Number of pairs of lines having integer intersection points

Given two integer arrays P[] and Q[], where p_i and q_j for each 0 <= i < size(P) and 0 <= j < size(Q) represents the line equations x - y = -p_i and x + y = q_j respectively. The task is to find the number of pairs from P[] and Q[] having integer intersection points.

Examples: 

Input: P[] = {1, 3, 2}, Q[] = {3, 0} 
Output: 3 
The pairs of lines (p, q) having integer intersection points are (1, 3), (2, 0) and (3, 3). Here p is the line parameter of P[] and q is the that of Q[].

Input: P[] = {1, 4, 3, 2}, Q[] = {3, 6, 10, 11} 
Output: 8 
 

Approach:  

• The problem can be solved easily by solving the two equations and analyzing the condition for integer intersection points.
• The two equations are x - y = -p and x + y = q.
• Solving for x and y we get, x = (q-p)/2 and y = (p+q)/2.
• It is clear that integer intersection point is possible if and only if p and q have same parity.
• Let p₀ and p₁ be the number of even and odd p_i respectively.
• Similarly, q₀ and q₁ for the number of even and odd q_i respectively.
• Therefore the required answers is p₀ * q₀ + p₁ * q₁.

Below is the implementation of the above approach: 

// C++ program to Number of pairs of lines
// having integer intersection points

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

// Count number of pairs of lines
// having integer intersection point
int countPairs(int* P, int* Q, int N, int M)
{
    // Initialize arrays to store counts
    int A[2] = { 0 }, B[2] = { 0 };

    // Count number of odd and even Pi
    for (int i = 0; i < N; i++)
        A[P[i] % 2]++;

    // Count number of odd and even Qi
    for (int i = 0; i < M; i++)
        B[Q[i] % 2]++;

    // Return the count of pairs
    return (A[0] * B[0] + A[1] * B[1]);
}

// Driver code
int main()
{
    int P[] = { 1, 3, 2 }, Q[] = { 3, 0 };
    int N = sizeof(P) / sizeof(P[0]);
    int M = sizeof(Q) / sizeof(Q[0]);

    cout << countPairs(P, Q, N, M);

    return 0;
}

// Java program to Number of pairs of lines
// having integer intersection points
class GFG
{

// Count number of pairs of lines
// having integer intersection point
static int countPairs(int []P, int []Q, 
                      int N, int M)
{
    // Initialize arrays to store counts
    int []A = new int[2], B = new int[2];

    // Count number of odd and even Pi
    for (int i = 0; i < N; i++)
        A[P[i] % 2]++;

    // Count number of odd and even Qi
    for (int i = 0; i < M; i++)
        B[Q[i] % 2]++;

    // Return the count of pairs
    return (A[0] * B[0] + A[1] * B[1]);
}

// Driver code
public static void main(String[] args)
{
    int []P = { 1, 3, 2 };
    int []Q = { 3, 0 };
    int N = P.length;
    int M = Q.length;

    System.out.print(countPairs(P, Q, N, M));
}
}

// This code is contributed by Rajput-Ji

# Python3 program to Number of pairs of lines
# having eger ersection pos

# Count number of pairs of lines
# having eger ersection po
def countPairs(P, Q, N, M):
    
    # Initialize arrays to store counts
    A = [0] * 2
    B = [0] * 2

    # Count number of odd and even Pi
    for i in range(N):
        A[P[i] % 2] += 1

    # Count number of odd and even Qi
    for i in range(M):
        B[Q[i] % 2] += 1

    # Return the count of pairs
    return (A[0] * B[0] + A[1] * B[1])

# Driver code

P = [1, 3, 2]
Q = [3, 0]
N = len(P)
M = len(Q)

print(countPairs(P, Q, N, M))

# This code is contributed by mohit kumar 29

// C# program to Number of pairs of lines
// having integer intersection points
using System;

class GFG
{
    
    // Count number of pairs of lines
    // having integer intersection point
    static int countPairs(int []P, int []Q, 
                        int N, int M)
    {
        // Initialize arrays to store counts
        int []A = new int[2];
        int []B = new int[2];
    
        // Count number of odd and even Pi
        for (int i = 0; i < N; i++)
            A[P[i] % 2]++;
    
        // Count number of odd and even Qi
        for (int i = 0; i < M; i++)
            B[Q[i] % 2]++;
    
        // Return the count of pairs
        return (A[0] * B[0] + A[1] * B[1]);
    }
    
    // Driver code
    public static void Main()
    {
        int []P = { 1, 3, 2 };
        int []Q = { 3, 0 };
        int N = P.Length;
        int M = Q.Length;
    
        Console.Write(countPairs(P, Q, N, M));
    }
}

// This code is contributed by AnkitRai01

<script>

// Javascript program to Number of 
// pairs of lines having integer 
// intersection points

// Count number of pairs of lines
// having integer intersection point
function countPairs(P, Q, N, M)
{
    
    // Initialize arrays to store counts
    var A = [0, 0], B = [0, 0];

    // Count number of odd and even Pi
    for(var i = 0; i < N; i++)
        A[P[i] % 2]++;

    // Count number of odd and even Qi
    for(var i = 0; i < M; i++)
        B[Q[i] % 2]++;

    // Return the count of pairs
    return(A[0] * B[0] + A[1] * B[1]);
}

// Driver code
var P = [ 1, 3, 2 ], Q = [ 3, 0 ];
var N = P.length;
var M = Q.length;

document.write(countPairs(P, Q, N, M));

// This code is contributed by rrrtnx

</script>

3

Time Complexity: O(P + Q)

Auxiliary Space: O(1)