Find the integer points (x, y) with Manhattan distance atleast N - GeeksforGeeks

03 Mar, 2022
Given a number N, the task is to find the integer points (x, y) such that 0 <= x, y <= N and Manhattan distance between any two points will be atleast N.Examples:  
Input: N = 3
Output: (0, 0) (0, 3) (3, 0) (3, 3)

Input: N = 4
Output: (0, 0) (0, 4) (4, 0) (4, 4) (2, 2)
 
Approach:  
Manhattan Distance between two points (x1, y1) and (x2, y2) is: |x1 – x2| + |y1 – y2|
Here for all pair of points this distance will be atleast N.
As 0 <= x <= N and 0 <= y <= N so we can imagine a square of side length N whose bottom left corner is (0, 0) and top right corner is (N, N).
So if we place 4 points in this corner then Manhattan distance will be atleast N.
Now as we have to maximize the number of the point we have to check is there any available point inside the square.
If N is even then middle point of the square which is (N/2, N/2) is integer point, otherwise, it will be float value as N/2 is not a integer when N is odd.
So the only available position is the middle point and we can put a point there only if N is even.
So number of points will be 4 if N is odd and if N is even then the number of points will be 5.
Below is the implementation of the above approach:  
C++
Java
Python3
C#
Javascript
// C++ code to Find the integer points (x, y)// with Manhattan distance atleast N #include <bits/stdc++.h>using namespace std; // C++ function to find all possible pointvector<pair<int, int> > FindPoints(int n){     vector<pair<int, int> > v;     // Find all 4 corners of the square    // whose side length is n    v.push_back({ 0, 0 });    v.push_back({ 0, n });    v.push_back({ n, 0 });    v.push_back({ n, n });     // If n is even then the middle point    // of the square will be an integer,    // so we will take that point    if (n % 2 == 0)        v.push_back({ n / 2, n / 2 });     return v;} // Driver Codeint main(){     int N = 8;     vector<pair<int, int> > v        = FindPoints(N);     // Printing all possible points    for (auto i : v) {        cout << "(" << i.first << ", "             << i.second << ") ";    }    return 0;}
// Java code to Find the integer points (x, y)// with Manhattan distance atleast Nimport java.util.*; class GFG{ static class pair{    int first, second;    public pair(int first, int second)    {        this.first = first;        this.second = second;    }} // Java function to find all possible pointstatic Vector<pair> FindPoints(int n){    Vector<pair> v = new Vector<pair>();     // Find all 4 corners of the square    // whose side length is n    v.add(new pair( 0, 0 ));    v.add(new pair( 0, n ));    v.add(new pair( n, 0 ));    v.add(new pair( n, n ));     // If n is even then the middle point    // of the square will be an integer,    // so we will take that point    if (n % 2 == 0)        v.add(new pair( n / 2, n / 2 ));     return v;} // Driver Codepublic static void main(String[] args){    int N = 8;     Vector<pair > v = FindPoints(N);     // Printing all possible points    for (pair i : v)    {        System.out.print("(" + i.first + ", " +                               i.second + ") ");    }}} // This code is contributed by PrinciRaj1992
# Python3 code to Find the integer points (x, y)# with Manhattan distance atleast N # function to find all possible pointdef FindPoints(n) :     v = [];     # Find all 4 corners of the square    # whose side length is n    v.append([ 0, 0 ]);    v.append([ 0, n ]);    v.append([ n, 0 ]);    v.append([ n, n ]);     # If n is even then the middle point    # of the square will be an integer,    # so we will take that point    if (n % 2 == 0) :        v.append([ n // 2, n // 2 ]);     return v; # Driver Codeif __name__ == "__main__" :     N = 8;     v = FindPoints(N);     # Printing all possible points    for element in v :        print("(", element[0],              ",", element[1], ")", end = " "); # This code is contributed by AnkitRai01
// C# code to Find the integer points (x, y)// with Manhattan distance atleast Nusing System;using System.Collections.Generic; class GFG{ class pair{    public int first, second;    public pair(int first, int second)    {        this.first = first;        this.second = second;    }} // Function to find all possible pointstatic List<pair> FindPoints(int n){    List<pair> v = new List<pair>();     // Find all 4 corners of the square    // whose side length is n    v.Add(new pair( 0, 0 ));    v.Add(new pair( 0, n ));    v.Add(new pair( n, 0 ));    v.Add(new pair( n, n ));     // If n is even then the middle point    // of the square will be an integer,    // so we will take that point    if (n % 2 == 0)        v.Add(new pair( n / 2, n / 2 ));     return v;} // Driver Codepublic static void Main(String[] args){    int N = 8;     List<pair > v = FindPoints(N);     // Printing all possible points    foreach (pair i in v)    {        Console.Write("(" + i.first + ", " +                            i.second + ") ");    }}} // This code is contributed by Rajput-Ji
<script> // Javascript code to Find the integer points (x, y)// with Manhattan distance atleast N // C++ function to find all possible pointfunction FindPoints(n){     var v = [];     // Find all 4 corners of the square    // whose side length is n    v.push([ 0, 0 ]);    v.push([ 0, n ]);    v.push([ n, 0 ]);    v.push([ n, n ]);     // If n is even then the middle point    // of the square will be an integer,    // so we will take that point    if (n % 2 == 0)        v.push([ n / 2, n / 2 ]);     return v;} // Driver Codevar N = 8;var v = FindPoints(N);// Printing all possible pointsv.forEach(i => {    document.write( "(" + i[0] + ", "         + i[1] + ") ");}); // This code is contributed by rrrtnx.</script>
(0, 0) (0, 8) (8, 0) (8, 8) (4, 4)
 
Time Complexity: O(1)
Auxiliary Space: O(1)
ankthon
princiraj1992
Rajput-Ji
rrrtnx
subhammahato348
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