asymptotic-notations.md

Asymptotic Notations

Asymptotic notations are mathematical tools used to describe the behavior of an algorithm as the input size grows. They help us analyze the efficiency of algorithms by providing a way to compare their running times without relying on specific hardware or software implementations. The most common asymptotic notations are:

Types of Asymptotic Notations:

1. Big O notation (O)

   • Represents the upper bound of the running time of an algorithm.
   • Defines the worst-case scenario.
   • Used to describe the maximum amount of time an algorithm could take to complete.
   • Notation: $$f(n) = O(g(n))$$

     • $$f(n)$$ is the actual running time of the algorithm.

     • $$g(n)$$ is a function that represents the upper bound.

     • There exist positive constants $$c$$ and $$n_0$$ such that $$f(n) <= c \cdot g(n)$$ for all $$n >= n_0$$.

       

2. Big Omega notation (Ω)

   • Represents the lower bound of the running time of an algorithm.
   • Defines the best-case scenario.
   • Used to describe the minimum amount of time an algorithm will take to complete.
   • Notation: $$f(n) = Ω(g(n))$$

     • $$f(n)$$ is the actual running time of the algorithm.

     • $$g(n)$$ is a function that represents the lower bound.

     • There exist positive constants $$c$$ and $$n_0$$ such that $$f(n) >= c \cdot g(n)$$ for all $$n >= n_0$$.

       

3. Big Theta notation (Θ)

   • Represents the tight bound of the running time of an algorithm.
   • Defines both the upper and lower bounds.
   • Used to describe the exact growth rate of an algorithm.
   • Used for analyzing the average-case scenario.
   • Notation: $$f(n) = Θ(g(n))$$

     • $$f(n)$$ is the actual running time of the algorithm.

     • $$g(n)$$ is a function that represents the tight bound.

     • There exist positive constants $$c_1$$, $$c_2$$, and $$n_0$$ such that $$c_1 \cdot g(n) <= f(n) <= c_2 \cdot g(n)$$ for all $$n >= n_0$$.

       

Example:

Consider the following functions:

• $$f(n) = 2n + 3$$
• $$g(n) = n$$

To determine the relationship between $$f(n)$$ and $$g(n)$$ using asymptotic notations:

1. Big O notation (O):
   • $$f(n) = O(g(n))$$ if there exist positive constants $$c$$ and $$n_0$$ such that $$f(n) <= c \cdot g(n)$$ for all $$n >= n_0$$.
   • In this case, $$f(n) = 2n + 3$$ and $$g(n) = n$$.
   • Let's choose $$c = 3$$ and $$n_0 = 1$$.
   • For $$n >= 1$$, we have:
     • $$f(n) = 2n + 3 <= 3n = c \cdot g(n)$$.
   • Therefore, $$f(n) = O(g(n))$$.
2. Big Omega notation (Ω):
   • $$f(n) = Ω(g(n))$$ if there exist positive constants $$c$$ and $$n_0$$ such that $$f(n) >= c \cdot g(n)$$ for all $$n >= n_0$$.
   • In this case, $$f(n) = 2n + 3 and g(n) = n$$.
   • Let's choose $$c = 1$$ and $$n_0 = 1$$.
   • For $$n >= 1$$, we have:
     • $$f(n) = 2n + 3 >= n = c \cdot g(n)$$.
   • Therefore, $$f(n) = Ω(g(n))$$.
3. Big Theta notation (Θ):
   • $$f(n) = Θ(g(n))$$ if there exist positive constants $$c_1, c_2,$$ and $$n_0$$ such that $$c_1 \cdot g(n) <= f(n) <= c_2 \cdot g(n)$$ for all $$n >= n_0$$.
   • In this case, $$f(n) = 2n + 3$$ and $$g(n) = n$$.
   • Let's choose $$c_1 = 1, c_2 = 3$$, and $$n_0 = 1$$.
   • For $$n >= 1$$, we have:
     • $$1n <= 2n + 3 <= 3n$$
   • Therefore, $$f(n) = Θ(g(n))$$.

In summary, the relationship between $$f(n) = 2n + 3$$ and $$g(n) = n$$ is: $$f(n) = O(g(n)), f(n) = Ω(g(n))$$, and $$f(n) = Θ(g(n))$$. This means that $$f(n)$$ grows linearly with $$n$$.

Key Points:

• Asymptotic notations help us analyze the efficiency of algorithms by providing a way to compare their running times.
• Big O notation represents the upper bound, Big Omega notation represents the lower bound, and Big Theta notation represents the tight bound of the running time of an algorithm.
• Understanding asymptotic notations is essential for evaluating the performance of algorithms and making informed decisions about algorithm selection and optimization.