Big-Oh Notation in Algorithms

<p>𝑂(𝑛3) (A), 𝑂(𝑛2) (B), 𝑂(𝑛4) (C)</p> Signup and view all the answers

<p>𝑂(𝑛2) (C), 𝑂(𝑛) (D)</p> Signup and view all the answers

<p>It specifies the minimum input size for which the upper bound is valid. (C)</p> Signup and view all the answers

<p>𝑂(log 𝑛) (A), 𝑂(𝑛) (D)</p> Signup and view all the answers

<p>It provides a more accurate and precise estimate of the algorithm's runtime behavior. (C)</p> Signup and view all the answers

<p>O(n!) (A)</p> Signup and view all the answers

<p>All of the above. (D)</p> Signup and view all the answers

<p>To check if a partial solution is feasible and can lead to a complete solution. (C)</p> Signup and view all the answers

<p>To describe the growth rate of an algorithm's resource consumption as the input size increases. (B)</p> Signup and view all the answers

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

<p>O(n) represents the upper bound, while Ω(n) represents the lower bound of an algorithm's running time. (C)</p> Signup and view all the answers

<p>To express the running time of a recursive algorithm in terms of the size of the problem. (A)</p> Signup and view all the answers

<p>A complete solution has been found. (B)</p> Signup and view all the answers

<p>To demonstrate that 3𝑛2 − 6 is bounded above by a constant multiple of 𝑛4 for sufficiently large 𝑛. (C)</p> Signup and view all the answers

<p>There exists a constant 𝑐 and a value 𝑛0 such that for all 𝑛 ≥ 𝑛0, 3𝑛2 − 6 is less than or equal to 𝑐𝑛4. (B)</p> Signup and view all the answers

<p>Divide both sides by 3 to obtain 2 ≤ 𝑛2. (C)</p> Signup and view all the answers

<p>It is an arbitrary constant that can be chosen to satisfy the inequality for sufficiently large 𝑛. (C)</p> Signup and view all the answers

<p>It is the minimum value of 𝑛 for which the inequality 3𝑛2 − 6 ≤ 𝑐𝑛4 holds true. (C)</p> Signup and view all the answers

<p>The growth of 3𝑛2 − 6 is asymptotically bounded above by 𝑛4. (C)</p> Signup and view all the answers

<p>$a^{m/n} = a^{m+n}$ (C)</p> Signup and view all the answers

<p>1 (C)</p> Signup and view all the answers

<p>Logarithmic (A)</p> Signup and view all the answers

<p>An 𝑂(𝑛4) function grows faster than an 𝑂(𝑛2) function as 𝑛 approaches infinity. (A)</p> Signup and view all the answers

<p>$2^n$ (D)</p> Signup and view all the answers

<p>$T(n)$ and $f(n)$ grow at the same rate (C)</p> Signup and view all the answers

<p>It ensures that 3𝑛2 − 6 is always positive. (B), It is a necessary condition for showing that 3𝑛2 − 6 = 𝑂(𝑛4). (D)</p> Signup and view all the answers

<p>4 (A), 2 (B), 3 (D)</p> Signup and view all the answers

<p>9 (D)</p> Signup and view all the answers

<p>10 (D)</p> Signup and view all the answers

<p>It is $O(n^3)$ (B), It is $\Theta(n^2)$ (C)</p> Signup and view all the answers

<p>3 (A)</p> Signup and view all the answers

<p>It is $\Theta(n^3)$ (B)</p> Signup and view all the answers

<p>$f(x)$ grows no faster than $g(x)$ (B)</p> Signup and view all the answers

<p>1 (A)</p> Signup and view all the answers

<p>The number of recursive calls (B)</p> Signup and view all the answers

<p>To initialize the recursive method (D)</p> Signup and view all the answers

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

<p>A queen's position on the chessboard (B)</p> Signup and view all the answers

<p>To find a specific arrangement of queens on a chessboard where no two queens threaten each other (C)</p> Signup and view all the answers

<p>It illustrates the process of trying different queen placements and backing up if a conflict occurs. (C)</p> Signup and view all the answers

<p>A way of tracing back the steps taken while placing queens on the board to identify conflicting positions (B)</p> Signup and view all the answers

<p>2 (A)</p> Signup and view all the answers

<p>2 (C)</p> Signup and view all the answers

<p>Both siblings of 𝑥.𝑐𝑖 must have 't - 1' keys. (C)</p> Signup and view all the answers

<p>136 (D)</p> Signup and view all the answers

<p>To ensure the new merged node has a median key. (A)</p> Signup and view all the answers

<p>Node 'x' has 1 key. (D)</p> Signup and view all the answers

<p>2 (C)</p> Signup and view all the answers

<p>Rule 3A is applied when the node has 't - 1' keys, while Rule 3B requires both siblings to have 't - 1' keys. (C)</p> Signup and view all the answers

Represents the behavior of algorithms, primarily running time and space.

Denotes an upper bound of a function's growth rate.

T(n) = O(f(n)) if there exist constants c > 0 and n0 such that 0 ≤ T(n) ≤ c * f(n) for all n ≥ n0.

The maximum growth rate of an algorithm’s running time in Big-Oh notation.

3n² - 6 = O(n⁴); shows what upper bound T(n) fits under.

Denotes a lower bound of a function's growth rate.

Denotes both upper and lower bounds, representing tight bounds on growth.

ω and o are additional notations not studied in this course.

A notation that describes the tight bound on the growth of a function.

Indicates a lower bound on the growth of a function, showing the best-case scenario.

Ordering functions based on their rate of growth.

Function growth where the rate becomes increasingly rapid, typically in the form 2^n.

Function growth described by n^2, where the output is proportional to the square of the input.

Function growth described by n, where output increases directly in proportion to input.

A function where the output remains the same regardless of the input, noted as c.

Function growth characterized by log(n), where output grows slowly compared to linear or quadratic.

Function growth characterized by n log(n), which is faster than linear but slower than quadratic.

A rule for merging nodes in a binary tree when both siblings have t - 1 keys.

The process of combining two sibling nodes due to insufficient keys.

Shifting a key from a parent node down to a merged node to establish a median.

The key that represents the middle value in a newly formed node after merging.

The node directly adjacent to another node in a tree.

The top node of a subtree that contains certain keys.

The minimum degree of a node in a B-tree, influencing key quantities.

A sequence of operations to remove a key from a data structure, like a binary tree.

A challenge to place 4 queens on a 4x4 chessboard without threats.

A visual representation of possible moves in a problem-solving process.

Functions that call themselves to solve subproblems.

Evaluating the efficiency of algorithms in terms of execution time.

Initial method used to set variables for recursive processes.

The potential moves a queen can make to threaten another piece.

The arrangement of pieces on a chessboard for a particular problem.

Initial values assigned to variables before execution starts.

A method for finding solutions by exploring all potential candidates and discarding those that fail to satisfy the conditions.

Tests whether a partial solution can be extended to a complete valid solution in backtracking.

A mathematical equation that describes a function in terms of its value at smaller inputs.

A framework for designing backtracking solutions to various problems.

A measure describing the amount of resources (time/space) required for an algorithm to complete.

A formal demonstration that shows the correctness of the algorithm's efficiency or outcome.

It signifies that the function 3𝑛² - 6 grows slower than or at the same rate as n⁴ as n approaches infinity.

Determining constants c and n₀ to satisfy the Big O definition: 0 ≤ f(n) ≤ c*g(n) for all n ≥ n₀.

Transforming inequalities to establish upper bounds in Big O analysis.

Establishing that one function does not grow faster than another provides a limit on its growth behavior.

Demonstrating that 3𝑛² - 6 fits into the category of O(n⁴) using inequalities.

A simplified form to find the lower limit of n for the analysis of 3𝑛² - 6.

An inequality showing how the function behaves regarding the chosen c and n.

A method to analyze how fast a function grows in relation to a polynomial of different degree.

A crucial aspect in determining Big O notation, focusing on how functions behave for large n.