## Find the winner by incrementing co-ordinates till Euclidean distance <= D in C++## Introduction:In this C++ approach, the objective is to determine a set of winning coordinates by systematically incrementing their values until the The algorithm starts with initial coordinates at the 'D'. and employs a origin (0,0) in a 2D plane strategy. The Euclidean distance, calculated using the distance formula, serves as the criterion for determining the winning coordinates. This method ensures a structured exploration of the coordinate space, incrementing both systematic incrementation and x values until the desired distance condition is met. The simplicity and clarity of this approach make it an effective way to find a winning point within the specified Euclidean distance systematically and predictably .y## Program:
Enter the maximum distance (D): 5 The winner coordinates are: (0, 0)
**Introduction:**
The code aims to find a winning set of coordinates by gradually incrementing the x and y values until the Euclidean distance from the **Function to Calculate Euclidean Distance:**
The c : sqrt((x2 - x1)^2 + (y2 - y1)^2).
**User Input:**
The program starts by prompting the user to input the maximum allowed **Nested Loops:**
The code uses nested loops to iterate through possible coordinate values. **Distance Calculation:**
For each set of coordinates using the calculateDistance function.origin (0,0)**Winning Condition Check:**
If the calculated distance is less than or equal to the specified maximum distance **Output:**
Finally, the program outputs the winning coordinates. **Infinite Loops:**
The use of
The time complexity of the code is not straightforward due to the presence of infinite loops. The nested loops increment the x and y coordinates until a winning condition is met, and this process may continue indefinitely until a suitable set of coordinates is found. Therefore, the time complexity is best described as unbounded.
The , distance, and the winner coordinates as x, y, D These variables occupy a constant amount of memory regardless of the input values. Thus, the space complexity is considered constant or x_winner and y_winner.O(1).## Approach 1: Using Linear Search with Step IncrementThe approach employs a systematic linear search strategy to find coordinates within a It starts from the coordinates by a fixed step size.both x and y## Program:
Enter the maximum distance (D): 5 The winner coordinates are: (0, 0)
**Initialization:**
distance, which the program will use to find the winning coordinates.maximum allowed**User Input:**
The user is prompted to input the maximum **Coordinate Initialization:**
in a 2D plane.starting point (origin)**Linear Search:**
The program enters an infinite loop winning coordinates.**For each iteration of the loop:**
The Euclidean distance from the calculated using the distance formula.coordinates (x, y) isIf the calculated distance is less than or equal to the specified The winning coordinates are then **Incrementation:**
If the distance condition is not met, the coordinates This incrementation ensures that the program 2D plane.**Output:**
The final output displays the winning coordinates, providing the point that satisfies the condition of being within the specified
The time complexity of the code is determined by the number of iterations in the Since the loop increments If we denote the number of iterations is O(N).
The space complexity of the code is constant The variables The space required for these variables does not grow with the size of the input. ## Approach 2: Using Quadratic Search with Diagonal MovementThis approach adopts a It starts from the covering more ground diagonally.quadratic pattern,## Program:
Enter the maximum distance (D): 5 The winner coordinates are: (0, 0)
**User Input:**
The user is prompted to input the (D).**Coordinate Initialization:**
2D plane.**Quadratic Search with Diagonal Movement:**
The program enters an infinite loop for the winning coordinates.diagonal movement**For each iteration of the loop:**
The Euclidean distance from the is calculated using the distance formula.(x, y)If the calculated distance is less than or equal to the specified maximum distance (D), the current coordinates are considered as the winner. The winning coordinates are then displayed, and the loop breaks. **Incrementation - Quadratic Movement:**
If the distance condition is not met, the The is added to (x and y) in each iteration.both x and yThis more quickly.2D plane**Output:**
The the point that satisfies the condition of being within the specified , providing from the origin.maximum distance
The while (true) loop.The loop performs a and xy.The time complexity can be considered quadratic, denoted as is the number of iterations needed to find the winning coordinates.N
The because the amount of memory used by the program remains fixed regardless of the input values.O(1),The The .size of the input## Approach 3: Random Search with ConvergenceThe approach uses a random search strategy to find coordinates within a It starts with random initial coordinates within a specified range and iteratively ## Program:
Enter the maximum distance (D): 6 The winner coordinates are: (-3, 5)
**User Input:**
The user inputs the distance (D).**Random Initialization:**
Random initial coordinates are chosen using the range [-D, D] function.rand()The **Random Search Iteration:**
In each iteration, the program calculates the If the calculated distance is less than or equal to D, the current coordinates are considered the winner, and the program terminates. Otherwise, it adjusts the coordinates randomly to explore different areas of the 2D plane. **Random Adjustment:**
Random adjustments are made to both in each iteration.yThe adjustments are within the **Convergence:**
The Over iterations, the randomness tends to converge towards coordinates that satisfy the **Output:**
Once the
The On average, the time complexity is expected to be relatively high because it depends on the number of iterations needed to find coordinates within the specified The as where O(N), is the average number of iterations required.N
The space complexity is constant, denoted as The variables D, x, y, distance, and the random number generation variables all occupy a constant amount of memory throughout the execution of the program. The space required for these variables does not grow with the size of the input. |

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