Hacks

  • Record your findings when testing the time elapsed of the different algorithms.
    • I realized that fibonacci code increases a very great amount for an increase in just one
    • Constant algorithms only take 0.0s
    • Base width increases both space and time complexity greatly
    • If it is asked to print a calulation from just random integers it is fast but if it is asked to calculate a number based on an output the time and space complexity is increased greatly
    • For i in j is very big
    • The time increases exponentially when the data is increased by just a bit
      • Makes it even more important to manege complexity
## This takes minutes to execute
def find_duplicates(arr):
    duplicates = []
    for i in range(len(arr)):
        for j in range(i + 1, len(arr)):
            if arr[i] == arr[j]:
                duplicates.append(arr[i])
    return duplicates

arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] * 1000000
duplicates = find_duplicates(arr)
print(duplicates)
# This takes 2.1 seconds to execute. This is why its important to manage complexity
def find_duplicates(arr):
    duplicates = []
    unique_set = set()
    for i in arr:
        if i not in unique_set:
            unique_set.add(i)
        else:
            duplicates.append(i)
    return duplicates

arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] * 1000000
duplicates = find_duplicates(arr)
print(duplicates)
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3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 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Hacks

  • Although we will go more in depth later, time complexity is a key concept that relates to the different sorting algorithms. Do some basic research on the different types of sorting algorithms and their time complexity.
    • Selection Sort Best: Ω(n^2)
      • Average: θ(n^2)
      • Worst: O(n^2)
    • Bubble Sort
      • Best: Ω(n)
      • Average: θ(n^2)
      • Worst: O(n^2)
    • Insertion Sort
      • Best: Ω(n)
      • Average: θ(n^2)
      • Worst: O(n^2)
    • Heap Sort
      • Best: Ω(n log(n))
      • Average: θ(n log(n))
      • Worst: O(n log(n))
    • Quick Sort
      • Best: Ω(n log(n))
      • Average: θ(n log(n))
      • Worst: O(n^2)
    • Merge Sort
      • Best: Ω(n log(n))
      • Average: θ(n log(n))
      • Worst: O(n log(n))
    • Bucket Sort
      • Best: Ω(n +k)
      • Average: θ(n +k)
      • Worst: O(n^2)
    • Radix Sort
      • Best: Ω(nk)
      • Average: θ(nk)
      • Worst: O(nk)
    • Count Sort
      • Best: Ω(n +k)
      • Average: θ(n +k)
      • Worst: O(n +k)
    • Shell Sort
      • Best: Ω(n log(n))
      • Average: θ(n log(n))
      • Worst: O(n^2)
    • Tim Sort
      • Best: Ω(n)
      • Average: θ(n log(n))
      • Worst: O(n log (n))
    • Tree Sort
      • Best: Ω(n log(n))
      • Average: θ(n log(n))
      • Worst: O(n^2)
    • Cube Sort
      • Best: Ω(n)
      • Average: θ(n log(n))
      • Worst: O(n log(n))
def selection_sort(arr):
    n = len(arr)
    for i in range(n):
        min_idx = i
        for j in range(i+1, n):
            if arr[j] < arr[min_idx]:
                min_idx = j
        arr[i], arr[min_idx] = arr[min_idx], arr[i]
    return arr

arr = [64, 25, 12, 22, 11]
sorted_arr = selection_sort(arr)
print(sorted_arr) # Output: [11, 12, 22, 25, 64]
[11, 12, 22, 25, 64]
def bubble_sort(arr):
    n = len(arr)
    for i in range(n):
        # Last i elements are already sorted
        for j in range(0, n-i-1):
            # Traverse the array from 0 to n-i-1
            # Swap if the element found is greater than the next element
            if arr[j] > arr[j+1]:
                arr[j], arr[j+1] = arr[j+1], arr[j]
    return arr
  • Why is time and space complexity important when choosing an algorithm?
    • Both time and space complexity are important because they have to do with efficiancy. Time complexity is how much time a program is going to take to run. Space complexity is how much space an algorithm takes up. The more space it takes up the more time it takes to run because there is more to run. The reason we have to understand this is because we can control it. The more efficiant we make code and the more presice it is the less time and space it will take up. We have to make these as low as possible as well as maintaining the original function for consumer experience.
  • Should you always use a constant time algorithm / Should you never use an exponential time algorithm? Explain?
    • While using a constant algorithm can make a function run faster and have a better space complexity, it doesn't always fit the function. Sometimes the function requires an exponential function to properly execute.
#
def fibonacci(n):
    if n <= 1:
        return n
    else:
        return fibonacci(n-1) + fibonacci(n-2)
import math
import random

def simulate_game(team1, team2):
    # Simulate a game between team1 and team2
    # Here we're just generating a random winner
    return random.choice([team1, team2])

def generate_bracket(num_teams):
    bracket = []
    num_rounds = int(math.log(num_teams, 2))
    teams = list(range(1, num_teams+1))
    for round_num in range(num_rounds):
        round_matches = []
        num_matches = num_teams // 2**(round_num+1)
        for match_num in range(num_matches):
            team1, team2 = teams[2*match_num], teams[2*match_num+1]
            winner = simulate_game(team1, team2)
            match = (team1, team2, winner)
            round_matches.append(match)
        bracket.append(round_matches)
        teams = [match[2] for match in round_matches]
    return bracket

num_teams = 8
bracket = generate_bracket(num_teams)
for i, round_matches in enumerate(bracket):
    print("Round", i+1)
    for match in round_matches:
        print("Game:", match[0], "vs", match[1])
        print("Winner:", match[2])
    print()
Round 1
Game: 1 vs 2
Winner: 2
Game: 3 vs 4
Winner: 4
Game: 5 vs 6
Winner: 5
Game: 7 vs 8
Winner: 7

Round 2
Game: 2 vs 4
Winner: 2
Game: 5 vs 7
Winner: 7

Round 3
Game: 2 vs 7
Winner: 2

  • What are some general patterns that you noticed to determine each algorithm's time and space complexity?
    • Patterns I've noticed are:
      • Nested Loops: O(n^2) calls for a nested loop or a loop inside of itself. The nested loops are inside of each other and n is the data given. So n is repeating inself.
      • Recursive: The algorithm uses recurrence relation. The more recurrence the higher the complexity and the longer it takes to run the function.
      • Data: The more data the longer it is going to take for a piece of code to run. It has more space and data to store and calculate so it takes longer to run

Practice Questions

Question 1

What is the time and space complexity of the following code?

a = 0
b = 0
for i in range(N):
  a = a + random()
 
for i in range(M):
  b = b + random()
  1. O(N * M) time, O(1) space
  2. O(N + M) time, O(N + M) space
  3. O(N + M) time, O(1) space
  4. O(N * M) time, O(N + M) space
  • Answer: 2
  • Correct Answer: 3
  • Reasoning: The reason I got it wrong was because I saw two addition signs so I thought that would match up with the answer. The correct answer was 3 because the algorithm in then constant and I thought it would be exponential.

Question 2

What is the time complexity of the following code?

a = 0
for i in range(N):
  for j in reversed(range(i,N)):
    a = a + i + j
  1. O(N)
  2. O(N*log(N))
  3. O(N * Sqrt(N))
  4. O(N*N)
  • Answer: 4
  • Correct Answer: 4
  • Reasoning: This is because the loop is nested i times j or n times n. It is times itself. This makes it execute longer

Question 3

What is the time complexity of the following code?

k = 0
for i in range(n//2, n):
  for j in range(2, n, pow(2,j)):
        k = k + n / 2
  1. O(n)
  2. O(nlog(n))
  3. O(n^2)
  4. O(n^2(log(n)))
  • Answer: 4
  • Correct Answer: 2
  • Reasoning: To me this code looked unfamiliar so I picked the most unfamiliar option. The reason it was two was because n was being logged in the equation and there is a nested loop.

Question 4

What does it mean when we say that an algorithm X is asymptotically more efficient than Y?

  1. X will always be a better choice for small inputs
  2. X will always be a better choice for large inputs
  3. Y will always be a better choice for small inputs
  4. X will always be a better choice for all inputs
  • Answer: 2
  • Correct Answer: 2
  • Reasoning: X isn't always necesarally beter but it cerainly is better for large outputs because it reduces the space and time complexity reduceing the time it takes to run and the space it takes up in your computer

Question 5

What is the time complexity of the following code?

a = 0
i = N
while (i > 0):
  a += i
  i //= 2
  1. O(N)
  2. O(Sqrt(N))
  3. O(N / 2)
  4. O(log N)
  • Answer: 1
  • Correct Answer: 4
  • Reasoning: This seemed like a normal n equation because i = n but in reality it was the los of n because of the while loop

Question 6

Which of the following best describes the useful criterion for comparing the efficiency of algorithms?

  1. Time
  2. Memory
  3. Both of the above
  4. None of the above
  • Answer: 3
  • Correct Answer: 3
  • Reasoning: Both of these are very important because both space and time need to be used efficiantly. Space can help save room anf time can make the algorithm faster.

Question 7

How is time complexity measured?

  1. By counting the number of algorithms in an algorithm.
  2. By counting the number of primitive operations performed by the algorithm on a given input size.
  3. By counting the size of data input to the algorithm.
  4. None of the above
  • Answer: 2
  • Correct Answer: 2
  • Reasoning: The number of primitive operatios can increase complexity

Question 8

What will be the time complexity of the following code?

for i in range(n):
  i=i*k
  1. O(n)
  2. O(k)
  3. O(logk(n))
  4. O(logn(k))
  • Answer: 3
  • Correct Answer: 3
  • Reasoning: Now when I see a loop I think of log. Log of k is not a function

Question 9

What will be the time complexity of the following code

value = 0
for i in range(n):
  for j in range(i):
    value=value+1
  1. O(n)
  2. O(n+1)
  3. O(n(n-1))
  4. O(n(n+1))
  • Answer: 4
  • Correct Answer: 3
  • Reasoning: I saw that n was times itself so I knew it was nested but I thought because of the plus sign it would be added but it would be subtracked like it a parabala equation

Question 10

Algorithm A and B have a worst-case running time of O(n) and O(logn), respectively. Therefore, algorithm B always runs faster than algorithm A.

  1. True
  2. False
  • Answer: 2
  • Correct Answer: 2
  • Reasoning: We should never say always runs faster because there can be an error or a slow computer. Also, it depends on the input. If the input for the first one is high and the second low then the second could be lower.