dsa hw 3

HOMEWORK-3 The closest pair of points algorithm using the Divide and Conquer approach step by step: 1. Base Case:  If the number of points is less than or equal to 3, use a brute-force approach to find the closest pair among them. 2. Sort Points:  Sort the list of points based on their x-coordinates. 3. Divide:  Divide the sorted points into two halves.  Find the midpoint of the sorted list. 4. Recursive Calls: Recursively find the closest pair in the left and right halves.  min_dist_left = closest_pair(left_half)  min_dist_right = closest_pair(right_half) 5. Combine Results: Find the minimum distance between points across the dividing line.  Merge the left and right halves.  Identify a "strip" of points within the minimum distance from the dividing line.  Sort this strip by y-coordinates.  Check the distance between each point in the strip and the next 7 points. 6. Final Result:  Return the minimum distance among the distances found in the left, right, and strip. Explanation: 1. Base Case: If there are three or fewer points, the algorithm uses a brute-force approach to find the closest pair among them. This is a simple and direct method suitable for a small number of points.

2. Sort Points: The list of points is sorted based on their x-coordinates. Sorting allows for efficient division of the points and simplifies subsequent computations. 3. Divide: The sorted list of points is divided into two halves, and the midpoint of the list is identified. This step sets the stage for recursive exploration of the left and right halves. 4. Recursive Calls: Recursively, the algorithm finds the closest pair in the left and right halves of the list of points. This step breaks down the problem into smaller subproblems. 5. Combine Results: The minimum distance found in the left and right halves is determined. Additionally, the algorithm identifies a "strip" of points close to the dividing line and sorts them by their y-coordinates. It then checks for the minimum distance between each point in the strip and the next 7 points, combining these results. 6. Final Result: The algorithm returns the minimum distance among the distances found in the left, right, and strip. This ensures that the overall closest pair of points is correctly identified and reported. The Divide and Conquer approach, in this case, allows for efficient and systematic exploration of the point set. Algorithm ClosestPair(points): 1. If the number of points in the set is less than or equal to 3, use a brute-force approach to find the closest pair. 2. Sort the points by their x-coordinates. 3. Divide the sorted points into two halves. 4. Recursively find the closest pair in each half. 5. Let d be the minimum distance found among the two halves. 6. Find the points in the strip of width 2d around the middle point of the sorted list. 7. Sort the points in the strip by their y-coordinates. 8. Compare each point in the strip with the next 7 points to find the minimum distance in the strip. 9. Return the minimum of the minimum distances found in the two halves and in the strip. Function strip_closest(strip, d): 1. Sort the points in the strip by their y-coordinates. 2. Initialize min_dist to d. 3. For each point i in the strip: a. Compare it with the next 7 points and update min_dist if a smaller distance is found. 4. Return min_dist. Function distance(point1, point2): >> Compute and return the Euclidean distance between point1 and point2. Pseudocode: import math

# Function to calculate the distance between two points def distance ( point1 , point2 ): return math . sqrt (( point1 [ 0 ] - point2 [ 0 ]) ** 2 + ( point1 [ 1 ] - point2 [ 1 ]) ** 2 ) # Function to find the minimum distance between points in a strip def strip_closest ( strip , d ): min_dist = d strip . sort ( key = lambda point : point [ 1 ]) # Compare each point in the strip with the next 7 points for i in range ( len ( strip )): j = i + 1 while j < len ( strip ) and ( strip [ j ][ 1 ] - strip [ i ][ 1 ]) < min_dist : min_dist = min ( min_dist , distance ( strip [ i ], strip [ j ])) j += 1 return min_dist # Function to find the minimum distance using divide and conquer def closest_pair ( points ): n = len ( points ) # Base case: If the number of points is less than or equal to 3, use a brute-force approach if n <= 3 : return min ( distance ( points [ i ], points [ j ]) for i in range ( n ) for j in range ( i + 1 , n )) # Sort points by x-coordinate points . sort () # Divide the points into two halves mid = n // 2 left_half = points [: mid ] right_half = points [ mid :] # Recursively find the minimum distance in each half min_dist_left = closest_pair ( left_half ) min_dist_right = closest_pair ( right_half )

# Find the minimum distance across the two halves min_dist = min ( min_dist_left , min_dist_right ) # Find points in the strip within the minimum distance from the dividing line strip = [ point for point in points if abs ( point [ 0 ] - points [ mid ][ 0 ]) < min_dist ] # Find the minimum distance in the strip min_dist_strip = strip_closest ( strip , min_dist ) return min ( min_dist , min_dist_strip ) # Example Usage points = [( 0 , 0 ), ( 1 , 1 ), ( 2 , 2 ), ( 3 , 3 ), ( 4 , 4 )] result = closest_pair ( points ) print ( result ) Explanation: This pseudocode defines a function closest_pair that takes a list of points and returns the minimum distance between any pair of points using the Divide and Conquer approach. The strip_closest function is used to find the minimum distance in the strip of points that crosses the dividing line between the left and right halves. Summary: The provided pseudocode implements a Divide and Conquer algorithm to find the closest pair of points in a set of points on a plane. It recursively divides the points into halves, computes the minimum distances in each half, and then considers points close to the dividing line. The minimum distance is efficiently found using a strip of points. The overall time complexity is O(n log n).