Introduction to Algorithms

<p>It describes a set of functions that are both upper and lower bounded. (D)</p> Signup and view all the answers

<p>To compare the theoretical limits of algorithm performance for various input sizes. (A)</p> Signup and view all the answers

<p>To add three numbers and print their sum (A)</p> Signup and view all the answers

<p>The amount of time required for the algorithm to run (B)</p> Signup and view all the answers

<p>The total time taken for execution (A)</p> Signup and view all the answers

<p>Each additional input step increases total time proportionally (C)</p> Signup and view all the answers

<p>Fixed part (C)</p> Signup and view all the answers

<p>Space complexity indicates memory needed throughout the algorithm's lifecycle (A)</p> Signup and view all the answers

<p>It must be printed or displayed (C)</p> Signup and view all the answers

<p>Three (B)</p> Signup and view all the answers

<p>A step-by-step procedure for solving a problem (C)</p> Signup and view all the answers

<p>Iterate (B)</p> Signup and view all the answers

<p>A clear problem definition (C)</p> Signup and view all the answers

<p>The solution must be within given constraints (A)</p> Signup and view all the answers

<p>The result of the addition (A)</p> Signup and view all the answers

<p>Sort (D)</p> Signup and view all the answers

<p>The inputs needed for the problem (A)</p> Signup and view all the answers

<p>An algorithm to remove an existing item from a data structure (B)</p> Signup and view all the answers

<p>Identifying frequently occurring elements (D)</p> Signup and view all the answers

<p>A Posterior Analysis (C)</p> Signup and view all the answers

<p>Defining the best, average, and worst-case performance (A)</p> Signup and view all the answers

<p>As a function $f(n)$ without any constants (B)</p> Signup and view all the answers

<p>It works in constant time (C)</p> Signup and view all the answers

<p>Tools that analyze an algorithm’s running time based on input size (C)</p> Signup and view all the answers

<p>To compare theoretical performance with actual performance (D)</p> Signup and view all the answers

<p>Best, average, and worst-case running times (A)</p> Signup and view all the answers

<p>The upper bound of the running time of an algorithm. (C)</p> Signup and view all the answers

<p>Theta Notation (Θ-notation) (C)</p> Signup and view all the answers

<p>Worst Case (B)</p> Signup and view all the answers

<p>To compare algorithms based on their worst-case complexity. (C)</p> Signup and view all the answers

<p>Time required averaged over all possible inputs. (D)</p> Signup and view all the answers

<p>It provides an upper bound on the running time for worst-case scenarios. (B)</p> Signup and view all the answers

<p>$0 ext{ ≤ } f(n) ext{ ≤ } c g(n)$ for all $n ext{ ≥ } n_0$. (C)</p> Signup and view all the answers

<p>It specifies the lower bound of the running time for the best case. (A)</p> Signup and view all the answers

A set of step-by-step instructions leading to a desired output.

Clear problem, constraints, input, output, and a solution within constraints are needed when creating an algorithm.

An algorithm used to find a specific item within a data structure.

An algorithm to arrange items in a specific order.

An algorithm enabling the addition of an item to a data structure.

An algorithm to modify an existing item in a data structure.

An algorithm for removing an existing item from a data structure.

Clearly defined problem, constraints, input, expected output, and a solution within the constraints are required to write an algorithm correctly.

Time taken for an algorithm to complete, measured as a function of input size.

Memory an algorithm uses during its execution, related to the input size.

A mathematical expression representing the algorithm's time requirements as a function of input size (n).

Memory always needed by the algorithm, regardless of input size.

Amount of data processed by the algorithm.

An example of an algorithm with a linear time complexity, where each bit consumes constant time.

The time an algorithm needs to complete, measured in computational steps.

Represents a tight bound on the growth rate of an algorithm's running time, meaning it describes both the upper and lower bounds.

Represents the lower bound of an algorithm's running time, indicating the best-case scenario for its efficiency.

The set of functions that are lower bounded by g(n), meaning they grow at least as fast as g(n) for large enough input sizes.

It helps us understand the minimum amount of time an algorithm will take, providing valuable insight into its efficiency.

The minimum amount of resources (e.g., time or memory) an algorithm requires to solve a problem.

Analyzing an algorithm's efficiency before implementing it. Focuses on identifying patterns and frequently occurring elements in data.

Analyzing an algorithm's efficiency after implementation. Measures actual performance like running time and memory usage.

Mathematical framework for understanding an algorithm's runtime performance, considering how it changes with input size.

Different scenarios in asymptotic analysis, describing the algorithm's performance under ideal, typical, and challenging conditions.

Asymptotic analysis primarily focuses on algorithm performance based on the size of the input data, treating other factors as constants.

Programming tools to analyze an algorithm's runtime growth as input size increases. Identifying the increasing behavior.

Representing the runtime of different operations in asymptotic analysis. f(n) grows linearly, while g(n2) grows exponentially.

An algorithm takes the same amount of time to complete, regardless of the input size.

The rate at which an algorithm's time or space requirements increase with the size of the input. It's a way to measure how efficient an algorithm is for large datasets.

A notation used to describe the upper bound of an algorithm's running time or space complexity. It provides a worst-case estimate of the algorithm's performance.

A notation that represents the tight bound of an algorithm's running time or space complexity. It describes both the upper and lower bounds, providing an average-case estimate.

Big-Oh notation measures the upper bound, or worst-case scenario, of an algorithm's running time or space complexity.

Theta notation measures the average-case complexity of an algorithm, providing a tight bound on its running time.

Algorithms are analyzed considering the best-case, average-case, and worst-case scenarios to understand their behavior across different inputs.

It helps us understand how the performance of an algorithm changes as the input size grows, allowing us to choose the most efficient algorithm for a given task.