(a) Cônsider the following algorithm segment. for i := 1 to8 for j:= 1 to i [Statements in body of inner loop. None contain branching statements that lead outside the loop.] next j next i How many times will the inner loop be iterated when i = 2? How many times will the inner loop be iterated when i = 3? How many times will the inner loop be iterated when the algorithm is implemented and run? (b) Let n be a positive integer, and consider the following algorithm segment. for i := 1 to n for j := 1 to i [Statements in body of inner loop. None contain branching statements that lead outside the loop.] next j next i How many times will the inner loop be iterated when the algorithm is implemented and run?

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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### Algorithm Analysis Exercise

#### (a) Consider the following algorithm segment:

```
for i := 1 to 8
    for j := 1 to i
        [Statements in body of inner loop.
         None contain branching statements
         that lead outside the loop.]
    next j
next i
```

1. **How many times will the inner loop be iterated when \( i = 2 \)?**

   **Answer:**

2. **How many times will the inner loop be iterated when \( i = 3 \)?**

   **Answer:**

3. **How many times will the inner loop be iterated when the algorithm is implemented and run?**

   **Answer:**

#### (b) Let \( n \) be a positive integer, and consider the following algorithm segment:

```
for i := 1 to n
    for j := 1 to i
        [Statements in body of inner loop.
         None contain branching statements
         that lead outside the loop.]
    next j
next i
```

4. **How many times will the inner loop be iterated when the algorithm is implemented and run?**

   **Answer:**

---

**Explanation:**

In part (a), the inner loop runs for each value of \( i \) from 1 to 8, iterating \( i \) times. You calculate the total iterations by summing up the series from 1 to 8.

In part (b), the loop runs similarly but for a variable \( n \). The total number of iterations of the inner loop is the sum of the first \( n \) positive integers.
Transcribed Image Text:### Algorithm Analysis Exercise #### (a) Consider the following algorithm segment: ``` for i := 1 to 8 for j := 1 to i [Statements in body of inner loop. None contain branching statements that lead outside the loop.] next j next i ``` 1. **How many times will the inner loop be iterated when \( i = 2 \)?** **Answer:** 2. **How many times will the inner loop be iterated when \( i = 3 \)?** **Answer:** 3. **How many times will the inner loop be iterated when the algorithm is implemented and run?** **Answer:** #### (b) Let \( n \) be a positive integer, and consider the following algorithm segment: ``` for i := 1 to n for j := 1 to i [Statements in body of inner loop. None contain branching statements that lead outside the loop.] next j next i ``` 4. **How many times will the inner loop be iterated when the algorithm is implemented and run?** **Answer:** --- **Explanation:** In part (a), the inner loop runs for each value of \( i \) from 1 to 8, iterating \( i \) times. You calculate the total iterations by summing up the series from 1 to 8. In part (b), the loop runs similarly but for a variable \( n \). The total number of iterations of the inner loop is the sum of the first \( n \) positive integers.
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