Find Four points such that they form a square whose sides are parallel to x and y axes
Given ‘n’ pair of points, the task is to find four points such that they form a square whose sides are parallel to x and y axes or print “No such square” otherwise.
If more than one square is possible then choose the one with the maximum area.
Examples:
Input : n = 6, points = (1, 1), (4, 4), (3, 4), (4, 3), (1, 4), (4, 1)
Output :
side of the square is : 3, points of the square are 1, 1 4, 1 1, 4 4, 4
Explanation: The points 1, 1 4, 1 1, 4 4, 4 form a square of side 3Input :n= 6, points= (1, 1), (4, 5), (3, 4), (4, 3), (7, 4), (3, 1)
Output :No such square
Simple approach: Choose all possible pairs of points with four nested loops and then see if the points form a square which is parallel to principal axes. If yes then check if it is the largest square so far in terms of the area and store the result, and then display the result at the end of the program.
Time Complexity: O(N^4)
Efficient Approach: Create a nested loop for top right and bottom left corner of the square and form a square with those two points, then check if the other two points which were assumed actually exist. To check if a point exists or not, create a map and store the points in the map to reduce the time to check whether the points exist. Also, keep in check the largest square by area so far and print it in the end.
Below is the implementation of the above approach:
CPP
// C++ implemenataion of the above approach#include <bits/stdc++.h>using namespace std; // find the largest squarevoid findLargestSquare(long long int points[][2], int n){ // map to store which points exist map<pair<long long int, long long int>, int> m; // mark the available points for (int i = 0; i < n; i++) { m[make_pair(points[i][0], points[i][1])]++; } long long int side = -1, x = -1, y = -1; // a nested loop to choose the opposite corners of square for (int i = 0; i < n; i++) { // remove the chosen point m[make_pair(points[i][0], points[i][1])]--; for (int j = 0; j < n; j++) { // remove the chosen point m[make_pair(points[j][0], points[j][1])]--; // check if the other two points exist if (i != j && (points[i][0]-points[j][0]) == (points[i][1]-points[j][1])){ if (m[make_pair(points[i][0], points[j][1])] > 0 && m[make_pair(points[j][0], points[i][1])] > 0) { // if the square is largest then store it if (side < abs(points[i][0] - points[j][0]) || (side == abs(points[i][0] - points[j][0]) && ((points[i][0] * points[i][0] + points[i][1] * points[i][1]) < (x * x + y * y)))) { x = points[i][0]; y = points[i][1]; side = abs(points[i][0] - points[j][0]); } } } // add the removed point m[make_pair(points[j][0], points[j][1])]++; } // add the removed point m[make_pair(points[i][0], points[i][1])]++; } // display the largest square if (side != -1) cout << "Side of the square is : " << side << ", \npoints of the square are " << x << ", " << y << " " << (x + side) << ", " << y << " " << (x) << ", " << (y + side) << " " << (x + side) << ", " << (y + side) << endl; else cout << "No such square" << endl;} //Driver codeint main(){ int n = 6; // given points long long int points[n][2] = { { 1, 1 }, { 4, 4 }, { 3, 4 }, { 4, 3 }, { 1, 4 }, { 4, 1 } }; // find the largest square findLargestSquare(points, n); return 0;} |
Python3
# Python3 implemenataion of the above approach # find the largest squaredef findLargestSquare(points,n): # map to store which points exist m = dict() # mark the available points for i in range(n): m[(points[i][0], points[i][1])] = \ m.get((points[i][0], points[i][1]), 0) + 1 side = -1 x = -1 y = -1 # a nested loop to choose the opposite corners of square for i in range(n): # remove the chosen point m[(points[i][0], points[i][1])]-=1 for j in range(n): # remove the chosen point m[(points[j][0], points[j][1])]-=1 # check if the other two points exist if (i != j and (points[i][0]-points[j][0]) == \ (points[i][1]-points[j][1])): if (m[(points[i][0], points[j][1])] > 0 and m[(points[j][0], points[i][1])] > 0): # if the square is largest then store it if (side < abs(points[i][0] - points[j][0]) or (side == abs(points[i][0] - points[j][0]) and ((points[i][0] * points[i][0] + points[i][1] * points[i][1]) < (x * x + y * y)))): x = points[i][0] y = points[i][1] side = abs(points[i][0] - points[j][0]) # add the removed point m[(points[j][0], points[j][1])] += 1 # add the removed point m[(points[i][0], points[i][1])] += 1 # display the largest square if (side != -1): print("Side of the square is : ",side ,", \npoints of the square are ",x,", ",y ," " ,(x + side),", ",y ," " ,(x),", ",(y + side) ," " ,(x + side),", ",(y + side)) else: print("No such square") # Driver coden = 6 # given pointspoints=[[ 1, 1 ],[ 4, 4 ],[ 3, 4 ],[ 4, 3 ],[ 1, 4 ],[ 4, 1 ] ] # find the largest squarefindLargestSquare(points, n) # This code is contributed by mohit kumar 29 |
Side of the square is : 3, points of the square are 1, 1 4, 1 1, 4 4, 4
Time Complexity: O(N^2)
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