Maximum distinct lines passing through a single point

Given N lines represented by two points (x1, y1) and (x2, y2). The task is to find maximum number of lines which can pass through a single point, without superimposing (or covering) any other line. We can move any line but not rotate it.

Examples:

Input : Line 1 : x1 = 1, y1 = 1, x2 = 2, y2 = 2
        Line 2 : x2 = 2, y1 = 2, x2 = 4, y2 = 10
Output : 2
There are two lines. These two lines are not 
parallel, so both of them will pass through
a single point.


Input : Line 1 : x1 = 1, y1 = 5, x2 = 1, y2 = 10
        Line 2 : x2 = 5, y1 = 1, x2 = 10, y2 = 1
Output : 2



  • Represent lines as pair (m, c) where line can be given as y=mx+c, called line slope form. We can now see that we can change the c for any line, but cannot modify m.
  • Lines having same value of m parallel, given that (c1 ≠ c2). Also no two parallel lines can pass through same point without superimposing to each other.
  • So, our problem reduces to finding different values of slopes from given set of lines.

We can calculate slope of a line as \frac{(y2-y1)}{(x2-x1)}, add them to a set and count the number of distinct values of slope in set. But we have to handle vertical lines separately.
So, if  x1 = x2 then, slope = INT_MAX.
Otherwise, slope = \frac{(y2-y1)}{(x2-x1)}.

Below is the implementation of the approach.

C++

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// C++ program to find maximum number of lines
// which can pass through a single point
#include <bits/stdc++.h>
using namespace std;
  
// function to find maximum lines which passes
// through a single point
int maxLines(int n, int x1[], int y1[], 
                int x2[], int y2[])
{
    unordered_set<double> s;
  
    double slope;
    for (int i = 0; i < n; ++i) {
        if (x1[i] == x2[i])
            slope = INT_MAX;
        else
            slope = (y2[i] - y1[i]) * 1.0 
                    / (x2[i] - x1[i]) * 1.0;
  
        s.insert(slope);
    }
  
    return s.size();
}
  
// Driver program
int main()
{
    int n = 2, x1[] = { 1, 2 }, y1[] = { 1, 2 },
            x2[] = { 2, 4 }, y2[] = { 2, 10 };
    cout << maxLines(n, x1, y1, x2, y2);
    return 0;
}
// This code is written by
// Sanjit_Prasad

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Java

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// Java program to find maximum number of lines
// which can pass through a single point
  
import java.util.*;
import java.lang.*;
import java.io.*;
  
class GFG{
  
// function to find maximum lines which passes
// through a single point
static int maxLines(int n, int x1[], int y1[], 
                    int x2[], int y2[])
{
    Set<Double> s=new HashSet<Double>();
  
    double slope;
    for (int i = 0; i < n; ++i) {
        if (x1[i] == x2[i])
            slope = Integer.MAX_VALUE;
        else
            slope = (y2[i] - y1[i]) * 1.0 
                    / (x2[i] - x1[i]) * 1.0;
  
        s.add(slope);
    }
  
    return s.size();
}
  
// Driver program
public static void main(String args[])
{
    int n = 2, x1[] = { 1, 2 }, y1[] = { 1, 2 },
            x2[] = { 2, 4 }, y2[] = { 2, 10 };
    System.out.print(maxLines(n, x1, y1, x2, y2));
}
}
// This code is written by
// Subhadeep

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Python3

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# Python3 program to find maximum number 
# of lines which can pass through a 
# single point
import sys
# function to find maximum lines 
# which passes through a single point
def maxLines(n, x1, y1, x2, y2):
  
    s = [];
  
    slope=sys.maxsize;
    for i in range(n):
        if (x1[i] == x2[i]):
            slope = sys.maxsize;
        else:
            slope = (y2[i] - y1[i]) * 1.0 /(x2[i] - x1[i]) * 1.0;
  
        s.append(slope);
  
    return len(s);
  
# Driver Code
n = 2;
x1 = [ 1, 2 ];
y1 = [1, 2];
x2 = [2, 4];
y2 = [2, 10];
print(maxLines(n, x1, y1, x2, y2));
  
# This code is contributed by mits

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C#

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// C# program to find maximum number of lines
// which can pass through a single point
using System;
using System.Collections.Generic;
  
class GFG
{
  
// function to find maximum lines which passes
// through a single point
static int maxLines(int n, int []x1, int []y1, 
                    int []x2, int []y2)
{
    HashSet<Double> s = new HashSet<Double>();
  
    double slope;
    for (int i = 0; i < n; ++i) 
    {
        if (x1[i] == x2[i])
            slope = int.MaxValue;
        else
            slope = (y2[i] - y1[i]) * 1.0
                    / (x2[i] - x1[i]) * 1.0;
  
        s.Add(slope);
    }
  
    return s.Count;
}
  
// Driver code
public static void Main()
{
    int n = 2;
    int []x1 = { 1, 2 }; int []y1 = { 1, 2 };
    int []x2 = { 2, 4 }; int []y2 = { 2, 10 };
    Console.Write(maxLines(n, x1, y1, x2, y2));
}
}
  
/* This code contributed by PrinciRaj1992 */

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PHP

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<?php
// PHP program to find maximum number 
// of lines which can pass through a 
// single point
  
// function to find maximum lines 
// which passes through a single point
function maxLines($n, $x1, $y1, $x2, $y2)
{
    $s = array();
  
    $slope;
    for ($i = 0; $i < $n; ++$i)
    {
        if ($x1[$i] == $x2[$i])
            $slope = PHP_INT_MAX;
        else
            $slope = ($y2[$i] - $y1[$i]) * 1.0 /
                     ($x2[$i] - $x1[$i]) * 1.0;
  
        array_push($s, $slope);
    }
  
    return count($s);
}
  
// Driver Code
$n = 2;
$x1 = array( 1, 2 );
$y1 = array(1, 2);
$x2 = array(2, 4);
$y2 = array(2, 10);
echo maxLines($n, $x1, $y1, $x2, $y2);
  
// This code is contributed by mits
?>

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Output:

2

Time Complexity: O(N)



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