### Pseudocode Function Analysis: BAR FunctionConsider the following pseudocode function BAR given below. (Note: the "end" statement simply signals the end of the relevant "for" loop.)```plaintext// BAR : ℕ → ℕFunction BAR(n): x ← 0 for k ∈ {1, 2, ..., n} do t ← 0 for i ∈ {1, 2, ..., k} do t ← t + i end x ← x + t end return x```#### Analysis:(a) A precondition for BAR is that \( n \in \mathbb{N} \). One postcondition for BAR is that \( x \in \mathbb{N} \), but this postcondition isn't very insightful. Provide a postcondition that accurately describes the value \( x \) in terms of the input \( n \).**Postcondition Analysis:**The outer loop runs from 1 to \( n \). For each \( k \), the inner loop calculates the sum of the first \( k \) natural numbers, which is given by the formula \( \frac{k(k+1)}{2} \). This sum is stored in \( t \) and then added to \( x \). Therefore, by the end of the function, \( x \) will be the sum of the series:\[ x = \sum_{k=1}^{n} \frac{k(k+1)}{2} \](b) Provide an exact value (in terms of \( n \)) for the total number of \( + \) operations performed by BAR.**Exact Value Calculation:**The number of \( + \) operations can be split into two parts:1. The \( + \) operations in the inner loop for calculating \( t \),2. The \( + \) operation for updating \( x \).For the inner loop:- When \( k = 1 \), there is 1 \( + \) operation.- When \( k = 2 \), there are 2 \( + \) operations.- When \( k = 3 \), there are 3 \( + \) operations.- ...- When \( k = n \), there are \( n \) \( + \) operations.Thus, the total for the inner loop is:\[ \sum_{k

Postcondition for the value of x in terms of input n: The value of x returned by the function BAR is…

Consider the following pseudocode function BAR given below. (Note: the "end" statement simply signals the end of the relevant "for" loop.) // BAR : N → N Function BAR(n): x 0 for k = {1, 2,...,n} do t←0 for i = {1, 2,..., k} do It+t+i end xx+t end return x (a) A precondition for BAR is that ne N. One postcondition for BAR is that x = N, but this postcondition isn't very insightful. Provide a postcondition that accurately describes the value x in terms of the input n. (b) Provide an exact value (in terms of n) for the total number of + operations performed by BAR. (c) Provide a Big-O estimate (in terms of n) for total the number of + operations performed by BAR.

Consider the following pseudocode function BAR given below. (Note: the "end" statement simply signals the end of the relevant "for" loop.) // BAR : N → N Function BAR(n): x 0 for k = {1, 2,...,n} do t←0 for i = {1, 2,..., k} do It+t+i end xx+t end return x (a) A precondition for BAR is that ne N. One postcondition for BAR is that x = N, but this postcondition isn't very insightful. Provide a postcondition that accurately describes the value x in terms of the input n. (b) Provide an exact value (in terms of n) for the total number of + operations performed by BAR. (c) Provide a Big-O estimate (in terms of n) for total the number of + operations performed by BAR.

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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Question

### Pseudocode Function Analysis: BAR Function

Consider the following pseudocode function BAR given below. (Note: the "end" statement simply signals the end of the relevant "for" loop.)

```plaintext
// BAR : ℕ → ℕ
Function BAR(n):
x ← 0
for k ∈ {1, 2, ..., n} do
t ← 0
for i ∈ {1, 2, ..., k} do
t ← t + i
end
x ← x + t
end
return x
```

#### Analysis:

(a) A precondition for BAR is that \( n \in \mathbb{N} \). One postcondition for BAR is that \( x \in \mathbb{N} \), but this postcondition isn't very insightful. Provide a postcondition that accurately describes the value \( x \) in terms of the input \( n \).

**Postcondition Analysis:**
The outer loop runs from 1 to \( n \). For each \( k \), the inner loop calculates the sum of the first \( k \) natural numbers, which is given by the formula \( \frac{k(k+1)}{2} \). This sum is stored in \( t \) and then added to \( x \). Therefore, by the end of the function, \( x \) will be the sum of the series:

\[ x = \sum_{k=1}^{n} \frac{k(k+1)}{2} \]

(b) Provide an exact value (in terms of \( n \)) for the total number of \( + \) operations performed by BAR.

**Exact Value Calculation:**
The number of \( + \) operations can be split into two parts:
1. The \( + \) operations in the inner loop for calculating \( t \),
2. The \( + \) operation for updating \( x \).

For the inner loop:
- When \( k = 1 \), there is 1 \( + \) operation.
- When \( k = 2 \), there are 2 \( + \) operations.
- When \( k = 3 \), there are 3 \( + \) operations.
- ...
- When \( k = n \), there are \( n \) \( + \) operations.

Thus, the total for the inner loop is:
\[ \sum_{k

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