Pairs with same Manhattan and Euclidean distance

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[xi] stores the number of points having their x-coordinate equal to xi
2) Map Y, where Y[yi] stores the number of points having their y-coordinate equal to yi
3) Map XY, where XY[(Xi, Yi)] stores the number of points coincident with point (xi, yi)

Now,
Let Xans be the Number of pairs with same X-coordinates = X[xi]2 for all distinct xi =
Let Yans be the Number of pairs with same Y-coordinates = Y[xi]2 for all distinct yi
Let XYans be the Number of coincident points = XY[{xi, yi}]2 for all distinct points (xi, yi)

Thus the required answer = Xans + Yans – XYans

Below is the implementation of the above approach:

C++

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// C++ implementtaion 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;
}

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Python3

# Python3 implementtaion 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:

2

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



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