diff --git a/divide_and_conquer/closest_pair_of_points.py b/divide_and_conquer/closest_pair_of_points.py new file mode 100644 index 000000000..cc5be428d --- /dev/null +++ b/divide_and_conquer/closest_pair_of_points.py @@ -0,0 +1,113 @@ +""" +The algorithm finds distance btw 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, +minimum distance is obtained recursively. + +>> closest points 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. +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) +""" + + +import math + + +def euclidean_distance_sqr(point1, point2): + return pow(point1[0] - point2[0], 2) + pow(point1[1] - point2[1], 2) + + +def column_based_sort(array, column = 0): + 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 + + """ + + 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) + + """ + + 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, points_counts): + """ divide and conquer approach + + Parameters : + points, points_count (list(tuple(int, int)), int) + + Returns : + (float): distance btw closest pair of points + + """ + + # base case + if points_counts <= 3: + return dis_between_closest_pair(points, 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_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: + if abs(point[0] - points[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 = [(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))) + + diff --git a/divide_and_conquer/max_subarray_sum.py b/divide_and_conquer/max_subarray_sum.py new file mode 100644 index 000000000..0428f4e13 --- /dev/null +++ b/divide_and_conquer/max_subarray_sum.py @@ -0,0 +1,75 @@ +""" +Given a array of length n, max_subarray_sum() finds +the maximum of sum of contiguous sub-array using divide and conquer method. + +Time complexity : O(n log n) + +Ref : INTRODUCTION TO ALGORITHMS THIRD EDITION +(section : 4, sub-section : 4.1, page : 70) + +""" + + +def max_sum_from_start(array): + """ This function finds the maximum contiguous sum of array from 0 index + + Parameters : + array (list[int]) : given array + + Returns : + max_sum (int) : maximum contiguous sum of array from 0 index + + """ + array_sum = 0 + max_sum = float("-inf") + for num in array: + array_sum += num + if array_sum > max_sum: + max_sum = array_sum + return max_sum + + +def max_cross_array_sum(array, left, mid, right): + """ This function finds the maximum contiguous sum of left and right arrays + + Parameters : + array, left, mid, right (list[int], int, int, int) + + Returns : + (int) : maximum of sum of contiguous sum of left and right arrays + + """ + + max_sum_of_left = max_sum_from_start(array[left:mid+1][::-1]) + max_sum_of_right = max_sum_from_start(array[mid+1: right+1]) + return max_sum_of_left + max_sum_of_right + + +def max_subarray_sum(array, left, right): + """ Maximum contiguous sub-array sum, using divide and conquer method + + Parameters : + array, left, right (list[int], int, int) : + given array, current left index and current right index + + Returns : + int : maximum of sum of contiguous sub-array + + """ + + # base case: array has only one element + if left == right: + return array[right] + + # Recursion + mid = (left + right) // 2 + left_half_sum = max_subarray_sum(array, left, mid) + right_half_sum = max_subarray_sum(array, mid + 1, right) + cross_sum = max_cross_array_sum(array, left, mid, right) + return max(left_half_sum, right_half_sum, cross_sum) + + +array = [-2, -5, 6, -2, -3, 1, 5, -6] +array_length = len(array) +print("Maximum sum of contiguous subarray:", max_subarray_sum(array, 0, array_length - 1)) +