if n< d T(n)= - laT(n/b)+ f(n) if n2 d 1. if f(n) is O(n" 2. if f(n) is O(n™*“ log* n), then T(n) is O(n*.“ log**" n)| 3. if f(n) is 2(n*. *“), then T(n) is O(f (n)), provided af(n/b) S F (n) for some & <1. log, a-e ), then T(n) is O(no") log, a+e Which of the three cases above applies to an algorithm, the running time of which is described using the following recurrence relation: T(n) = 3 T(n/3) + n^/a Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2" Type N/A if the Master theorem is not applicable to the recurrence relation Answer:

if n< d T(n)= - laT(n/b)+ f(n) if n2 d 1. if f(n) is O(n" 2. if f(n) is O(n™“ log n), then T(n) is O(n*.“ log**" n)| 3. if f(n) is 2(n. “), then T(n) is O(f (n)), provided af(n/b) S F (n) for some & <1. log, a-e ), then T(n) is O(no") log, a+e Which of the three cases above applies to an algorithm, the running time of which is described using the following recurrence relation: T(n) = 3 T(n/3) + n^/a Type 1 for case 1, and so on. For case 2, also specify a k value after a space, e.g., "2 2" Type N/A if the Master theorem is not applicable to the recurrence relation Answer:

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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T(n) = T(n - 1) + 3n and T(0)=0. Then T(n) is e (n³1 ө (п) e (n lg n) e (2")
Master theorem: if n< d T(n) = |aT(n/b)+ f (n) if n2 d 1. if f(n) is O(n - ), then T(n) is O(nos") 2. if f(n) is O(n' 3. if f(n) is 2(n® provided af (n/b)S (n) for some 8
T(n) = nT(n/2) +0(1) T(n) = 3T(n/3) +n T(n) = 8T(n/2) +n³
O(1) if n = 1, T (n) 8T (n/2) + O(n²) if n > 1 .
n 0, T(n - а) + 1 п>а. T(n} =
Reminder: f € O(g) if there exist integers C > 0 and k > 0 such that f(n) : Select all pairs (C, k) that prove f E O(g) if f(n) = 8n2 + 4n and g(n) = n' O k=2, C=10 O k=1, C=8 k=2, C=6 O k=4, C=9 k=1, C=12
Use & for •, v for V, > for Ɔ, and = for = What can you derive by applying De Morgan's (DM) to this formula: ~(N • E) Ɔ G (N v E) > G O (-N v -E) > G O (N & E) v G O (N & E) v G O (N v E) > G O (N & E) > G
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You are given the following definitions to help you: O(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ f(n) ≤ cg(n) for all n ≥ no} . N(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ cg (n) ≤ f(n) for all n ≥ no} . ©(g(n)) = {f(n): there exist positive constants C₁, C2, and no such that 0 ≤ c₁g(n) ≤ f(n) ≤ c₂g (n) for all n ≥ no} . Using above definitions, prove that: 5. T(n) = 6n + 4n+ 3 € 0 (n) 6. T(n) = 100n + 10000 € O(n²) 7. T(n) = 5n² - 2n + 16 € O(n³) 8. T(n)=5n² - 2n + 16 is not € O(n) 9. T(n) = n³ + 20n € №(n² ) 10. T(n) = 2n³ - 7n + 1 € ☺(n³)
Let A := {0* 1"|n > 0}. And B := {w\wis an even number}. Show that A
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Given the following NDFSM M, the correct statements are: q0 8 b a q1 94 a b a a q2 q5 a 93 L(M) = {w € {a, b}*: w contains aaa or bab or w| is even). OL(M) = {w € {a, b}*: w contains aaa or bab or w| is odd}. L(M) = {w € {a,b}*: The 2nd to the last character of w is a or|w| is even}. L(M) = {w = {a, b}*: The 2nd to the last character of w is a or |w| is odd}.