Number of lines from given N points not parallel to X or Y axis

Given N distinct integers points on 2D Plane. The task is to count the number of lines which are formed from given N points and not parallel to X or Y-axis.

Examples:

Input: points[][] = {{1, 2}, {1, 5}, {1, 15}, {2, 10}} 
Output: 3 
Chosen pairs are {(1, 2), (2, 10)}, {(1, 5), (2, 10)}, {(1, 15), (2, 10)}.

Input: points[][] = {{1, 2}, {2, 5}, {3, 15}} 
Output: 3 
Choose any pair of points. 

Approach:

1. We know that 
   
   • Line formed by connecting any two-points will be parallel to X-axis if they have the same Y coordinates
   • It will be parallel to Y-axis if they have the same X coordinates.
2. Total number of line segments that can formed from N points = 
   
3. Now we will exclude those line segments which are parallel to the X-axis or the Y-axis.
4. For each X coordinate and Y coordinate, calculate the number of points and exclude those line segments at the end.

Below is the implementation of above approach:  

C++

// C++ program to find the number
// of lines which are formed from
// given N points and not parallel
// to X or Y axis

#include <bits/stdc++.h>
using namespace std;

// Function to find the number of lines
// which are formed from given N points
// and not parallel to X or Y axis
int NotParallel(int p[][2], int n)
{
    // This will store the number of points has
    // same x or y coordinates using the map as
    // the value of coordinate can be very large
    map<int, int> x_axis, y_axis;

    for (int i = 0; i < n; i++) {

        // Counting frequency of each x and y
        // coordinates
        x_axis[p[i][0]]++;
        y_axis[p[i][1]]++;
    }

    // Total number of pairs can be formed
    int total = (n * (n - 1)) / 2;

    for (auto i : x_axis) {
        int c = i.second;

        // We can not choose pairs from these as
        // they have same x coordinatethus they
        // will result line segment
        // parallel to y axis
        total -= (c * (c - 1)) / 2;
    }

    for (auto i : y_axis) {
        int c = i.second;

        // we can not choose pairs from these as
        // they have same y coordinate thus they
        // will result line segment
        // parallel to x-axis
        total -= (c * (c - 1)) / 2;
    }

    // Return the required answer
    return total;
}

// Driver Code
int main()
{

    int p[][2] = { { 1, 2 },
                   { 1, 5 },
                   { 1, 15 },
                   { 2, 10 } };

    int n = sizeof(p) / sizeof(p[0]);

    // Function call
    cout << NotParallel(p, n);

    return 0;
}

Java

// Java program to find the number
// of lines which are formed from
// given N points and not parallel
// to X or Y axis
import java.util.*;

class GFG{

// Function to find the number of lines
// which are formed from given N points
// and not parallel to X or Y axis
static int NotParallel(int p[][], int n)
{
    // This will store the number of points has
    // same x or y coordinates using the map as
    // the value of coordinate can be very large
    HashMap<Integer,Integer> x_axis = new HashMap<Integer,Integer>();
    HashMap<Integer,Integer> y_axis = new HashMap<Integer,Integer>();

    for (int i = 0; i < n; i++) {

        // Counting frequency of each x and y
        // coordinates
        if(x_axis.containsKey(p[i][0]))
            x_axis.put(p[i][0], x_axis.get(p[i][0])+1);
        else
            x_axis.put(p[i][0], 1);
        if(y_axis.containsKey(p[i][1]))
            y_axis.put(p[i][1], y_axis.get(p[i][1])+1);
        else
            y_axis.put(p[i][1], 1);
    }

    // Total number of pairs can be formed
    int total = (n * (n - 1)) / 2;

    for (Map.Entry<Integer,Integer> i : x_axis.entrySet()) {
        int c = i.getValue();

        // We can not choose pairs from these as
        // they have same x coordinatethus they
        // will result line segment
        // parallel to y axis
        total -= (c * (c - 1)) / 2;
    }

    for (Map.Entry<Integer,Integer> i : y_axis.entrySet()) {
        int c = i.getValue();

        // we can not choose pairs from these as
        // they have same y coordinate thus they
        // will result line segment
        // parallel to x-axis
        total -= (c * (c - 1)) / 2;
    }

    // Return the required answer
    return total;
}

// Driver Code
public static void main(String[] args)
{

    int p[][] = { { 1, 2 },
                   { 1, 5 },
                   { 1, 15 },
                   { 2, 10 } };

    int n = p.length;

    // Function call
    System.out.print(NotParallel(p, n));

}
}

// This code is contributed by PrinciRaj1992

C#

// C# program to find the number
// of lines which are formed from
// given N points and not parallel
// to X or Y axis
using System;
using System.Collections.Generic;

class GFG{

// Function to find the number of lines
// which are formed from given N points
// and not parallel to X or Y axis
static int NotParallel(int [,]p, int n)
{
    // This will store the number of points has
    // same x or y coordinates using the map as
    // the value of coordinate can be very large
    Dictionary<int,int> x_axis = new Dictionary<int,int>();
    Dictionary<int,int> y_axis = new Dictionary<int,int>();

    for (int i = 0; i < n; i++) {

        // Counting frequency of each x and y
        // coordinates
        if(x_axis.ContainsKey(p[i, 0]))
            x_axis[p[i, 0]] = x_axis[p[i, 0]] + 1;
        else
            x_axis.Add(p[i, 0], 1);
        if(y_axis.ContainsKey(p[i, 1]))
            y_axis[p[i, 1]] = y_axis[p[i, 1]] + 1;
        else
            y_axis.Add(p[i, 1], 1);
    }

    // Total number of pairs can be formed
    int total = (n * (n - 1)) / 2;

    foreach (KeyValuePair<int,int> i in x_axis) {
        int c = i.Value;

        // We can not choose pairs from these as
        // they have same x coordinatethus they
        // will result line segment
        // parallel to y axis
        total -= (c * (c - 1)) / 2;
    }

    foreach (KeyValuePair<int,int> i in y_axis) {
        int c = i.Value;

        // we can not choose pairs from these as
        // they have same y coordinate thus they
        // will result line segment
        // parallel to x-axis
        total -= (c * (c - 1)) / 2;
    }

    // Return the required answer
    return total;
}

// Driver Code
public static void Main(String[] args)
{

    int [,]p = { { 1, 2 },
                   { 1, 5 },
                   { 1, 15 },
                   { 2, 10 } };

    int n = p.GetLength(0);

    // Function call
    Console.Write(NotParallel(p, n));
}
}

// This code is contributed by Princi Singh

Python3

# Python3 program to find the number
# of lines which are formed from
# given N points and not parallel
# to X or Y axis

# Function to find the number of lines
# which are formed from given N points
# and not parallel to X or Y axis
def NotParallel(p, n) :

    # This will store the number of points has
    # same x or y coordinates using the map as
    # the value of coordinate can be very large
    x_axis  = {}; y_axis = {};

    for i in range(n) :

        # Counting frequency of each x and y
        # coordinates
        if p[i][0] not in x_axis :
            x_axis[p[i][0]] = 0;

        x_axis[p[i][0]] += 1;

        if p[i][1] not in y_axis :
            y_axis[p[i][1]] = 0;

        y_axis[p[i][1]] += 1;

    # Total number of pairs can be formed
    total = (n * (n - 1)) // 2;

    for i in x_axis :
        c =  x_axis[i];

        # We can not choose pairs from these as
        # they have same x coordinatethus they
        # will result line segment
        # parallel to y axis
        total -= (c * (c - 1)) // 2;

    for i in y_axis :
        c = y_axis[i];

        # we can not choose pairs from these as
        # they have same y coordinate thus they
        # will result line segment
        # parallel to x-axis
        total -= (c * (c - 1)) // 2;

    # Return the required answer
    return total;


# Driver Code
if __name__ == "__main__" :

    p = [ [ 1, 2 ],
            [1, 5 ],
            [1, 15 ],
            [ 2, 10 ] ];

    n = len(p);

    # Function call
    print(NotParallel(p, n));

    # This code is contributed by AnkitRai01

Javascript

<script>

// Javascript program to find the number
// of lines which are formed from
// given N points and not parallel
// to X or Y axis

// Function to find the number of lines
// which are formed from given N points
// and not parallel to X or Y axis
function NotParallel(p, n)
{
    // This will store the number of points has
    // same x or y coordinates using the map as
    // the value of coordinate can be very large
    var x_axis = new Map(), y_axis = new Map();

    for (var i = 0; i < n; i++) {

        // Counting frequency of each x and y
        // coordinates
        if(x_axis.has(p[i][0]))
            x_axis.set(p[i][0], x_axis.get(p[i][0])+1)
        else
            x_axis.set(p[i][0], 1)

        if(y_axis.has(p[i][1]))
            y_axis.set(p[i][1], y_axis.get(p[i][1])+1)
        else
            y_axis.set(p[i][1], 1)
    }

    // Total number of pairs can be formed
    var total = (n * (n - 1)) / 2;

    x_axis.forEach((value, key) => {

        var c = value;

        // We can not choose pairs from these as
        // they have same x coordinatethus they
        // will result line segment
        // parallel to y axis
        total -= (c * (c - 1)) / 2;
    });

    y_axis.forEach((value, key) => {

        var c = value;

        // we can not choose pairs from these as
        // they have same y coordinate thus they
        // will result line segment
        // parallel to x-axis
        total -= (c * (c - 1)) / 2;
    });

    // Return the required answer
    return total;
}

// Driver Code
var p = [ [ 1, 2 ],
               [ 1, 5 ],
               [ 1, 15 ],
               [ 2, 10 ] ];
var n = p.length;

// Function call
document.write( NotParallel(p, n));

// This code is contributed by itsok.
</script>

Output:

3

Time Complexity: O(N)

Auxiliary Space: O(N), since N extra space has been taken.