Update page 'Day 7 homework - Karen'

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 **Today's homework is to do some performance analysis of Python's hash tables, which are a key data structure used by the `dict` class.  We're also going to do some miscellaneous tasks to help practice writing Python.**
 **1. Use the `time.perf_counter_ns()` method to measure how many nanoseconds it takes for `time.sleep()` to run for the following durations: 10 milliseconds, 100 milliseconds, 1 second.  Repeat for 100 times for each of those durations.  The variation in timing is called *jitter*.  Can you find any pattern to how the jitter varies for each of the different sleep durations?  Binning the results and creating a histogram may help.**
 ```
 import time
 durations = [0.01, 0.1, 1.0]
 num_repetitions = 100
 # Jitter for 10 milliseconds, 100 milliseconds, and 1 second
 jitter_10ms = []
 jitter_100ms = []
 jitter_1s = []
 def measure_jitter(duration):
    start_time = time.perf_counter_ns()
    time.sleep(duration)
    end_time = time.perf_counter_ns()
    return end_time - start_time
 # Measure jitter for each duration and repeat for num_repetitions times
 for duration in durations:
    for _ in range(num_repetitions):
        jitter = measure_jitter(duration)
        if duration == 0.01:
            jitter_10ms.append(jitter)
        elif duration == 0.1:
            jitter_100ms.append(jitter)
        elif duration == 1.0:
            jitter_1s.append(jitter)
 # Create histograms for each duration's jitter
 import matplotlib.pyplot as plt
 plt.hist(jitter_10ms, bins=10, color='r', alpha=0.5, label='10ms')
 plt.hist(jitter_100ms, bins=10, color='g', alpha=0.5, label='100ms')
 plt.hist(jitter_1s, bins=10, color='b', alpha=0.5, label='1s')
 plt.xlabel('Jitter (ns)')
 plt.ylabel('Frequency')
 plt.title('Jitter of time.sleep() for Different Durations')
 plt.legend()
 plt.show()
 ```
 ![Jitter of `time.sleep()`](https://gogs.gtsplab.com:5432/j/python-bootcamp/src/Branch_Karen/jitter.png)
 From the histogram shown above, we can say that jitter is expected to be more fluctuated for longer sleep duration, and the relationship between the sleep duration and jitter can fit a polynomial curve of degree 2 as being shown below. A possible reason for this may be the overhead of the function call and operating system scheduling.
 ![Jitter & Duration relationship](https://gogs.gtsplab.com:5432/j/python-bootcamp/src/Branch_Karen/jitter%20relationship.png)
 **2. Create a dict with 1 million integer keys, each with 64 bytes of random values.**
 ```
 import random
 import secrets
 my_dict = {}
 for i in range(1000000):
    # Generate a random 64-byte value as bytes
    random_value = secrets.token_bytes(64)
    # Convert the bytes value to hexadecimal representation
    hex_value = random_value.hex()
    # Add the key-value pair to the dictionary
    my_dict[i] = hex_value
 print("Dictionary created with 1 million integer keys.")
 ```
 **3. Now that we know how to precisely time how long it takes for Python to do something, measure how long it takes to look up 1,000, 10,000, and 100,000 random integer keys, and repeat each experiment 100 times.  Does the amount of time per key lookup change?**
 ```
 def measure_time(num_lookups):
    total_time = 0
    for i in range(100):
        keys = random.sample(range(1000000), num_lookups)
        start_time = time.perf_counter()
        for key in keys:
            value = my_dict[key]
        end_time = time.perf_counter()
        total_time += end_time - start_time
    return total_time / 100
 # Measure time taken for 1,000 key lookups
 time_1000 = measure_time(1000)
 print(f"Average time taken for 1,000 key lookups: {time_1000:.9f} seconds")
 # Measure time taken for 10,000 key lookups
 time_10000 = measure_time(10000)
 print(f"Average time taken for 10,000 key lookups: {time_10000:.9f} seconds")
 # Measure time taken for 100,000 key lookups
 time_100000 = measure_time(100000)
 print(f"Average time taken for 100,000 key lookups: {time_100000:.9f} seconds")
 ```
 Average time taken for 1,000 key lookups: 0.000496019 seconds
 Average time taken for 10,000 key lookups: 0.004768387 seconds
 Average time taken for 100,000 key lookups: 0.055429382 seconds
 Obviously, the average time becomes longer when looking up more keys in a dic. The reason for this fact should relate to the nature of dictionary data structure in Python as it is implemented as a hash table, which is an unordered collection of key and value pairs. As more keys need to be looked up, the hash collisions may increase that cause the average time to become longer.
 **4. The ABC module can be used to create so-called *virtual base classes*, similar to the `Simulator.py` file from day 4.  Look around you, and find two related objects.  Create a virtual base class by inheriting from the ABC module and using the `@abstractmethod` and `@property` decorators to define the shared methods and properties of the class, and non-abstract methods and properties which are the same for the two objects.**
 Just use the pets "Cat" and "Dog" as an example, and the class `FunnyPets` is defined to inherit from `ABC`.
 ```
 from abc import ABC, abstractmethod
 class FunnyPets(ABC):
    @abstractmethod
    def make_sound(self):
        pass
    @abstractmethod
    def play(self):
        pass
    @property
    def color(self):
        return "Multicolor"
    @property
    def size(self):
        return "Small"
    def common_snack(self):
        return "This is a common snack for Cat and Dog objects"
 ```
 **5. Create *concrete classes* which inherit from your virtual base class from the previous task, and define the non-abstract methods and properties which are different for each of the two objects.  I'm deliberately omitting an example here to incentivize you to read the documentation on the ABC module: https://docs.python.org/3/library/abc.html#module-abc**
 ```
 class Cat(FunnyPets):
    def make_sound(self):
        return "Meow!"
    def play(self):
        return "Cat is playing with a little mouse toy."
    @property
    def color(self):
        return "Tabby"
    @property
    def size(self):
        return "Small"
    def unique_snack(self):
        return "This is a dehydrated fish for Cat object"
 class Dog(FunnyPets):
    def make_sound(self):
        return "Woof!"
    def play(self):
        return "Dog is playing fetch with a frisbee."
    @property
    def color(self):
        return "Gray & white"
    @property
    def size(self):
        return "Small, but larger than cat"
    def unique_snack(self):
        return "This is a bone for Dog object"
 ```
 **6. What happens when you try to call an abstract method, or access an abstract property in the base class?**
 It reports a TypeError saying "can't instantiate abstract class FunnyPets with abstract methods".
 To solve this issue, we need to create instances of Cat and Dog first and then call the overridden methods. For example, Alice is the cat living in my apartment and Bibi is my friend's dog, then I can get their sounds like:
 ```
 Alice = Cat()
 Bibi = Dog()
 print(cat.make_sound())  # Output: "Meow"
 print(dog.make_sound())  # Output: "Woof"
 ```