Check if given point lies on the Square Spiral Path

Given a 2D Graph, present initially at (0, 0). Print “YES” if the given point (X, Y) lies on the spiral path, otherwise “NO“. The spiral path is to be followed ln a sequential manner as given below:

• Move one unit right side from the current position.
• Move two units upwards from the current position.
• Move three units left side from the current position.
• Move four units downwards and so on.

Examples:

Input: X = 0, Y = 0
Output: YES
Explanation:

Graphical Explanation of input case 1

The point (X, Y) = (0, 0) denotes the origin. The point is at the path. Therefore, the output is YES.

Input: X = -2, Y = -2
Output: YES
Explanation: 

Graphical Explanation of input case 2

It can be verified that the point lies on the square spiral path.

Input: X = 1, Y = -1
Output: NO
Explanation: It can be verified that the given point doesn’t lies on the square spiral path.

Approach: This can be solved using the following idea:

The problem is observation based and can be solved by using some mathematical observation. For solving the problem there are some conditions at which the problem has a solution:

• y ≤ 0,  y % 2 = 0, x ≥ y and x ≤ (-1 * y) + 1
• x > 0, x % 2 = 1, y ≤ x + 1 and y ≥ (-1 * x) + 1
• y > 0, y % 2 = 0, x ≥ (-1 * y) and x < y
• x < 0, x % 2 = 0, y ≥ x and y ≤ (-1 * x)

All rest of the conditions will not have any solution.

Follow the steps mentioned below to implement the idea:

• Check for the conditions mentioned above
• If any one of the conditions is satisfied, then the point lies in the path.
• Else the point lies outside the path.

Below is the implementation for the above approach:

C++

// C++ code to implement the approach

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

// User defined function for checking if
// point lies in square spiral path or not
string PointLies(int x, int y)
{
    if (y <= 0 && y % 2 == 0 && x >= y
        && x <= (-1 * y) + 1) {
        return "YES";
    }
    else if (x > 0 && x % 2 == 1 && y <= x + 1
             && y >= (-1 * x) + 1) {
        return "YES";
    }
    else if (y > 0 && y % 2 == 0 && x >= (-1 * y)
             && x < y) {
        return "YES";
    }
    else if (x < 0 && x % 2 == 0 && y >= x
             && y <= (-1 * x)) {
        return "YES";
    }
    return "NO";
}

int main()
{
    // Input values of X and Y coordinates
    int X = -2, Y = -2;

    // Function call
    cout << PointLies(X, Y);
}

Java

// Java code to implement the approach

import java.io.*;
import java.lang.*;
import java.util.*;

class GFG {

    // Driver code
    public static void main(String[] args)
        throws java.lang.Exception
    {
        // Input values of X and Y coordinates
        int X = -2, Y = -2;

        // Function call
        System.out.println(PointLies(X, Y));
    }

    // User defined function for checking if
    // point lies in square spiral path or not
    static String PointLies(int x, int y)
    {
        if (y <= 0 && y % 2 == 0 && x >= y
            && x <= (-1 * y) + 1) {
            return "YES";
        }
        else if (x > 0 && x % 2 == 1 && y <= x + 1
                 && y >= (-1 * x) + 1) {
            return "YES";
        }
        else if (y > 0 && y % 2 == 0 && x >= (-1 * y)
                 && x < y) {
            return "YES";
        }
        else if (x < 0 && x % 2 == 0 && y >= x
                 && y <= (-1 * x)) {
            return "YES";
        }
        return "NO";
    }
}

Python3

def PointLies(x, y):
    if y <= 0 and y % 2 == 0 and x >= y and x <= (-1 * y) + 1:
        return "YES"
    elif x > 0 and x % 2 == 1 and y <= x + 1 and y >= (-1 * x) + 1:
        return "YES"
    elif y > 0 and y % 2 == 0 and x >= (-1 * y) and x < y:
        return "YES"
    elif x < 0 and x % 2 == 0 and y >= x and y <= (-1 * x):
        return "YES"
    return "NO"


# Input values of X and Y coordinates
X = -2
Y = -2

# Function call
print(PointLies(X, Y))

C#

using System;

public class GFG {

    static void Main(string[] args)
    {
        // Input values of X and Y coordinates
        int X = -2, Y = -2;

        // Function call
        Console.WriteLine(PointLies(X, Y));
    }

    // User defined function for checking if
    // point lies in square spiral path or not
    static string PointLies(int x, int y)
    {
        if (y <= 0 && y % 2 == 0 && x >= y
            && x <= (-1 * y) + 1) {
            return "YES";
        }
        else if (x > 0 && x % 2 == 1 && y <= x + 1
                 && y >= (-1 * x) + 1) {
            return "YES";
        }
        else if (y > 0 && y % 2 == 0 && x >= (-1 * y)
                 && x < y) {
            return "YES";
        }
        else if (x < 0 && x % 2 == 0 && y >= x
                 && y <= (-1 * x)) {
            return "YES";
        }
        return "NO";
    }
}

Javascript

// JavaScript code for the above approach

// User defined function for checking if
// point lies in square spiral path or not
function PointLies(x, y) {
    if (y <= 0 && y % 2 === 0 && x >= y && x <= (-1 * y) + 1) {
        return "YES";
    } else if (x > 0 && x % 2 === 1 && y <= x + 1 && y >= (-1 * x) + 1) {
        return "YES";
    } else if (y > 0 && y % 2 === 0 && x >= (-1 * y) && x < y) {
        return "YES";
    } else if (x < 0 && x % 2 === 0 && y >= x && y <= (-1 * x)) {
        return "YES";
    }
    return "NO";
}

// Input values of X and Y coordinates
var X = -2, Y = -2;

// Function call
console.log(PointLies(X, Y));

// This code is contributed by divya_p123.

Output

YES

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