# 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)^0.5where 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)| = 1Input: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 - (2 * xyAns));` ` ` `/* we are subtracting 2 * xyAns because we have counted` ` ` `let say A,B coinciding points two times in xAns and` ` ` `yAns which should not be add to the final answer so` ` ` `we are subtracting xyAns 2 times. */` `}` `// Driver Code` `int` `main()` `{` ` ` `pair<` `int` `, ` `int` `> arr[]` ` ` `= { { 1, 2 }, { 1, 2 }, { 4, 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 ` `-` `2` `*` `xyAns)` ` ` `# we are subtracting 2 * xyAns because we have counted let say A,B coinciding points two times` ` ` `# in xAns and yAns which should not be add to the final answer so we are subtracting xyAns 2 times.` `# Driver Code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `arr ` `=` `[[` `1` `, ` `2` `], [` `1` `,` `2` `], [` `4` `, ` `3` `], [` `1` `, ` `3` `]]` ` ` ` ` `n ` `=` `len` `(arr)` ` ` `print` `(findManhattanEuclidPair(arr, n))` ` ` `# This code is contributed by Rituraj Jain` |

**Output**

3

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