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 3

Input :n= 6, points= (1, 1), (4, 5), (3, 4), (4, 3), (7, 4), (3, 1)
Output :No such square

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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:

 // C++ implemenataion of the above approach #include using namespace std;    // find the largest square void findLargestSquare(long long int points[], int n) {     // map to store which points exist     map, int> m;        // mark the available points     for (int i = 0; i < n; i++) {         m[make_pair(points[i], points[i])]++;     }     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], points[i])]--;         for (int j = 0; j < n; j++) {                // remove the chosen point             m[make_pair(points[j], points[j])]--;                // check if the other two points exist             if (i != j                    && (points[i]-points[j]) == (points[i]-points[j])){                 if (m[make_pair(points[i], points[j])] > 0                      && m[make_pair(points[j], points[i])] > 0) {                        // if the square is largest then store it                     if (side < abs(points[i] - points[j])                           || (side == abs(points[i] - points[j])                              && ((points[i] * points[i]                                     + points[i] * points[i])                                        < (x * x + y * y)))) {                         x = points[i];                         y = points[i];                         side = abs(points[i] - points[j]);                     }                 }             }                // add the removed point             m[make_pair(points[j], points[j])]++;         }            // add the removed point         m[make_pair(points[i], points[i])]++;     }        // 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 code int main() {     int n = 6;        // given points     long long int points[n]       = { { 1, 1 }, { 4, 4 }, { 3, 4 }, { 4, 3 }, { 1, 4 }, { 4, 1 } };        // find the largest square     findLargestSquare(points, n);        return 0; }

Output:

Side of the square is : 3,
points of the square are 1, 1 4, 1 1, 4 4, 4

Time Complexity: O(N^2)

My Personal Notes arrow_drop_up Second year Department of Information Technology Jadavpur University

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