### Solving Recurrence Relations**Objective**: Obtain the exact solution to the recurrence for $ n = 2^k $ (restricted solution). Use this to find the time complexity of $ T(n) $ when $ n = 2^k $ (restricted complexity).#### Given Recurrence Relation\[ T(1) = \frac{1}{2} \]\[ T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + \frac{1}{n(n+1)} \]#### Conditions- The recurrence relation must be solved for $ n \geq 0 $.This problem involves solving a recurrence relation where the input size $ n $ reduces by half in each step, which is a common pattern in divide-and-conquer algorithms like merge sort or binary tree operations. The recurrence provides a base case for $ n = 1 $ and defines how $ T(n) $ is computed based on smaller subproblems.

Given: recurrence relation Tn = Tn2+1nn+1T1 = 12 ∵n≥0Tn = Tn2+1nn+1…

Answered: obtain the exact solution to the…

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

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### Solving Recurrence Relations

**Objective**: Obtain the exact solution to the recurrence for $ n = 2^k $ (restricted solution). Use this to find the time complexity of $ T(n) $ when $ n = 2^k $ (restricted complexity).

#### Given Recurrence Relation

\[ T(1) = \frac{1}{2} \]

\[ T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + \frac{1}{n(n+1)} \]

#### Conditions

- The recurrence relation must be solved for $ n \geq 0 $.

This problem involves solving a recurrence relation where the input size $ n $ reduces by half in each step, which is a common pattern in divide-and-conquer algorithms like merge sort or binary tree operations. The recurrence provides a base case for $ n = 1 $ and defines how $ T(n) $ is computed based on smaller subproblems.

Expert Solution

Step 1

Given:

recurrence relation $T (n) = T (\frac{n}{2}) + \frac{1}{n (n + 1)} T (1) = \frac{1}{2} ∵ n \geq 0 T (n) = T (\frac{n}{2}) + \frac{1}{n (n + 1)} (1)$

Now,

$T (\frac{n}{2}) = place n with \frac{n}{2}$

to equation (1).

$T (\frac{n}{2}) = T (\frac{\frac{n}{2}}{2}) + \frac{1}{\frac{n}{2} (\frac{n}{2} + 1)} T (\frac{n}{2}) = T (\frac{n}{4}) + \frac{1}{\frac{n}{2} (\frac{n + 2}{2})} (2)$

Now, replace $T (\frac{n}{2})$ the value in equation (1)

$T (n) = T (\frac{n}{4}) + \frac{1}{\frac{n}{2} (\frac{n + 2}{2})} + \frac{1}{n (n + 1)} T (n) = T (\frac{n}{2^{2}}) + \frac{1}{\frac{n}{2} (\frac{n + 2}{2})} + \frac{1}{n (n + 1)} (3)$

Now $T (\frac{n}{2^{2}}) = T (\frac{n}{2^{3}}) + \frac{1}{\frac{n}{2^{2}} (\frac{\frac{n}{2} + 1}{2})}$

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