1.8.2 Asymptotic Notations - Big Oh - Omega - Theta #2

Summary

The video explains key concepts of asymptotic notations—Big O, Omega, and Theta—used to describe algorithmic time complexity. It uses examples like polynomial functions and factorials to demonstrate how to determine upper and lower bounds for functions. Theta notation is highlighted as the most precise, while Big O and Omega provide broader estimates. The importance of selecting the nearest time complexity for meaningful analysis is emphasized, with analogies for clearer understanding. The video also prepares viewers for a subsequent discussion on the properties of these notations.

Takeaways

• 📐 The function 2n^2 + 3n + 4 is bounded by 9n^2 and n^2, indicating it is O(n^2) and Ω(n^2) but not Θ(n^2).
• 🔍 For the function n^2 log n + n, it is both O(n^2 log n) and Ω(n log n), making it Θ(n^2 log n).
• 📈 The function n! (n factorial) does not have a tight bound and is described as O(n^n) and Ω(1).
• 📊 log n! (log of n factorial) is bounded by n log n, making it O(n log n) and Ω(1), without a tight bound.
• 🌐 The importance of using the tightest possible bound is emphasized, with Θ notation being the most desirable for its precision.
• 📉 When a tight bound cannot be found, using upper bounds O and lower bounds Ω is recommended.
• 🔑 The script illustrates how to compare functions to determine their asymptotic relationships using inequalities.
• 📚 Understanding the concept of asymptotic notations is crucial for analyzing algorithm performance.
• 📝 The script provides practical examples to help viewers grasp the concept of asymptotic notations and their application.
• 🎯 The takeaway is that while Θ provides an exact measure, O and Ω offer insights into the upper and lower bounds of a function's growth.

Q & A

What does the symbol 'Big O' represent in the context of the transcript?

In the transcript, 'Big O' notation is used to describe the upper bound of a function, indicating that one function grows at most as fast as another function. It is denoted as f(n) = O(g(n)) where f(n) is less than or equal to c*g(n) for some constant c and for all n greater than or equal to some value.

Can you explain the example given for the function 2n² + 3n + 4 in terms of Big O notation?

The transcript explains that the function 2n² + 3n + 4 is less than 2n² + 3n² + 4n², which simplifies to 9n². Therefore, 2n² + 3n + 4 is O(n²), meaning it is bounded above by a constant times n² for sufficiently large n.

What is meant by 'Omega' in the transcript?

In the transcript, 'Omega' notation is used to describe the lower bound of a function, indicating that one function grows at least as fast as another function. It is denoted as f(n) = Ω(g(n)) where f(n) is greater than or equal to c*g(n) for some constant c and for all n greater than or equal to some value.

How is the function 2n² + 3n + 4 described using Omega notation according to the transcript?

The transcript states that 2n² + 3n + 4 is greater than or equal to 1*n², which means it is Ω(n²). This indicates that the function grows at least as fast as n².

What is the significance of 'Theta' notation as discussed in the transcript?

Theta notation in the transcript is used when a function is both Big O and Omega of another function, indicating that it grows at the same rate. It is denoted as f(n) = Θ(g(n)) where f(n) is bounded by c*g(n) and c'*g(n) for some constants c and c' and for all n greater than or equal to some value.

How does the transcript describe the function n² log n + n in terms of Big O and Theta notation?

The transcript explains that n² log n + n is less than or equal to 10n² log n and also greater than n^s log n for some s. This places it between n^s log n and n^s log n, making it both Big O of n² log n and Theta of n² log n.

What does the transcript suggest about the use of Big O, Omega, and Theta notations?

The transcript suggests that Theta notation is preferable when a tight bound can be found, as it provides an exact figure for the time complexity. However, when a tight bound cannot be found, Big O and Omega notations are used to describe the upper and lower bounds, respectively.

Why is it difficult to define a Theta notation for n factorial as per the transcript?

The transcript explains that n factorial does not have a tight bound or average bound that can be represented by Theta notation because it grows faster than any polynomial function. For small values of n, it may be closer to 1, but for larger values, it grows towards n^n, making it impossible to define a consistent Theta relationship.

How does the transcript describe the function log n factorial in terms of upper and lower bounds?

The transcript states that log n factorial is less than or equal to n log n, which means its upper bound is log n^n and the lower bound is 1. Since there is no tight bound for log n factorial, it cannot be represented by Theta notation.

What is the importance of choosing the 'right' value when using Big O notation as explained in the transcript?

The transcript emphasizes the importance of choosing the 'right' value when using Big O notation because it should be meaningful and useful. For example, if the function is n², then stating it as n² is the most useful and meaningful, whereas stating it as n log n or n might be correct but less useful.

What advice does the transcript give regarding the use of asymptotic notations?

The transcript advises that when using asymptotic notations, one should aim to use a tight bound if possible, which is represented by Theta notation. If a tight bound cannot be found, then Big O and Omega notations should be used, with an attempt to provide the nearest value for the upper or lower bound.