Number of ordered points pair satisfying line equation

Given an array of n integers, slope of a line i. e., m and the intercept of the line i.e c, Count the number of ordered pairs(i, j) of points where i ≠ j, such that point (Ai, Aj) satisfies the line formed with given slope and intercept.
Note : The equation of the line is y = mx + c, where m is the slope of the line and c is the intercept.

Examples :

Input : m = 1, c = 1, arr[] = [ 1, 2, 3, 4, 2 ]
Output : 5 ordered points
Explanation : The equation of the line with given slope and intercept is : y = x + 1. The Number of pairs (i, j), for which (arri, arrj) satisfies the above equation of the line are : (1, 2), (1, 5), (2, 3), (3, 4), (5, 3).



Input : m = 2, c = 1, arr[] = [ 1, 2, 3, 4, 2, 5 ]
Output : 3 ordered points

Method 1 (Brute Force):
Generate all possible pairs (i, j) and check if a particular ordered pair (i, j) is such that, (arri, arrj) satisfies the given equation of the line y = mx + c, and i ≠ j. If the point is valid(a point is valid if the above condition is satisfied), increment the counter which stores the total number of valid points.

C++

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// CPP code to count the number of ordered
// pairs satisfying Line Equation
#include <bits/stdc++.h>
  
using namespace std;
  
/* Checks if (i, j) is valid, a point (i, j)
   is valid if point (arr[i], arr[j])
   satisfies the equation y = mx + c And 
   i is not equal to j*/
bool isValid(int arr[], int i, int j, 
             int m, int c)
{
  
    // check if i equals to j
    if (i == j) 
        return false;
      
      
    // Equation LHS = y, and RHS = mx + c
    int lhs = arr[j];    
    int rhs = m * arr[i] + c;
  
    return (lhs == rhs);
}
  
/* Returns the number of ordered pairs
   (i, j) for which point (arr[i], arr[j])
   satisfies the equation of the line 
   y = mx + c */
int findOrderedPoints(int arr[], int n, 
                      int m, int c)
{
  
    int counter = 0;
  
    // for every possible (i, j) check
    // if (a[i], a[j]) satisfies the 
    // equation y = mx + c
    for (int i = 0; i < n; i++) 
    {
        for (int j = 0; j < n; j++) 
        {
            // (firstIndex, secondIndex) 
            // is same as (i, j)
            int firstIndex = i, secondIndex = j;
  
            // check if (firstIndex, 
            // secondIndex) is a valid point
            if (isValid(arr, firstIndex, secondIndex, m, c)) 
                counter++;
        }
    }
    return counter;
}
  
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4, 2 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    // equation of line is y = mx + c
    int m = 1, c = 1;
    cout << findOrderedPoints(arr, n, m, c);
    return 0;
}

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Java

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// Java code to find number of ordered
// points satisfying line equation
import java.io.*;
  
public class GFG {
      
    // Checks if (i, j) is valid,
    // a point (i, j) is valid if 
    // point (arr[i], arr[j]) 
    // satisfies the equation 
    // y = mx + c And 
    // i is not equal to j
    static boolean isValid(int []arr, int i, 
                        int j, int m, int c)
    {
      
        // check if i equals to j
        if (i == j) 
            return false;
          
          
        // Equation LHS = y,
        // and RHS = mx + c
        int lhs = arr[j]; 
        int rhs = m * arr[i] + c;
      
        return (lhs == rhs);
    }
      
    /* Returns the number of ordered pairs
    (i, j) for which point (arr[i], arr[j])
    satisfies the equation of the line 
    y = mx + c */
    static int findOrderedPoints(int []arr, 
                       int n, int m, int c)
    {
      
        int counter = 0;
      
        // for every possible (i, j) check
        // if (a[i], a[j]) satisfies the 
        // equation y = mx + c
        for (int i = 0; i < n; i++) 
        {
            for (int j = 0; j < n; j++) 
            {
                  
                // (firstIndex, secondIndex) 
                // is same as (i, j)
                int firstIndex = i,
                   secondIndex = j;
      
                // check if (firstIndex, 
                // secondIndex) is a 
                // valid point
                if (isValid(arr, firstIndex, 
                         secondIndex, m, c)) 
                    counter++;
            }
        }
        return counter;
    }
      
    // Driver Code
    public static void main(String args[])
    {
        int []arr = { 1, 2, 3, 4, 2 };
        int n = arr.length;
      
        // equation of line is y = mx + c
        int m = 1, c = 1;
        System.out.print(
           findOrderedPoints(arr, n, m, c));
    }
}
  
// This code is contributed by
// Manish Shaw (manishshaw1)

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Python3

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# Python code to count the number of ordered
# pairs satisfying Line Equation
  
# Checks if (i, j) is valid, a point (i, j)
# is valid if point (arr[i], arr[j])
# satisfies the equation y = mx + c And 
# i is not equal to j
def isValid(arr, i, j, m, c) :
  
    # check if i equals to j
    if (i == j) :
        return False
      
      
    # Equation LHS = y, and RHS = mx + c
    lhs = arr[j];
    rhs = m * arr[i] + c
  
    return (lhs == rhs)
  
# Returns the number of ordered pairs
# (i, j) for which point (arr[i], arr[j])
# satisfies the equation of the line 
# y = mx + c */
def findOrderedPoints(arr, n, m, c) :
  
    counter = 0
  
    # for every possible (i, j) check
    # if (a[i], a[j]) satisfies the 
    # equation y = mx + c
    for i in range(0, n) :
        for j in range(0, n) :
            # (firstIndex, secondIndex) 
            # is same as (i, j)
            firstIndex = i
            secondIndex = j
  
            # check if (firstIndex, 
            # secondIndex) is a valid point
            if (isValid(arr, firstIndex,
                      secondIndex, m, c)) : 
                counter = counter + 1
  
    return counter
  
# Driver Code
arr = [ 1, 2, 3, 4, 2 ]
n = len(arr)
  
# equation of line is y = mx + c
m = 1
c = 1
print (findOrderedPoints(arr, n, m, c))
  
# This code is contributed by Manish Shaw
# (manishshaw1)

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C#

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// C# code to find number of ordered
// points satisfying line equation
using System;
class GFG {
      
    // Checks if (i, j) is valid,
    // a point (i, j) is valid if 
    // point (arr[i], arr[j]) 
    // satisfies the equation 
    // y = mx + c And 
    // i is not equal to j
    static bool isValid(int []arr, int i, 
                     int j, int m, int c)
    {
      
        // check if i equals to j
        if (i == j) 
            return false;
          
          
        // Equation LHS = y,
        // and RHS = mx + c
        int lhs = arr[j]; 
        int rhs = m * arr[i] + c;
      
        return (lhs == rhs);
    }
      
    /* Returns the number of ordered pairs
      (i, j) for which point (arr[i], arr[j])
      satisfies the equation of the line 
       y = mx + c */
    static int findOrderedPoints(int []arr, int n, 
                                     int m, int c)
    {
      
        int counter = 0;
      
        // for every possible (i, j) check
        // if (a[i], a[j]) satisfies the 
        // equation y = mx + c
        for (int i = 0; i < n; i++) 
        {
            for (int j = 0; j < n; j++) 
            {
                  
                // (firstIndex, secondIndex) 
                // is same as (i, j)
                int firstIndex = i, secondIndex = j;
      
                // check if (firstIndex, 
                // secondIndex) is a valid point
                if (isValid(arr, firstIndex, secondIndex, m, c)) 
                    counter++;
            }
        }
        return counter;
    }
      
    // Driver Code
    public static void Main()
    {
        int []arr = { 1, 2, 3, 4, 2 };
        int n = arr.Length;
      
        // equation of line is y = mx + c
        int m = 1, c = 1;
        Console.Write(findOrderedPoints(arr, n, m, c));
    }
}
  
// This code is contributed by
// Manish Shaw (manishshaw1)

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PHP

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<?php
// PHP code to count the 
// number of ordered pairs
// satisfying Line Equation
  
/* Checks if (i, j) is valid,
a point (i, j) is valid if 
point (arr[i], arr[j]) satisfies 
the equation y = mx + c And i 
is not equal to j*/
function isValid($arr, $i
                 $j, $m, $c)
{
    // check if i equals to j
    if ($i == $j
        return false;
      
    // Equation LHS = y, and
    // RHS = mx + c
    $lhs = $arr[$j];
    $rhs = $m * $arr[$i] + $c;
  
    return ($lhs == $rhs);
}
  
/* Returns the number of 
ordered pairs (i, j) for 
which point (arr[i], arr[j]) 
satisfies the equation of 
the line y = mx + c */
function findOrderedPoints($arr, $n
                           $m, $c)
{
  
    $counter = 0;
  
    // for every possible (i, j) 
    // check if (a[i], a[j]) 
    // satisfies the equation 
    // y = mx + c
    for ($i = 0; $i < $n; $i++) 
    {
        for ($j = 0; $j < $n; $j++) 
        {
            // (firstIndex, secondIndex) 
            // is same as (i, j)
            $firstIndex = $i; $secondIndex = $j;
  
            // check if (firstIndex, 
            // secondIndex) is a valid point
            if (isValid($arr, $firstIndex
                        $secondIndex, $m, $c)) 
                $counter++;
        }
    }
    return $counter;
}
  
// Driver Code
$arr = array( 1, 2, 3, 4, 2 );
$n = count($arr);
  
// equation of line 
// is y = mx + c
$m = 1; $c = 1;
echo (findOrderedPoints($arr, $n, $m, $c));
  
// This code is contributed by 
// Manish Shaw (manishshaw1)
?>

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Output :

5

Time Complexity : O(n2)

Method 2 (Efficient) :
Given a x coordinate of a point, for each x there is a unique value of y and the value of y is nothing but m * x + c. So, for each possible x coordinate of the array arr, calculate how many times the unique value of y which satisfies the equation of the line occurs in that array. Store count of all the integers of array, arr in a map. Now, for each value, arri, add to the answer, the number of occurrences of m * arri + c. For a given i, m * a[i] + c occurs x times in the array, then, add x to our counter for total valid points, but need to check that if a[i] = m * a[i] + c then, it is obvious that since this occurs x times in the array then one occurrence is at the ith index and rest (x – 1) occurrences are the valid y coordinates so add (x – 1) to our points counter.

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// CPP code to find number of ordered
// points satisfying line equation
#include <bits/stdc++.h>
using namespace std;
  
/* Returns the number of ordered pairs
   (i, j) for which point (arr[i], arr[j])
   satisfies the equation of the line 
   y = mx + c */
int findOrderedPoints(int arr[], int n, 
                      int m, int c)
{
    int counter = 0;
  
    // map stores the frequency 
    // of arr[i] for all i
    unordered_map<int, int> frequency;
  
    for (int i = 0; i < n; i++) 
        frequency[arr[i]]++;
  
    for (int i = 0; i < n; i++) 
    {
        int xCoordinate = arr[i];
        int yCoordinate = (m * arr[i] + c);
  
        // if for a[i](xCoordinate), 
        // a yCoordinate exists in the map
        // add the frequency of yCoordinate
        // to the counter
  
        if (frequency.find(yCoordinate) != 
            frequency.end()) 
            counter += frequency[yCoordinate];
  
        // check if xCoordinate = yCoordinate,
        // if this is true then since we only
        // want (i, j) such that i != j, decrement
        // the counter by one to avoid points 
        // of type (arr[i], arr[i])
        if (xCoordinate == yCoordinate) 
            counter--;
    }
    return counter;
}
  
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4, 2 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int m = 1, c = 1;
    cout << findOrderedPoints(arr, n, m, c);
    return 0;
}

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Output:

5

Time Complexity : O(n)



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