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:
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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:
$$f(n) = O(g(n))$$
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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:
$$f(n) = Ω(g(n))$$
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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:
$$f(n) = Θ(g(n))$$
Consider the following functions:
$$f(n) = 2n + 3$$ $$g(n) = n$$
To determine the relationship between
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Big O notation (O):
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$$f(n) = O(g(n))$$ 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:-
$$f(n) = 2n + 3 <= 3n = c \cdot g(n)$$ .
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- Therefore,
$$f(n) = O(g(n))$$ .
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Big Omega notation (Ω):
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$$f(n) = Ω(g(n))$$ 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:-
$$f(n) = 2n + 3 >= n = c \cdot g(n)$$ .
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- Therefore,
$$f(n) = Ω(g(n))$$ .
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Big Theta notation (Θ):
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$$f(n) = Θ(g(n))$$ 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,
$$f(n) = Θ(g(n))$$ .
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In summary, the relationship between
- 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.