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Math Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

To prove: f ( x ) = O ( log a x ) ⇒ f ( x ) = O ( log b x ) (1) For real a , b b o t h > 1 , (2)

To prove: f ( x ) = O ( log a x ) ⇒ f ( x ) = O ( log b x ) (1) For real a , b b o t h > 1 , (2)

Solution Summary: The author explains how to prove that f(x)=O(mathrmlog_abx=

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 3.2, Problem 40E

To determine

To prove:

f(x)=O(logax)⇒f(x)=O(logbx) (1)

For real a, b both >1, (2)

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Chapter 3.2, Problem 39E

Chapter 3.2, Problem 41E

Chapter 3 Solutions

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

Ch. 3.1 - List all the steps used by Algorithm 1 to find the...Ch. 3.1 - Determine which characteristics of an algorithm...Ch. 3.1 - Devise an algorithm that finds the sum of all the...Ch. 3.1 - Describe an algorithm that takes as input a list...Ch. 3.1 - Describe an algorithm that takes as input a list...Ch. 3.1 - Describe an algorithm that takes as input a list...Ch. 3.1 - Describe an algorithm that takes as input a list...Ch. 3.1 - Describe an algorithm that takes as input a list...Ch. 3.1 - Apalindromeis a string that reads the same forward...Ch. 3.1 - Devise an algorithm to computexn, wherexis a real...

Ch. 3.1 - Describe an algorithm that interchanges the values...Ch. 3.1 - cribe an algorithm that uses only assignment...Ch. 3.1 - List all the steps used to search for 9 in the...Ch. 3.1 - List all the steps used to search for 7 in the...Ch. 3.1 - cribe an algorithm that inserts an integerxin the...Ch. 3.1 - Describe an algorithm for finding the smallest...Ch. 3.1 - Describe an algorithm that locates the first...Ch. 3.1 - Describe an algorithm that locates the last...Ch. 3.1 - Describe an algorithm that produces the maximum,...Ch. 3.1 - Describe an algorithm for finding both the largest...Ch. 3.1 - Describe an algorithm that puts the first three...Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Describe an algorithm that determines whether a...Ch. 3.1 - Describe an algorithm that will count the number...Ch. 3.1 - nge Algorithm 3 so that the binary search...Ch. 3.1 - Theternary search algorithmlocates an element in a...Ch. 3.1 - Specify the steps of an algorithm that locates an...Ch. 3.1 - Devise an algorithm that finds a mode in a list of...Ch. 3.1 - Devise an algorithm that finds all modes. (Recall...Ch. 3.1 - Two strings areanagramsif each can be formed from...Ch. 3.1 - ennreal numbersx1,x2,...,xn , find the two that...Ch. 3.1 - Devise an algorithm that finds the first term of a...Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Use the bubble sort to sort 6, 2, 3, 1, 5, 4,...Ch. 3.1 - Use the bubble sort to sort 3, 1, 5, 7, 4, showing...Ch. 3.1 - Use the bubble sort to sortd,f,k,m,a,b, showing...Ch. 3.1 - Adapt the bubble sort algorithm so that it stops...Ch. 3.1 - Use the insertion sort to sort the list in...Ch. 3.1 - Use the insertion sort to sort the list in...Ch. 3.1 - Use the insertion sort to sort the list in...Ch. 3.1 - Sort these lists using the selection sort....Ch. 3.1 - Write the selection sort algorithm in pseudocode.Ch. 3.1 - Describe an algorithm based on the linear search...Ch. 3.1 - Describe an algorithm based on the binary search...Ch. 3.1 - How many comparisons does the insertion sort use...Ch. 3.1 - How many comparisons does the insertion sort use...Ch. 3.1 - Show all the steps used by the binary insertion...Ch. 3.1 - Compare the number of comparisons used by the...Ch. 3.1 - Prob. 51E Ch. 3.1 - Devise a variation of the insertion sort that uses...Ch. 3.1 - Prob. 53E Ch. 3.1 - List all the steps the naive string matcher uses...Ch. 3.1 - List all the steps the naive string matcher uses...Ch. 3.1 - Use the cashier’s algorithm to make change using...Ch. 3.1 - Use the cashier’s algorithm to make change using...Ch. 3.1 - Use the cashier’s algorithm to make change using...Ch. 3.1 - Prob. 59E Ch. 3.1 - Show that if there were a coin worth 12 cents, the...Ch. 3.1 - Prob. 61E Ch. 3.1 - Prob. 62E Ch. 3.1 - Devise a greedy algorithm that determines the...Ch. 3.1 - Suppose we have three menm1,m2, andm3and three...Ch. 3.1 - Write the deferred acceptance algorithm in...Ch. 3.1 - Prob. 66E Ch. 3.1 - Prob. 67E Ch. 3.1 - Prob. 68E Ch. 3.1 - Prove that the Boyer-Moore majority vote algorithm...Ch. 3.1 - Show that the problem of determining whether a...Ch. 3.1 - Prob. 71E Ch. 3.1 - Show that the problem of deciding whether a...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Prob. 11E Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - Exercises 1—14, to establish a big-Orelationship,...Ch. 3.2 - ermine whetherx3isO(g(x))for each of these...Ch. 3.2 - Explain what it means for a function to be 0(1)Ch. 3.2 - w that iff(x)isO(x)thenf(x)isO(x2).Ch. 3.2 - Suppose thatf(x),g(x), andh(x)are functions such...Ch. 3.2 - kbe a positive integer. Show...Ch. 3.2 - Prob. 19E Ch. 3.2 - To simplify:(3a5)3 27a15 Given information:(3a5)3....Ch. 3.2 - ange the functionsn, 1000 logn,nlogn,2n!,2n,3n,...Ch. 3.2 - Arrange the...Ch. 3.2 - Suppose that you have two different algorithms for...Ch. 3.2 - Suppose that you have two different algorithms for...Ch. 3.2 - Give as good a big-Oestimate as possible for each...Ch. 3.2 - e a big-Oestimate for each of these functions. For...Ch. 3.2 - Give a big-Oestimate for each of these functions....Ch. 3.2 - each function in Exercise 1, determine whether...Ch. 3.2 - Prob. 29E Ch. 3.2 - Show that each of these pairs of functions are of...Ch. 3.2 - Prob. 31E Ch. 3.2 - w thatf(x)andg(x)are functions from the set of...Ch. 3.2 - Prob. 33E Ch. 3.2 - Show that3x2+x+1is(3x2)by directly finding the...Ch. 3.2 - Prob. 35E Ch. 3.2 - lain what it means for a function to be(1).Ch. 3.2 - Prob. 37E Ch. 3.2 - Give a big-Oestimate of the product of the...Ch. 3.2 - Prob. 39E Ch. 3.2 - Prob. 40E Ch. 3.2 - Prob. 41E Ch. 3.2 - pose thatf(x)isO(g(x)). Does it follow...Ch. 3.2 - Prob. 43E Ch. 3.2 - pose thatf(x),g(x), andh(x)are functions such...Ch. 3.2 - Prob. 45E Ch. 3.2 - Prob. 46E Ch. 3.2 - Prob. 47E Ch. 3.2 - ress the relationshipf(x)is(g(x))using a picture....Ch. 3.2 - Prob. 49E Ch. 3.2 - w that iff(x)=anxn+an1xn1++a1x+a0,...Ch. 3.2 - Prob. 51E Ch. 3.2 - Prob. 52E Ch. 3.2 - Prob. 53E Ch. 3.2 - w thatx5y3+x4y4+x3y5is(x3y3).Ch. 3.2 - w thatxyisO(xy).Ch. 3.2 - w thatxyis(xy).Ch. 3.2 - Prob. 57E Ch. 3.2 - Prob. 58E Ch. 3.2 - Prob. 59E Ch. 3.2 - Prob. 60E Ch. 3.2 - Prob. 61E Ch. 3.2 - (Requires calculus) Prove or disprove that (2n)!...Ch. 3.2 - Prob. 63E Ch. 3.2 - Prob. 64E Ch. 3.2 - Prob. 65E Ch. 3.2 - Prob. 66E Ch. 3.2 - Prob. 67E Ch. 3.2 - Prob. 68E Ch. 3.2 - Prob. 69E Ch. 3.2 - Prob. 70E Ch. 3.2 - Prob. 71E Ch. 3.2 - Prob. 72E Ch. 3.2 - Show thatnlognisO(logn!).Ch. 3.2 - Prob. 74E Ch. 3.2 - Prob. 75E Ch. 3.2 - Prob. 76E Ch. 3.2 - (Requires calculus) For each of these pairs of...Ch. 3.3 - Give a big-Oestimate for the number of operations...Ch. 3.3 - Give a big-Oestimate for the number additions used...Ch. 3.3 - Give a big-Oestimate for the number of operations,...Ch. 3.3 - Give a big-Oestimate for the number of operations,...Ch. 3.3 - Prob. 5E Ch. 3.3 - Use pseudocode to describe the algorithm that puts...Ch. 3.3 - Suppose that an element is known to be among the...Ch. 3.3 - Prob. 8E Ch. 3.3 - Give a big-Oestimate for the number of comparisons...Ch. 3.3 - Show that this algorithm determines the number of...Ch. 3.3 - pose we havensubsetsS1,S2, ...,Snof the set {1, 2,...Ch. 3.3 - Consider the following algorithm, which takes as...Ch. 3.3 - The conventional algorithm for evaluating a...Ch. 3.3 - re is a more efficient algorithm (in terms of the...Ch. 3.3 - t is the largestnfor which one can solve within...Ch. 3.3 - What is the largestnfor which one can solve within...Ch. 3.3 - What is the largestnfor which one can solve within...Ch. 3.3 - How much time does an algorithm take to solve a...Ch. 3.3 - Prob. 19E Ch. 3.3 - What is the effect in the time required to solve a...Ch. 3.3 - Prob. 21E Ch. 3.3 - Determine the least number of comparisons, or...Ch. 3.3 - Analyze the average-case performance of the linear...Ch. 3.3 - An algorithm is calledoptimalfor the solution of a...Ch. 3.3 - Describe the worst-case time complexity, measured...Ch. 3.3 - Prob. 26E Ch. 3.3 - Prob. 27E Ch. 3.3 - Prob. 28E Ch. 3.3 - Analyze the worst-case time complexity of the...Ch. 3.3 - Analyze the worst-case time complexity of the...Ch. 3.3 - Analyze the worst-case time complexity of the...Ch. 3.3 - Prob. 32E Ch. 3.3 - Prob. 33E Ch. 3.3 - Prob. 34E Ch. 3.3 - Determine a big-O estimate for the worst-case...Ch. 3.3 - Determine the number of character comparisons used...Ch. 3.3 - Determine a big-Oestimate of the number of...Ch. 3.3 - Prob. 38E Ch. 3.3 - Prob. 39E Ch. 3.3 - Show that the greedy algorithm for making change...Ch. 3.3 - rcises 41 and 42 deal with the problem of...Ch. 3.3 - rcises 41 and 42 deal with the problem of...Ch. 3.3 - Prob. 43E Ch. 3.3 - Prob. 44E Ch. 3.3 - Prob. 45E Ch. 3.3 - Prob. 46E Ch. 3.3 - Prob. 47E Ch. 3.3 - Prob. 48E Ch. 3.3 - Prob. 49E Ch. 3 - Define the termalgorithm. What are the different...Ch. 3 - Describe, using English, an algorithm for finding...Ch. 3 - Prob. 3RQ Ch. 3 - Prob. 4RQ Ch. 3 - Prob. 5RQ Ch. 3 - Define what the worst-case time complexity,...Ch. 3 - Prob. 7RQ Ch. 3 - Describe the bubble sort algorithm. Use the bubble...Ch. 3 - Describe the insertion sort algorithm. Use the...Ch. 3 - Explain the concept of a greedy algorithm. Provide...Ch. 3 - Prob. 11RQ Ch. 3 - Describe an algorithm for locating the last...Ch. 3 - Prob. 2SE Ch. 3 - Give an algorithm to determine whether a bit...Ch. 3 - Suppose that a list contains integers that are in...Ch. 3 - Prob. 5SE Ch. 3 - Prob. 6SE Ch. 3 - Prob. 7SE Ch. 3 - Prob. 8SE Ch. 3 - Prob. 9SE Ch. 3 - Prob. 10SE Ch. 3 - Show the steps used by the shaker sort to sort the...Ch. 3 - Express the shaker sort in pseudocode.Ch. 3 - Prob. 13SE Ch. 3 - Prob. 14SE Ch. 3 - Prob. 15SE Ch. 3 - w that8x3+12x+100logxisO(x3).Ch. 3 - Prob. 17SE Ch. 3 - Prob. 18SE Ch. 3 - Prob. 19SE Ch. 3 - w thatnnis notO(n!).Ch. 3 - Prob. 21SE Ch. 3 - Prob. 22SE Ch. 3 - Prob. 23SE Ch. 3 - Prob. 24SE Ch. 3 - Arrange the...Ch. 3 - Prob. 26SE Ch. 3 - Prob. 27SE Ch. 3 - Show that if the denominations of coins arec0,c1,...Ch. 3 - Prob. 29SE Ch. 3 - Prob. 30SE Ch. 3 - Prob. 31SE Ch. 3 - Show that the deferred acceptance algorithm given...Ch. 3 - Prob. 33SE Ch. 3 - Show that when woman do the proposing in the...Ch. 3 - Prob. 35SE Ch. 3 - Prob. 36SE Ch. 3 - Prob. 37SE Ch. 3 - Prob. 38SE Ch. 3 - Prob. 39SE Ch. 3 - Prob. 40SE Ch. 3 - Prob. 41SE Ch. 3 - Exercises 4246 we will study the problem of load...Ch. 3 - Prob. 43SE Ch. 3 - Prob. 44SE Ch. 3 - Prob. 45SE Ch. 3 - Prove that the algorithm from Exercise 44 is a...Ch. 3 - Prob. 1CP Ch. 3 - Prob. 2CP Ch. 3 - Prob. 3CP Ch. 3 - Prob. 4CP Ch. 3 - Prob. 5CP Ch. 3 - Prob. 6CP Ch. 3 - Prob. 7CP Ch. 3 - Given an integern, use the cashier’s algorithm to...Ch. 3 - Prob. 9CP Ch. 3 - Prob. 10CP Ch. 3 - Prob. 11CP Ch. 3 - Prob. 1CAE Ch. 3 - Prob. 2CAE Ch. 3 - Using a generator of random orderings of the...Ch. 3 - Prob. 4CAE Ch. 3 - Write a program that animates the progress of all...Ch. 3 - Examine the history of the wordalgorithmand...Ch. 3 - Prob. 2WP Ch. 3 - Explain how sorting algorithms can be classified...Ch. 3 - Prob. 4WP Ch. 3 - Prob. 5WP Ch. 3 - Prob. 6WP Ch. 3 - Describe the historic trends in how quickly...Ch. 3 - Develop a detailed list of algorithmic paradigms...Ch. 3 - Explain what the Turing Award is and describe the...Ch. 3 - Prob. 10WP Ch. 3 - Prob. 11WP Ch. 3 - Describe six different NP-complete problems.Ch. 3 - Prob. 13WP

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