Problem 1: A set of points in the plane, p1 = (x1; y1); p2 = (x2; y2); : : : ; pn = (xn; yn) aregiven, Give an algorithm to find the closest pair of points: that is, the pair pi 6 = pj for whichthe distance between pi and pj , that is, √(xi − xj )2 + (yi − yj )2; is minimized.
Question
Problem 1: A set of points in the plane, p1 = (x1; y1); p2 = (x2; y2); : : : ; pn = (xn; yn) aregiven, Give an algorithm to find the closest pair of points: that is, the pair pi 6 = pj for whichthe distance between pi and pj , that is, √(xi − xj )2 + (yi − yj )2; is minimized.
Solution
The problem you're asking about is a classic computational geometry problem known as the "closest pair of points problem". Here's a simple algorithm to solve it:
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Sort points according to their x-coordinates.
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Divide the set of points in half.
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Find the smallest distance in both halves. Let this be 'd'.
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Find the smallest distance between the points in the two halves. This will be the minimum of the smallest distances in the two halves and the distance between the points on either side of the dividing line that are less than 'd' apart.
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The smallest of these distances is the smallest distance between any two points.
This algorithm is known as the "divide and conquer" algorithm for the closest pair of points problem. It has a time complexity of O(n log n), which is much more efficient than the naive approach of checking the distance between every pair of points, which has a time complexity of O(n^2).
Similar Questions
Which algorithm is used to find the closest pair of points in a set of points? Question 31Select one: Cosine Similarity Search algorithm Greedy algorithm Dynamic Programming algorithm Brute Force algorithm
The distance D between two points with coordinates (x1,y1) and (x2,y2) on a plane is given byD = [(x2- x1)2 + (y2 – y1)2]1/2The distance between two points (x1,y1,z1) and (x2,y2,z2) in a three dimensional space is given byD = [(x2-x1)2 +(y2-y1)2 + (z2-z1)2]1/2Develop a program in C++ using function overloading by writing the same function ComputeDistance() with different signatures. Round off distance to two decimal pointsNote: Syntax to print 'x' decimal places of variable 'a'include <iomanip>usecout<<fixed<<setprecision(x)<<a;Input formatx coordinate of point 1y coordinate of point 1z coordinate of point 1x coordinate of point 2y coordinate of point 2z coordinate of point 2Output formatDistance in two dimensional planeDistance in three dimensional space
Classical Optimization TechniquesOptimization is the method of findingans.the minimum pointthe maximum pointAll of the aboveThe best available point Previous Marked for Review Next
Which pair of coordinate points have the least distance between them?REASONING(-6, -11) and (14, 10)(17, 9) and (7, 33)(3, -7) and (11, 8)(9, 2) and (16, -22)
So, the optimal point is located at x = 9/5, y = -1/5, with λ = 8/5, and f(x, y) = 4. give these in 3dp
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