Maximum Manhattan distance between a distinct pair from N coordinates

Given an array arr[] consisting of N integer coordinates, the task is to find the maximum Manhattan Distance between any two distinct pairs of coordinates.

The Manhattan Distance between two points (X1, Y1) and (X2, Y2) is given by |X1 – X2| + |Y1 – Y2|.

Examples:

Input: arr[] = {(1, 2), (2, 3), (3, 4)}
Output: 4
Explanation:
The maximum Manhattan distance is found between (1, 2) and (3, 4) i.e., |3 – 1| + |4- 2 | = 4.

Input: arr[] = {(-1, 2), (-4, 6), (3, -4), (-2, -4)}
Output: 17
Explanation:
The maximum Manhattan distance is found between (-4, 6) and (3, -4) i.e.,  |-4 – 3| + |6 – (-4)| = 17.

Naive Approach: The simplest approach is to iterate over the array, and for each coordinate, calculate its Manhattan distance from all remaining points. Keep updating the maximum distance obtained after each calculation. Finally, print the maximum distance obtained.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to calculate the maximum
// Manhattan distance
void MaxDist(vector<pair<int, int> >& A, int N)
{
    // Stores the maximum distance
    int maximum = INT_MIN;

    for (int i = 0; i < N; i++) {

        int sum = 0;

        for (int j = i + 1; j < N; j++) {

            // Find Manhattan distance
            // using the formula
            // |x1 - x2| + |y1 - y2|
            sum = abs(A[i].first - A[j].first)
                  + abs(A[i].second - A[j].second);

            // Updating the maximum
            maximum = max(maximum, sum);
        }
    }

    cout << maximum;
}

// Driver Code
int main()
{
    int N = 3;

    // Given Co-ordinates
    vector<pair<int, int> > A
        = { { 1, 2 }, { 2, 3 }, { 3, 4 } };

    // Function Call
    MaxDist(A, N);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;

class GFG {

    // Pair class
    public static class Pair {
        int x;
        int y;

        Pair(int x, int y)
        {
            this.x = x;
            this.y = y;
        }
    }

    // Function to calculate the maximum
    // Manhattan distance
    static void MaxDist(ArrayList<Pair> A, int N)
    {

        // Stores the maximum distance
        int maximum = Integer.MIN_VALUE;

        for (int i = 0; i < N; i++) {
            int sum = 0;

            for (int j = i + 1; j < N; j++) {

                // Find Manhattan distance
                // using the formula
                // |x1 - x2| + |y1 - y2|
                sum = Math.abs(A.get(i).x - A.get(j).x)
                      + Math.abs(A.get(i).y - A.get(j).y);

                // Updating the maximum
                maximum = Math.max(maximum, sum);
            }
        }
        System.out.println(maximum);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int n = 3;

        ArrayList<Pair> al = new ArrayList<>();

        // Given Co-ordinates
        Pair p1 = new Pair(1, 2);
        al.add(p1);

        Pair p2 = new Pair(2, 3);
        al.add(p2);

        Pair p3 = new Pair(3, 4);
        al.add(p3);

        // Function call
        MaxDist(al, n);
    }
}

// This code is contributed by bikram2001jha

Python3

# Python3 program for the above approach
import sys

# Function to calculate the maximum
# Manhattan distance


def MaxDist(A, N):

    # Stores the maximum distance
    maximum = - sys.maxsize

    for i in range(N):
        sum = 0

        for j in range(i + 1, N):

            # Find Manhattan distance
            # using the formula
            # |x1 - x2| + |y1 - y2|
            Sum = (abs(A[i][0] - A[j][0]) +
                   abs(A[i][1] - A[j][1]))

            # Updating the maximum
            maximum = max(maximum, Sum)

    print(maximum)


# Driver code
N = 3

# Given co-ordinates
A = [[1, 2], [2, 3], [3, 4]]

# Function call
MaxDist(A, N)

# This code is contributed by divyeshrabadiya07

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG {

    // Pair class
    public class Pair {
        public int x;
        public int y;

        public Pair(int x, int y)
        {
            this.x = x;
            this.y = y;
        }
    }

    // Function to calculate the maximum
    // Manhattan distance
    static void MaxDist(List<Pair> A, int N)
    {

        // Stores the maximum distance
        int maximum = int.MinValue;

        for (int i = 0; i < N; i++) {
            int sum = 0;

            for (int j = i + 1; j < N; j++) {

                // Find Manhattan distance
                // using the formula
                // |x1 - x2| + |y1 - y2|
                sum = Math.Abs(A[i].x - A[j].x)
                      + Math.Abs(A[i].y - A[j].y);

                // Updating the maximum
                maximum = Math.Max(maximum, sum);
            }
        }
        Console.WriteLine(maximum);
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int n = 3;

        List<Pair> al = new List<Pair>();

        // Given Co-ordinates
        Pair p1 = new Pair(1, 2);
        al.Add(p1);

        Pair p2 = new Pair(2, 3);
        al.Add(p2);

        Pair p3 = new Pair(3, 4);
        al.Add(p3);

        // Function call
        MaxDist(al, n);
    }
}

// This code is contributed by Amit Katiyar

Javascript

<script>

// JavaScript program for the above approach

// Function to calculate the maximum
// Manhattan distance
function MaxDist(A, N){

    // Stores the maximum distance
    let maximum = Number.MIN_VALUE
    for(let i = 0; i < N; i++)
    {
        let Sum = 0

        for(let j = i + 1; j < N; j++)
        {

            // Find Manhattan distance
            // using the formula
            // |x1 - x2| + |y1 - y2|
            Sum = (Math.abs(A[i][0] - A[j][0]) +
                   Math.abs(A[i][1] - A[j][1]))

            // Updating the maximum
            maximum = Math.max(maximum, Sum)
        }
    }

    document.write(maximum)
}


// Driver code
let N = 3

// Given co-ordinates
let A = [[1, 2], [2, 3], [3, 4]]

// Function call
MaxDist(A, N)

// This code is contributed by shinjanpatra

</script>

Output

4

Time Complexity: O(N²), where N is the size of the given array.
Auxiliary Space: O(1)

Efficient Approach: The idea is to use store sums and differences between X and Y coordinates and find the answer by sorting those differences. Below are the observations to the above problem statement:

• Manhattan Distance between any two points (X_i, Y_i) and (X_j, Y_j) can be written as follows:

 |X_i – X_j| + |Y_i – Y_j| = max(X_i – X_j -Y_i + Y_j, 
                                         -X_i + X_j + Y_i – Y_j, 
                                         -X_i + X_j – Y_i + Y_j, 
                                          X_i – Xj + Y_i – Y_j).

• The above expression can be rearranged as:

|X_i – X_j| + |Y_i – Y_j| = max((X_i – Y_i) – (X_j – Y_j), 
                                          (-X_i + Y_i) – (-X_j + Y_j), 
                                          (-X_i – Y_i) – (-X_j – Y_j), 
                                          (X_i + Y_i) – (X_j + Y_j))

• It can be observed from the above expression, that the answer can be found by storing the sum and differences of the coordinates.

Follow the below steps to solve the problem:

1. Initialize two arrays sum[] and diff[].
2. Store the sum of X and Y coordinates i.e., X_i + Y_i in sum[] and their difference i.e., X_i – Y_i in diff[].
3. Sort the sum[] and diff[] in ascending order.
4. The maximum of the values (sum[N-1] – sum[0]) and (diff[N-1] – diff[0]) is the required result.

 Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to calculate the maximum
// Manhattan distance
void MaxDist(vector<pair<int, int> >& A, int N)
{
    // Vectors to store maximum and
    // minimum of all the four forms
    vector<int> V(N), V1(N);

    for (int i = 0; i < N; i++) {
        V[i] = A[i].first + A[i].second;
        V1[i] = A[i].first - A[i].second;
    }

    // Sorting both the vectors
    sort(V.begin(), V.end());
    sort(V1.begin(), V1.end());

    int maximum
        = max(V.back() - V.front(), V1.back() - V1.front());

    cout << maximum << endl;
}

// Driver Code
int main()
{
    int N = 3;

    // Given Co-ordinates
    vector<pair<int, int> > A
        = { { 1, 2 }, { 2, 3 }, { 3, 4 } };

    // Function Call
    MaxDist(A, N);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;

class GFG {

    // Pair class
    public static class Pair {
        int x;
        int y;

        Pair(int x, int y)
        {
            this.x = x;
            this.y = y;
        }
    }

    // Function to calculate the maximum
    // Manhattan distance
    static void MaxDist(ArrayList<Pair> A, int N)
    {

        // ArrayLists to store maximum and
        // minimum of all the four forms
        ArrayList<Integer> V = new ArrayList<>();
        ArrayList<Integer> V1 = new ArrayList<>();

        for (int i = 0; i < N; i++) {
            V.add(A.get(i).x + A.get(i).y);
            V1.add(A.get(i).x - A.get(i).y);
        }

        // Sorting both the ArrayLists
        Collections.sort(V);
        Collections.sort(V1);

        int maximum
            = Math.max((V.get(V.size() - 1) - V.get(0)),
                       (V1.get(V1.size() - 1) - V1.get(0)));

        System.out.println(maximum);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int n = 3;

        ArrayList<Pair> al = new ArrayList<>();

        // Given Co-ordinates
        Pair p1 = new Pair(1, 2);
        al.add(p1);
        Pair p2 = new Pair(2, 3);
        al.add(p2);
        Pair p3 = new Pair(3, 4);
        al.add(p3);

        // Function call
        MaxDist(al, n);
    }
}

// This code is contributed by bikram2001jha

Python3

# Python3 program for the above approach

# Function to calculate the maximum
# Manhattan distance


def MaxDist(A, N):

    # List to store maximum and
    # minimum of all the four forms
    V = [0 for i in range(N)]
    V1 = [0 for i in range(N)]

    for i in range(N):
        V[i] = A[i][0] + A[i][1]
        V1[i] = A[i][0] - A[i][1]

    # Sorting both the vectors
    V.sort()
    V1.sort()

    maximum = max(V[-1] - V[0],
                  V1[-1] - V1[0])

    print(maximum)


# Driver code
if __name__ == "__main__":

    N = 3

    # Given Co-ordinates
    A = [[1, 2],
         [2, 3],
         [3, 4]]

    # Function call
    MaxDist(A, N)

# This code is contributed by rutvik_56

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG {

    // Pair class
    class Pair {
        public int x;
        public int y;

        public Pair(int x, int y)
        {
            this.x = x;
            this.y = y;
        }
    }

    // Function to calculate the maximum
    // Manhattan distance
    static void MaxDist(List<Pair> A, int N)
    {

        // Lists to store maximum and
        // minimum of all the four forms
        List<int> V = new List<int>();
        List<int> V1 = new List<int>();

        for (int i = 0; i < N; i++) {
            V.Add(A[i].x + A[i].y);
            V1.Add(A[i].x - A[i].y);
        }

        // Sorting both the Lists
        V.Sort();
        V1.Sort();

        int maximum = Math.Max((V[V.Count - 1] - V[0]),
                               (V1[V1.Count - 1] - V1[0]));

        Console.WriteLine(maximum);
    }

    // Driver Code
    public static void Main(String[] args)
    {
        int n = 3;

        List<Pair> al = new List<Pair>();

        // Given Co-ordinates
        Pair p1 = new Pair(1, 2);
        al.Add(p1);
        Pair p2 = new Pair(2, 3);
        al.Add(p2);
        Pair p3 = new Pair(3, 4);
        al.Add(p3);

        // Function call
        MaxDist(al, n);
    }
}

// This code is contributed by Amit Katiyar

Javascript

<script>

// JavaScript program for the above approach

// Function to calculate the maximum
// Manhattan distance
function MaxDist(A, N)
{

    // List to store maximum and
    // minimum of all the four forms
    let V = new Array(N).fill(0)
    let V1 = new Array(N).fill(0)

    for(let i = 0; i < N; i++){
        V[i] = A[i][0] + A[i][1]
        V1[i] = A[i][0] - A[i][1]
    }

    // Sorting both the vectors
    V.sort()
    V1.sort()

    let maximum = Math.max(V[V.length-1] - V[0],
                  V1[V1.length-1] - V1[0])

    document.write(maximum,"</br>")
}

// Driver code
let N = 3

// Given Co-ordinates
let A = [[1, 2],
         [2, 3],
         [3, 4]]

// Function call
MaxDist(A, N)

// This code is contributed by shinjanpatra

</script>

Output

4

Time Complexity: O(N*log N)
Auxiliary Space: O(N)

Improving the Efficient Approach: Instead of storing the sums and differences in an auxiliary array, then sorting the arrays to determine the minimum and maximums, it is possible to keep a running total of the extreme sums and differences.

C++

// C++ program for the above approach

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

// Function to calculate the maximum
// Manhattan distance
void MaxDist(vector<pair<int, int> >& A, int N)
{
    // Variables to track running extrema
    int minsum, maxsum, mindiff, maxdiff;

    minsum = maxsum = A[0].first + A[0].second;
    mindiff = maxdiff = A[0].first - A[0].second;
    for (int i = 1; i < N; i++) {
        int sum = A[i].first + A[i].second;
        int diff = A[i].first - A[i].second;
        if (sum < minsum)
            minsum = sum;
        else if (sum > maxsum)
            maxsum = sum;
        if (diff < mindiff)
            mindiff = diff;
        else if (diff > maxdiff)
            maxdiff = diff;
    }

    int maximum = max(maxsum - minsum, maxdiff - mindiff);

    cout << maximum << endl;
}

// Driver Code
int main()
{
    int N = 3;

    // Given Co-ordinates
    vector<pair<int, int> > A
        = { { 1, 2 }, { 2, 3 }, { 3, 4 } };

    // Function Call
    MaxDist(A, N);

    return 0;
}

Java

// Java program for the above approach
public class GFG {

    // Function to calculate the maximum
    // Manhattan distance
    static void MaxDist(int[][] A, int N)
    {
        // Variables to track running extrema
        int minsum, maxsum, mindiff, maxdiff;

        minsum = maxsum = A[0][0] + A[0][1];
        mindiff = maxdiff = A[0][0] - A[0][1];
        for (int i = 1; i < N; i++) {
            int sum = A[i][0] + A[i][1];
            int diff = A[i][0] - A[i][1];
            if (sum < minsum)
                minsum = sum;
            else if (sum > maxsum)
                maxsum = sum;
            if (diff < mindiff)
                mindiff = diff;
            else if (diff > maxdiff)
                maxdiff = diff;
        }

        int maximum
            = Math.max(maxsum - minsum, maxdiff - mindiff);

        System.out.println(maximum);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 3;

        // Given Co-ordinates
        int[][] A = { { 1, 2 }, { 2, 3 }, { 3, 4 } };

        // Function Call
        MaxDist(A, N);
    }
}

// The code is contributed by Gautam goel (gautamgoel962)

Python3

# Python program for the above approach

# Function to calculate the maximum
# Manhattan distance
def MaxDist(A, N):

    # Variables to track running extrema
    minsum = maxsum = A[0][0] + A[0][1]
    mindiff = maxdiff = A[0][0] - A[0][1]

    for i in range(1,N):
        sum = A[i][0] + A[i][1]
        diff = A[i][0] - A[i][1]
        if (sum < minsum):
            minsum = sum
        elif (sum > maxsum):
            maxsum = sum
        if (diff < mindiff):
            mindiff = diff
        elif (diff > maxdiff):
            maxdiff = diff

    maximum = max(maxsum - minsum, maxdiff - mindiff)

    print(maximum)

# Driver Code
N = 3

# Given Co-ordinates
A = [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ] ]

# Function Call
MaxDist(A, N)

# This code is contributed by shinjanpatra

C#

// Include namespace system
using System;

// C# program for the above approach
public class GFG
{
  // Function to calculate the maximum
  // Manhattan distance
  public static void MaxDist(int[,] A, int N)
  {

    // Variables to track running extrema
    int minsum;
    int maxsum;
    int mindiff;
    int maxdiff;
    minsum = maxsum = A[0,0] + A[0,1];
    mindiff = maxdiff = A[0,0] - A[0,1];
    for (int i = 1; i < N; i++)
    {
      var sum = A[i,0] + A[i,1];
      var diff = A[i,0] - A[i,1];
      if (sum < minsum)
      {
        minsum = sum;
      }
      else if (sum > maxsum)
      {
        maxsum = sum;
      }
      if (diff < mindiff)
      {
        mindiff = diff;
      }
      else if (diff > maxdiff)
      {
        maxdiff = diff;
      }
    }
    var maximum = Math.Max(maxsum - minsum,maxdiff - mindiff);
    Console.WriteLine(maximum);
  }

  // Driver Code
  public static void Main(String[] args)
  {
    var N = 3;

    // Given Co-ordinates
    int[,] A = {{1, 2}, {2, 3}, {3, 4}};

    // Function Call
    GFG.MaxDist(A, N);
  }
}

Javascript

<script>

// JavaScript program for the above approach


// Function to calculate the maximum
// Manhattan distance
function MaxDist(A, N)
{
    // Variables to track running extrema
    let minsum, maxsum, mindiff, maxdiff;

    minsum = maxsum = A[0][0] + A[0][1];
    mindiff = maxdiff = A[0][0] - A[0][1];
    for (let i = 1; i < N; i++) {
        let sum = A[i][0] + A[i][1];
        let diff = A[i][0] - A[i][1];
        if (sum < minsum)
            minsum = sum;
        else if (sum > maxsum)
            maxsum = sum;
        if (diff < mindiff)
            mindiff = diff;
        else if (diff > maxdiff)
            maxdiff = diff;
    }

    let maximum = Math.max(maxsum - minsum, maxdiff - mindiff);

    document.write(maximum,"</br>");
}

// Driver Code

let N = 3;

// Given Co-ordinates
let A = [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ] ];

// Function Call
MaxDist(A, N);

// code is contributed by shinjanpatra

</script>

Output

4

Time Complexity: O(N)
Auxiliary Space: O(1)

The ideas presented here are in two dimensions, but can be extended to further dimensions.  Each additional dimension requires double the amount of computations on each point.  For example, in 3-D space, the result is the maximum difference between the four extrema pairs computed from x+y+z, x+y-z, x-y+z, and x-y-z.