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# Algorithmic Efficiency When comparing algorithms, we primarily focus on their efficiency, especially as the problem size increases. Efficiency is typically measured by: * **Execution time:** How long the algorithm takes to run. * **Memory usage:** How much memory the algorithm requires. To...

# Algorithmic Efficiency When comparing algorithms, we primarily focus on their efficiency, especially as the problem size increases. Efficiency is typically measured by: * **Execution time:** How long the algorithm takes to run. * **Memory usage:** How much memory the algorithm requires. To quantify these, we use **asymptotic analysis**, which evaluates performance as the input size (n) grows arbitrarily large. ## Big-O Notation Big-O notation provides an upper bound on the growth rate of an algorithm's time or space complexity. It describes the **worst-case scenario**. * **Definition:** $f(n) = O(g(n))$ if there exist positive constants c and $n_0$ such that $f(n) \le c \cdot g(n)$ for all $n \ge n_0$. * $f(n)$ is the actual growth rate of the algorithm. * $g(n)$ is the upper bound. * **Common complexities:** * $O(1)$ - Constant * $O(log n)$ - Logarithmic * $O(n)$ - Linear * $O(n log n)$ - Linearithmic * $O(n^2)$ - Quadratic * $O(n^3)$ - Cubic * $O(2^n)$ - Exponential * $O(n!)$ - Factorial ## Big-Omega Notation Big-Omega notation provides a lower bound on the growth rate of an algorithm. It describes the **best-case scenario**. * **Definition:** $f(n) = \Omega(g(n))$ if there exist positive constants c and $n_0$ such that $f(n) \ge c \cdot g(n)$ for all $n \ge n_0$. * $f(n)$ is the actual growth rate of the algorithm. * $g(n)$ is the lower bound. ## Big-Theta Notation Big-Theta notation describes a tight bound on the growth rate of an algorithm. It describes the **average-case scenario** and indicates that the algorithm's growth rate is both bounded above and below by the same function. * **Definition:** $f(n) = \Theta(g(n))$ if there exist positive constants $c_1$, $c_2$, and $n_0$ such that $c_1 \cdot g(n) \le f(n) \le c_2 \cdot g(n)$ for all $n \ge n_0$. * $f(n)$ is the actual growth rate of the algorithm. * $g(n)$ is the tight bound. ## Visual Representation The image shows a graph that visually represents Big-O, Big-Omega, and Big-Theta notations. The x-axis represents the input size *n*, and the y-axis represents the time or space complexity *f(n)*. * **Big-O (O(g(n)))**: The graph of *f(n)* is below the curve of $c \cdot g(n)$ for $n > n_0$, indicating an upper bound * **Big-Omega ($\Omega$(g(n)))**: The graph of *f(n)* is above the curve of $c \cdot g(n)$ for $n > n_0$, indicating a lower bound. * **Big-Theta ($\Theta$(g(n)))**: The graph of *f(n)* is between the curves of $c_1 \cdot g(n)$ and $c_2 \cdot g(n)$ for $n > n_0$, indicating a tight bound.