Use the recursion tree method to solve the following recurrence T(n) by finding the tightest functionf(n) such that T(n) = O(f(n)).T(n) ≤ 4.T(n/3) +0(n³)

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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Use the recursion tree method to solve the following recurrence T(n) by finding the tightest function
f(n) such that T(n) = O(f(n)).
T(n) ≤ 4.T(n/3) +0(n³)

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Step 1

Answer:

To solve the recurrence relation T(n) = 4T(n / 3) + O(n ^ 3) using the recursion tree method, we can draw a tree in which each node represents the cost of a subproblem at a certain level of recursion. The root of the tree represents the original problem of size n, and each level of the tree represents a smaller subproblem of size n / 3.

At each level of the tree, we have four subproblems, each of size n / 3. Therefore, the total cost of each level of the tree is O((n / 3) ^ 3) = O(n ^ 3 / 27). We can represent the cost of each level by a geometric series:

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Can you try to draw out the tree to make it more clear.

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