Count squares possible from M and N straight lines parallel to X and Y axis respectively

Given two arrays X[] and Y[] consisting of N and M integers such that there are N lines parallel to the y-axis and M lines parallel to the x-axis. The task is to find the total number of squares formed by these lines on a coordinate plane.

Each integer(say a) in the array X[] denotes lines having equation x = a, parallel to y-axis
Each integer(say b) in the array Y[] denotes lines having equation y = b, parallel to x-axis

Examples: 

Input: N = 3, M = 4, X[] = {1, 3, 7}, Y[] = {2, 4, 6, 1} 
Output:
Explanation: 
3 lines are parallel to y-axis for x = 1, x = 3 and x = 7. 
4 lines are parallel to x-axis for y = 2, y = 4, y = 6 and y = 1. 
 



From the above image, below are three possible squares formed: 
1) square CDEF (x = 1, x = 3, y = 2, y = 4), side = 2 units. 
2) square ABDC (x = 1, x = 3, y = 4, y = 6), side = 2 units. 
3) square BGHF (x = 3, x = 7, y = 2, y = 6), side = 4 units. 

Input: N = 5, M = 4, X[] = {1, 9, 2, 3, 7}, Y[] = {1, 2, 4, 6} 
Output: 8  

 

Approach: Follow the steps below to solve the problem: 

  • Find the distance between all pairs in X[] array and store the count in a Map, say M1.
  • Find the distance between all pairs in Y[] array and store the count in a Map M2.
  • If the distance of pairs of M1 is present in M2, then a square can be made by using both the pairs.
  • Therefore, the total count of squares can be calculated by adding all the counts of distances stored in M1 as well as in M2.
  • Print the total count of squares after completing the above steps.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count all the possible
// squares with given lines parallel
// to both the X and Y axis
int numberOfSquares(int X[], int Y[],
                    int N, int M)
{
    // Stores the count of all possible
    // distances in X[] & Y[] respectively
    unordered_map<int, int> m1, m2;
    int i, j, ans = 0;
 
    // Find distance between all
    // pairs in the array X[]
    for (i = 0; i < N; i++) {
        for (j = i + 1; j < N; j++) {
 
            int dist = abs(X[i] - X[j]);
 
            // Add the count to m1
            m1[dist]++;
        }
    }
 
    // Find distance between all
    // pairs in the array Y[]
    for (i = 0; i < M; i++) {
        for (j = i + 1; j < M; j++) {
 
            int dist = abs(Y[i] - Y[j]);
 
            // Add the count to m2
            m2[dist]++;
        }
    }
 
    // Find sum of m1[i] * m2[i]
    // for same distance
    for (auto i = m1.begin();
         i != m1.end(); i++) {
 
        // Find current count in m2
        if (m2.find(i->first)
            != m2.end()) {
 
            // Add to the total count
            ans += (i->second
                    * m2[i->first]);
        }
    }
 
    // Return the final count
    return ans;
}
 
// Driver Code
int main()
{
    // Given lines
    int X[] = { 1, 3, 7 };
    int Y[] = { 2, 4, 6, 1 };
 
    int N = sizeof(X) / sizeof(X[0]);
 
    int M = sizeof(Y) / sizeof(Y[0]);
 
    // Function Call
    cout << numberOfSquares(X, Y, N, M);
 
    return 0;
}

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Java

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// Java program for the above approach
import java.io.*;
import java.util.*;
 
class GFG{
 
// Function to count all the possible
// squares with given lines parallel
// to both the X and Y axis
static int numberOfSquares(int[] X, int[] Y, int N,
                           int M)
{
     
    // Stores the count of all possible
    // distances in X[] & Y[] respectively
    HashMap<Integer,
            Integer> m1 = new HashMap<Integer,
                                      Integer>();
    HashMap<Integer,
            Integer> m2 = new HashMap<Integer,
                                      Integer>();
 
    int i, j, ans = 0;
 
    // Find distance between all
    // pairs in the array X[]
    for(i = 0; i < N; i++)
    {
        for(j = i + 1; j < N; j++)
        {
            int dist = Math.abs(X[i] - X[j]);
 
            // Add the count to m1
            m1.put(dist, m1.getOrDefault(dist, 0) + 1);
        }
    }
 
    // Find distance between all
    // pairs in the array Y[]
    for(i = 0; i < M; i++)
    {
        for(j = i + 1; j < M; j++)
        {
            int dist = Math.abs(Y[i] - Y[j]);
 
            // Add the count to m2
            m2.put(dist, m2.getOrDefault(dist, 0) + 1);
        }
    }
 
    // Find sum of m1[i] * m2[i]
    // for same distance
    for(Map.Entry<Integer,
                  Integer> entry : m1.entrySet())
    {
         
        // Find current count in m2
        if (m2.containsKey(entry.getKey()))
        {
             
            // Add to the total count
            ans += (entry.getValue() *
             m2.get(entry.getKey()));
        }
    }
 
    // Return the final count
    return ans;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given lines
    int X[] = { 1, 3, 7 };
    int Y[] = { 2, 4, 6, 1 };
 
    int N = X.length;
 
    int M = Y.length;
 
    // Function call
    System.out.println(numberOfSquares(X, Y, N, M));
}
}
 
// This code is contributed by akhilsaini

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Python3

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# Python3 program for the above approach
 
# Function to count all the possible
# squares with given lines parallel
# to both the X and Y axis
def numberOfSquares(X, Y, N, M):
 
    # Stores the count of all possible
    # distances in X[] & Y[] respectively
    m1 = {}
    m2 = {}
    ans = 0
 
    # Find distance between all
    # pairs in the array X[]
    for i in range(0, N):
        for j in range(i + 1, N):
            dist = abs(X[i] - X[j])
 
            # Add the count to m1
            if dist in m1:
                m1[dist] = m1[dist] + 1
            else:
                m1[dist] = 1
 
    # Find distance between all
    # pairs in the array Y[]
    for i in range(0, M):
        for j in range(i + 1, M):
            dist = abs(Y[i] - Y[j])
 
            # Add the count to m2
            if dist in m2:
                m2[dist] = m2[dist] + 1
            else:
                m2[dist] = 1
 
    # Find sum of m1[i] * m2[i]
    # for same distance
    for key in m1:
         
        # Find current count in m2
        if key in m2:
             
            # Add to the total count
            ans = ans + (m1[key] * m2[key])
 
    # Return the final count
    return ans
 
# Driver Code
if __name__ == "__main__":
     
    # Given lines
    X = [ 1, 3, 7 ]
    Y = [ 2, 4, 6, 1 ]
 
    N = len(X)
 
    M = len(Y)
 
    # Function call
    print(numberOfSquares(X, Y, N, M))
 
# This code is contributed by akhilsaini

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C#

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// C# program for the above approach
using System;
using System.Collections.Generic;
 
class GFG{
 
// Function to count all the possible
// squares with given lines parallel
// to both the X and Y axis
static int numberOfSquares(int[] X, int[] Y, int N,
                           int M)
{
     
    // Stores the count of all possible
    // distances in X[] & Y[] respectively
    Dictionary<int,
               int> m1 = new Dictionary<int,
                                        int>();
      Dictionary<int,
                 int> m2 = new Dictionary<int,
                                          int>();
 
    int i, j, ans = 0;
 
    // Find distance between all
    // pairs in the array X[]
    for(i = 0; i < N; i++)
    {
        for(j = i + 1; j < N; j++)
        {
            int dist = Math.Abs(X[i] - X[j]);
 
            // Add the count to m1
              if (m1.ContainsKey(dist))
                m1[dist]++;
              else
                m1.Add(dist, 1);
        }
    }
 
    // Find distance between all
    // pairs in the array Y[]
    for(i = 0; i < M; i++)
    {
        for(j = i + 1; j < M; j++)
        {
            int dist = Math.Abs(Y[i] - Y[j]);
 
            // Add the count to m2
            if (m2.ContainsKey(dist))
                m2[dist]++;
              else
                m2.Add(dist, 1);
        }
    }
 
    // Find sum of m1[i] * m2[i]
    // for same distance
    foreach(KeyValuePair<int, int> entry in m1)
    {
         
        // Find current count in m2
        if (m2.ContainsKey(entry.Key))
        {
             
            // Add to the total count
            ans += (entry.Value *
                 m2[entry.Key]);
        }
    }
 
    // Return the final count
    return ans;
}
 
// Driver Code
public static void Main()
{
     
    // Given lines
    int[] X = { 1, 3, 7 };
    int[] Y = { 2, 4, 6, 1 };
 
    int N = X.Length;
 
    int M = Y.Length;
 
    // Function call
    Console.WriteLine(numberOfSquares(X, Y, N, M));
}
}
 
// This code is contributed by akhilsaini

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Output: 

3


 

Time Complexity: O(N2)
Auxiliary Space: O(N)
 

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