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Recursion Tree Method for T(n) Analysis

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Questions and Answers

What is the first step in applying the recursion tree method to the recurrence $T(n) = 2T(\frac{n}{2}) + 4n$?

Draw the recursion tree. (correct)

Estimate the height of the tree.

Identify the base case.

Calculate the total contributions from leaf nodes.

What is the contribution from the leaves at the bottom of the recursion tree for $T(n) = 2T(\frac{n}{2}) + 4n$?

$4n$

$4n^{\log_2 2}$

$4n^2$

$4n \log n$ (correct)

How does one compute the height of the recursion tree for the recurrence $T(n) = 2T(\frac{n}{2}) + 4n$?

$\log n$ (correct)

$n$

$\frac{n}{2}$

$2n$

What characterizes the expansion of the recursion tree for $T(n) = 2T(\frac{n}{2}) + 4n$?

<p>Each level doubles the number of recursive calls. (A)</p> Signup and view all the answers

What is the total time complexity for the recurrence $T(n) = 2T(\frac{n}{2}) + 4n$ after applying the recursion tree method?

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

Flashcards

Recurrence Relation

An equation that defines a function in terms of its own previous values.

T(n)

A function that represents the time complexity of an algorithm.

Divide-and-conquer

Algorithm design technique. Breaking a problem into smaller subproblems, recursively solving them, and combining the results.

2T(n/2)

Recursive part of the recurrence. It represents solving two subproblems of half the size.

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4n

Non-recursive part of the recurrence. Represents the time taken to divide the problem and combine the solutions.

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Study Notes

Example 1 - Recursion Tree Method

Solve the recurrence relation: T(n) = 2T(n/2) + 4n

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Description

This quiz covers the application of the recursion tree method for analyzing the recurrence relation T(n) = 2T(n/2) + 4n. It addresses key steps such as calculating contributions from leaves, determining the height of the recursion tree, and deriving the total time complexity. Perfect for students looking to understand recursive algorithms more comprehensively.