06cc \u06a9\u0646\u0646\u062f.<\/p>\n