Pairs with same Manhattan and Euclidean distance

Difficulty Level : Easy
Last Updated : 05 Aug, 2021

In a given Cartesian plane, there are N points. The task is to find the Number of Pairs of points(A, B) such that

Point A and Point B do not coincide.
Manhattan Distance and the Euclidean Distance between the points should be equal.

Note: Pair of 2 points(A, B) is considered same as Pair of 2 points(B, A).

Manhattan Distance = |x2-x1|+|y2-y1|
Euclidean Distance = ((x2-x1)^2 + (y2-y1)^2)^0.5 where points are (x1, y1) and (x2, y2).

Examples:

Input: N = 3, Points = {{1, 2}, {2, 3}, {1, 3}}
Output: 2
Pairs are:
1) (1, 2) and (1, 3)
Euclidean distance of (1, 2) and (1, 3) = &root;((1 – 1)² + (3 – 2)²) = 1
Manhattan distance of (1, 2) and (1, 3) = |(1 – 1)| + |(2 – 3)| = 1
2) (1, 3) and (2, 3)
Euclidean distance of (1, 3) and (2, 3) = &root;((1 – 2)² + (3 – 3)²) = 1
Manhattan distance of (1, 3) and (2, 3) = |(1 – 2)| + |(3 – 3)| = 1
Input: N = 3, Points = { {1, 1}, {2, 3}, {1, 1} }
Output: 0
Here none of the pairs satisfy the above two conditions

Approach: On solving the equation

|x2-x1|+|y2-y1| = sqrt((x2-x1)^2+(y2-y1)^2)

we get , x2 = x1 or y2 = y1.
Consider 3 maps,
1) Map X, where X[x_i] stores the number of points having their x-coordinate equal to x_i
2) Map Y, where Y[y_i] stores the number of points having their y-coordinate equal to y_i
3) Map XY, where XY[(X_i, Y_i)] stores the number of points coincident with point (x_i, y_i)
Now,
Let Xans be the Number of pairs with same X-coordinates = ^X[x_ⁱ^]₂ for all distinct x_i =
Let Yans be the Number of pairs with same Y-coordinates = ^Y[x_ⁱ^]₂ for all distinct y_i
Let XYans be the Number of coincident points = ^XY[{x_ⁱ^{, y}_ⁱ^}]₂ for all distinct points (x_i, y_i)
Thus the required answer = Xans + Yans – XYans
Below is the implementation of the above approach:

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the number of non coincident
// pairs of points with manhattan distance
// equal to euclidean distance
int findManhattanEuclidPair(pair<int, int> arr[], int n)
{
    // To store frequency of all distinct Xi
    map<int, int> X;
 
    // To store Frequency of all distinct Yi
    map<int, int> Y;
 
    // To store Frequency of all distinct
    // points (Xi, Yi);
    map<pair<int, int>, int> XY;
 
    for (int i = 0; i < n; i++) {
        int xi = arr[i].first;
        int yi = arr[i].second;
 
        // Hash xi coordinate
        X[xi]++;
 
        // Hash yi coordinate
        Y[yi]++;
 
        // Hash the point (xi, yi)
        XY[arr[i]]++;
    }
 
    int xAns = 0, yAns = 0, xyAns = 0;
 
    // find pairs with same Xi
    for (auto xCoordinatePair : X) {
        int xFrequency = xCoordinatePair.second;
 
        // calculate ((xFrequency) C2)
        int sameXPairs =
             (xFrequency * (xFrequency - 1)) / 2;
        xAns += sameXPairs;
    }
 
    // find pairs with same Yi
    for (auto yCoordinatePair : Y) {
        int yFrequency = yCoordinatePair.second;
 
        // calculate ((yFrequency) C2)
        int sameYPairs =
                (yFrequency * (yFrequency - 1)) / 2;
        yAns += sameYPairs;
    }
 
    // find pairs with same (Xi, Yi)
    for (auto XYPair : XY) {
        int xyFrequency = XYPair.second;
  
        // calculate ((xyFrequency) C2)
        int samePointPairs =
             (xyFrequency * (xyFrequency - 1)) / 2;
        xyAns += samePointPairs;
    }
 
    return (xAns + yAns - xyAns);
}
 
// Driver Code
int main()
{
    pair<int, int> arr[] = {
        { 1, 2 },
        { 2, 3 },
        { 1, 3 }
    };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    cout << findManhattanEuclidPair(arr, n) << endl;
    return 0;
}

Python3

# Python3 implementation of the
# above approach
from collections import defaultdict
 
# Function to return the number of
# non coincident pairs of points with
# manhattan distance equal to
# euclidean distance
def findManhattanEuclidPair(arr, n):
 
    # To store frequency of all distinct Xi
    X = defaultdict(lambda:0)
 
    # To store Frequency of all distinct Yi
    Y = defaultdict(lambda:0)
 
    # To store Frequency of all distinct
    # points (Xi, Yi)
    XY = defaultdict(lambda:0)
 
    for i in range(0, n):
        xi = arr[i][0]
        yi = arr[i][1]
 
        # Hash xi coordinate
        X[xi] += 1
 
        # Hash yi coordinate
        Y[yi] += 1
 
        # Hash the point (xi, yi)
        XY[tuple(arr[i])] += 1
     
    xAns, yAns, xyAns = 0, 0, 0
 
    # find pairs with same Xi
    for xCoordinatePair in X:
        xFrequency = X[xCoordinatePair]
 
        # calculate ((xFrequency) C2)
        sameXPairs = (xFrequency *
                     (xFrequency - 1)) // 2
        xAns += sameXPairs
     
    # find pairs with same Yi
    for yCoordinatePair in Y:
        yFrequency = Y[yCoordinatePair]
 
        # calculate ((yFrequency) C2)
        sameYPairs = (yFrequency *
                     (yFrequency - 1)) // 2
        yAns += sameYPairs
 
    # find pairs with same (Xi, Yi)
    for XYPair in XY:
        xyFrequency = XY[XYPair]
     
        # calculate ((xyFrequency) C2)
        samePointPairs = (xyFrequency *
                         (xyFrequency - 1)) // 2
        xyAns += samePointPairs
     
    return (xAns + yAns - xyAns)
 
# Driver Code
if __name__ == "__main__":
 
    arr = [[1, 2], [2, 3], [1, 3]]
     
    n = len(arr)
 
    print(findManhattanEuclidPair(arr, n))
     
# This code is contributed by Rituraj Jain

Output:

Time Complexity: O(NlogN), where N is the number of points
Space Complexity: O(N)

My Personal Notes

Related Articles

Table of Contents

Pairs with same Manhattan and Euclidean distance

C++

Python3

Start Your Coding Journey Now!