Problem Description

In this problem, we're asked to design a simple parking system for a parking lot with three different types of parking spaces: big, medium, and small. Each type of parking space has a fixed number of slots that can be occupied by cars of that specific size. The parking system needs to be able to handle two operations:

1. Initializing the parking system with the number of slots for each type of parking space.
2. Adding a car to the parking lot, which is subject to there being an available slot for the car's type.

When a car tries to park, the parking system checks if there is an available slot for that particular size of the car. If an appropriate slot is available, the car parks (i.e., the count of available slots of that type reduces by one), and the system returns true. If no slot is available for that car's type, the system returns false.

The key to solving the problem is to keep track of the number of available slots for each car type in an efficient way that allows quick updates and queries.

Intuition

The solution approach is straightforward. Since there are only three types of car slots available, we can use an array with three elements, where each element corresponds to the count of available slots for each car type.

1. Initialization: We initialize an array of size four, where indices 1, 2, and 3 represent 'big', 'medium', and 'small' slots respectively. The reason for choosing index 1 to 3 instead of 0 to 2 is to map the carType directly to the array index, as carType is defined to be 1, 2, or 3 in the problem description. We leave index 0 unused. Each element in this array stores the number of available spaces for that type of car.

2. Adding a Car: The addCar function is called with a carType, which is used as the index to directly access the corresponding count in the array. We first check if there's at least one slot available of the given car type by checking if the counter at that index is greater than zero. If it is, we decrement the counter as we've now occupied a slot and return true. If the counter is already at zero, it means there are no available slots for that car type and we return false.

This array-based system allows constant-time operations for both adding cars and initializing the parking system, which means the time complexity for each operation is O(1), providing us with a very efficient solution.

Solution Approach

The implementation of the solution can be broken down into two parts, following the two major functionalities of the ParkingSystem class:

Part 1: Initialization

In the constructor __init__, we initialize an instance variable called self.cnt. This variable is a list that stores the count of available spots for each car type.

• Big car slots are stored at index 1, hence self.cnt[1] = big.
• Medium car slots are stored at index 2, hence self.cnt[2] = medium.
• Small car slots are stored at index 3, hence self.cnt[3] = small.

The array is initialized with the number of slots for each type of parking space given as arguments to the constructor. The index 0 of the array is not used in this problem.

def __init__(self, big: int, medium: int, small: int):
    # Initializing the array with an extra index for convenience in accessing by carType directly
    self.cnt = [0, big, medium, small]

Part 2: Adding a Car

The next part of our solution is the addCar function. This function's purpose is to process the request of adding a car to the parking lot based on the car's type and the available space.

Here's the step-by-step process of what happens when addCar is called:

1. Check if there are available slots for the given carType by directly accessing the self.cnt array using carType as the index.
2. If self.cnt[carType] is greater than 0, it implies an available slot. We decrease the count by one using self.cnt[carType] -= 1 — indicating that we've filled one slot — and return True.
3. If self.cnt[carType] equals 0, it means there are no slots available, and we return False.

def addCar(self, carType: int) -> bool:
    # Check if the car type has an available slot
    if self.cnt[carType] == 0:
        # If not, return False
        return False
    # If there is a slot, decrement the counter and return True
    self.cnt[carType] -= 1
    return True

The data structure used in this solution, a list in Python (also called an array in some programming languages), is the optimal choice for this scenario due to:

• The fixed number of car types, which corresponds to a fixed number of list indices.
• The need for constant-time access and update operations, both of which lists provide.

Algorithmically, the solution is simple and does not involve complex patterns or algorithms. It leverages direct indexing for fast operations, avoiding any iterations or searches. Through this method, both initialization and adding cars are performed with a time complexity of O(1).

Example Walkthrough

Let's go through an example to illustrate how the solution works.

Suppose the parking lot has the following number of slots for each car type:

• Big: 1
• Medium: 2
• Small: 3

And the sequence of cars that arrive are as follows:

1. A big car
2. A medium car
3. Another medium car
4. A small car
5. Another small car
6. A big car again

We will walk through how the ParkingSystem would handle this sequence of cars.

Step 1: Initialization

First, we initialize the parking system with the available slots. Using the solution's __init__ method:

parking_system = ParkingSystem(1, 2, 3)

After initialization, our self.cnt array looks like this: [0, 1, 2, 3]

Step 2: Adding Cars

• When the first big car arrives, we call addCar(1). Since self.cnt[1] is 1 (there's one big slot available), the car is parked, self.cnt gets updated to [0, 0, 2, 3], and True is returned.
• The medium car arrives, we call addCar(2). Since self.cnt[2] is 2, the car is parked, self.cnt is now [0, 0, 1, 3], and True is returned.
• Another medium car arrives, we call addCar(2) again. Now self.cnt[2] is 1, so the car is parked, self.cnt becomes [0, 0, 0, 3], and True is returned.
• A small car arrives, we call addCar(3). self.cnt[3] is 3, so the car is parked, self.cnt updates to [0, 0, 0, 2], and True is returned.
• Another small car arrives, addCar(3) is called. self.cnt[3] is now 2, so this car is also parked, updating self.cnt to [0, 0, 0, 1], and True is returned.
• Finally, another big car tries to park, so we call addCar(1). But self.cnt[1] is 0 because there are no more big slots available after the first car parked, so False is returned.

Throughout these operations, each addCar call checks and updates the self.cnt array in constant time, illustrating both the efficiency and simplicity of the approach.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of both the __init__ method and the addCar method is O(1). This is because both methods perform a constant number of operations regardless of the input size:

• __init__: Initializes the cnt array with three integers, which is a constant-time operation as it involves setting up three fixed indices.
• addCar: Accesses and modifies the cnt array at the index corresponding to carType, which is a constant-time operation due to direct array indexing.

Therefore, overall, the time complexity is O(1) for initialization and each car parking attempt.

Space Complexity

The space complexity of the ParkingSystem class is O(1). This constant space is due to the cnt array which always contains three elements regardless of the number of operations performed or the size of input parameters. The space occupied by the ParkingSystem object remains constant throughout its lifetime.

Learn more about how to find time and space complexity quickly using problem constraints.