Question 1 True or False? For two functions f(n) and g(n), if f(n) = 0(g(n)), then g(n) 2(f(n)). O True O False
Question 1 True or False? For two functions f(n) and g(n), if f(n) = 0(g(n)), then g(n) 2(f(n)). O True O False
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Question 1**
True or False? For two functions \( f(n) \) and \( g(n) \), if \( f(n) = O(g(n)) \), then \( g(n) = \Omega(f(n)) \).
- ○ True
- ○ False
---
**Question 2**
True or False? For positive valued functions \( g \) and \( h \), if \( f(n) = O(g(n)) \) and \( g(n) = O(h(n)) \), then \( f(n) = O(h(n)) \).
- ○ True
- ○ False
---
**Question 3**
True or False? If \( g_1(n) = O(f(n)) \) and \( g_2(n) = O(f(n)) \), then \( g_1 + g_2 = O(f(n)) \).
- ○ True
- ○ False
![### Question 4
True or False? If \( g_1(n) = O(f_1(n)) \) and \( g_2 = O(f_2(n)) \), then \( g_1 \times g_2 = O(f_1(n) \times f_2(n)) \).
- [ ] True
- [ ] False
---
### Question 5
True or False? If \( f(n) = \log_a n \) and \( g(n) = \log_b n \), then \( f(n) = \Theta(g(n)) \).
- [ ] True
- [ ] False
---
### Question 6
Consider the following variation of the MergeSort algorithm, which we will call MergeSort-3Way. Instead of dividing the input array \( A[1 : n] \) into 2 halves like in MergeSort, in MergeSort-3Way, we divide \( A \) into 3 sub-arrays: \( A[1 : n/3] \), \( A[n/3 + 1 : 2n/3] \), and \( A[2n/3 + 1 : n] \). These sub-arrays are sorted recursively, and the sorted sub-arrays are merged into a sorted array—much like in the MERGE procedure described in class, but now it is a 3-way merge instead of a 2-way merge.
The running time of MergeSort-3Way is still \( \Theta(n \log_2 n) \). True or False?
- [ ] True
- [ ] False](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc83b18c8-597b-42a8-bb3a-a714328b9fe4%2F51d29675-02a7-43eb-800d-596176826ed0%2Fupjxown_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 4
True or False? If \( g_1(n) = O(f_1(n)) \) and \( g_2 = O(f_2(n)) \), then \( g_1 \times g_2 = O(f_1(n) \times f_2(n)) \).
- [ ] True
- [ ] False
---
### Question 5
True or False? If \( f(n) = \log_a n \) and \( g(n) = \log_b n \), then \( f(n) = \Theta(g(n)) \).
- [ ] True
- [ ] False
---
### Question 6
Consider the following variation of the MergeSort algorithm, which we will call MergeSort-3Way. Instead of dividing the input array \( A[1 : n] \) into 2 halves like in MergeSort, in MergeSort-3Way, we divide \( A \) into 3 sub-arrays: \( A[1 : n/3] \), \( A[n/3 + 1 : 2n/3] \), and \( A[2n/3 + 1 : n] \). These sub-arrays are sorted recursively, and the sorted sub-arrays are merged into a sorted array—much like in the MERGE procedure described in class, but now it is a 3-way merge instead of a 2-way merge.
The running time of MergeSort-3Way is still \( \Theta(n \log_2 n) \). True or False?
- [ ] True
- [ ] False
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