Recursion Tree Method in Algorithm Analysis

True (A)

3^i

<p>T(n) = 3T(n/4) + n^2 (B)</p> Signup and view all the answers

<p>n^2 / 16^i</p> Signup and view all the answers

<p>Recurrence relation = An equation that defines a sequence recursively, meaning each term is expressed in terms of previous terms. Base case = The condition that terminates the recursion. Recursion tree method = A visual technique used to solve recurrence relations by representing the recursive calls as a tree. Node = A representation of a recursive call in the recursion tree.</p> Signup and view all the answers

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

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

<p>3^i * n^2 / 16^i</p> Signup and view all the answers

<p>Node Count (Layer 0) = 1 Impact (Layer 1) = 3n^2/16 Node Count (Layer 1) = 3 Impact (Layer 2) = 9n^2/256</p> Signup and view all the answers

<p>Use induction on n to prove T(n) ≤ cn - d for constants c, d &gt; 0. (C)</p> Signup and view all the answers

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

<p>Mathematical Induction</p> Signup and view all the answers

<p>linear</p> Signup and view all the answers

<p>⌊n/2⌋ = The largest integer less than or equal to n/2 ⌈n/2⌉ = The smallest integer greater than or equal to n/2 c = A constant that determines the upper bound of the function d = A constant that represents the additional factor in the upper bound T(n) = The time complexity function defining the time taken for an algorithm to run on an input of size n</p> Signup and view all the answers

<p>recursive</p> Signup and view all the answers

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

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

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

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

<p>evolves</p> Signup and view all the answers

<p>It provides a visual representation of the recursive calls and their associated costs.</p> Signup and view all the answers

<p>T(n) = Time complexity function n^2 = Cost of a single recursive call 3T(n/4) = Recursive calls T(1) = 1 = Base case</p> Signup and view all the answers

<p>n^2</p> Signup and view all the answers

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

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

<p>The work done at each level of the recursion tree</p> Signup and view all the answers

<p>Node Impact = The amount of work done at a particular node Layer Count = The number of nodes at a specific level of the tree Layer = A specific level in the recursion tree</p> Signup and view all the answers

<p>T(n) = 3T(n/4) + n^2 (D)</p> Signup and view all the answers

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

<p>1</p> Signup and view all the answers

<p>16n</p> Signup and view all the answers

<p>16 (D)</p> Signup and view all the answers

<p>T(n) = Total time complexity n^2 = Node impact per layer log4 n = Logarithmic term in the function i = Layer counter in the recursion tree</p> Signup and view all the answers

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

<p>n^(2) * 3^i</p> Signup and view all the answers

A visual method for analyzing the time complexity of recursive algorithms.

A specific recurrence relation representing time complexity in algorithms.

The contribution of each node's calculation to the overall complexity.

Levels in the recursion tree indicating stages of recursive calls.

The initial layer where the primary input size affects the complexity.

The condition that stops the recursion; here T(1) = 1.

The levels in the recursion tree until the base case is reached.

Smaller instances of the original problem at each level of the tree.

The amount of work performed at each level of recursion.

The sum of work done at all levels of the recursion tree.

A mathematical notation that describes the limiting behavior of a function when the argument tends towards infinity.

Determining the time and space complexity of the recursive function.

An equation that defines a sequence based on previous terms.

The function representing the time complexity for input size n.

Big O notation indicating linear time complexity.

A method of proving statements true using induction principle.

Assumption made for the inductive step that the statement holds for all k<n.

Specific values used to make time complexity claims tighter.

A more specific claim about the upper bound of T(n).

The sum of impacts of all nodes at a particular layer in the recursion tree.

T(n) = 3T(n/4) + n^2 describes a recurrence relation.

T(1) = 1 is the base case for the recursion.

Each layer of the recursion tree has n^2 impact.

Node impact increases exponentially in the recursion tree.

The depth determines how many times the problem is divided.

Total cost is the sum of all node impacts across layers.

log4 n reflects a logarithmic impact in the layer contributions.

n^2 describes polynomial growth affecting the cost calculation.