Discrete Mathematics and Its Applications
8th Edition
ISBN: 9781260501759
Author: ROSEN
Publisher: MCG
expand_more
expand_more
format_list_bulleted
Question
Chapter 3, Problem 41SE
To determine
(a)
Devise a brute-force algorithm for solving the knapsack problem.
To determine
(b)
Solve the knapsack problem when the capacity of the knapsack is 18 kg and there are five items: a 5-kg sleeping bag, an 8-kg tent, a 7-kg food pack, a 4-kg container of water, and an 11-kg portable stove.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Suppose a company manufactures different electronic components for a computer. Component A requires 2hours of fabrication and 1hour of assembly; Component B requires 3hours of fabrication and 1hour of assembly and component C requires 2hours of fabrication and 2hours of assembly.
The company has up 1000 Labour hours for fabrication and 800 labour hours assembly each week. If proof on each component A,B, and C is K7,K8 and K10 respectively.How many of each should be produced to maximize?
In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important.
The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest
nanometer. The red range is 675-700 nm and the blue range is 450-500 nm. Let A denote the event that
PAR occurs in the red range, and let B denote the event that PAR occurs in the blue range.
Choose the correct answer for event A nB
a. Let w denote the wavelength. The sample space is {w | w = 0, 1, 2, .}
AnB = {w | w = 450, 451, ., 500, 675, 676, .., 700 nm}
O b. None among the choices
Oc. Let w denote the wavelength. The sample space is {w | w = 0, 1, 2, .}
ANB = {w | w = 450, 451, ., 500 nm}
d. Let w denote the wavelength. The sample space is {w | w = 0, 1, 2, .}
AnB = {w | w = 675, 676, ., 700 nm}
How many ways are there to pack ten identical moon cakes into three indistinguishable boxes so that each box contains at least two moon cakes? Show your calculations.Comment: for this problem, a solution based on a “brute-force” enumeration of the relevant configurations is acceptable.
Chapter 3 Solutions
Discrete Mathematics and Its Applications
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. 22ECh. 3.1 - Prob. 23ECh. 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. 34ECh. 3.1 - Prob. 35ECh. 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. 51ECh. 3.1 - Devise a variation of the insertion sort that uses...Ch. 3.1 - Prob. 53ECh. 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. 59ECh. 3.1 - Show that if there were a coin worth 12 cents, the...Ch. 3.1 - Prob. 61ECh. 3.1 - Prob. 62ECh. 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. 66ECh. 3.1 - Prob. 67ECh. 3.1 - Prob. 68ECh. 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. 71ECh. 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. 11ECh. 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. 19ECh. 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. 29ECh. 3.2 - Show that each of these pairs of functions are of...Ch. 3.2 - Prob. 31ECh. 3.2 - w thatf(x)andg(x)are functions from the set of...Ch. 3.2 - Prob. 33ECh. 3.2 - Show that3x2+x+1is(3x2)by directly finding the...Ch. 3.2 - Prob. 35ECh. 3.2 - lain what it means for a function to be(1).Ch. 3.2 - Prob. 37ECh. 3.2 - Give a big-Oestimate of the product of the...Ch. 3.2 - Prob. 39ECh. 3.2 - Prob. 40ECh. 3.2 - Prob. 41ECh. 3.2 - pose thatf(x)isO(g(x)). Does it follow...Ch. 3.2 - Prob. 43ECh. 3.2 - pose thatf(x),g(x), andh(x)are functions such...Ch. 3.2 - Prob. 45ECh. 3.2 - Prob. 46ECh. 3.2 - Prob. 47ECh. 3.2 - ress the relationshipf(x)is(g(x))using a picture....Ch. 3.2 - Prob. 49ECh. 3.2 - w that iff(x)=anxn+an1xn1++a1x+a0,...Ch. 3.2 - Prob. 51ECh. 3.2 - Prob. 52ECh. 3.2 - Prob. 53ECh. 3.2 - w thatx5y3+x4y4+x3y5is(x3y3).Ch. 3.2 - w thatxyisO(xy).Ch. 3.2 - w thatxyis(xy).Ch. 3.2 - Prob. 57ECh. 3.2 - Prob. 58ECh. 3.2 - Prob. 59ECh. 3.2 - Prob. 60ECh. 3.2 - Prob. 61ECh. 3.2 - (Requires calculus) Prove or disprove that (2n)!...Ch. 3.2 - Prob. 63ECh. 3.2 - Prob. 64ECh. 3.2 - Prob. 65ECh. 3.2 - Prob. 66ECh. 3.2 - Prob. 67ECh. 3.2 - Prob. 68ECh. 3.2 - Prob. 69ECh. 3.2 - Prob. 70ECh. 3.2 - Prob. 71ECh. 3.2 - Prob. 72ECh. 3.2 - Show thatnlognisO(logn!).Ch. 3.2 - Prob. 74ECh. 3.2 - Prob. 75ECh. 3.2 - Prob. 76ECh. 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. 5ECh. 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. 8ECh. 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. 19ECh. 3.3 - What is the effect in the time required to solve a...Ch. 3.3 - Prob. 21ECh. 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. 26ECh. 3.3 - Prob. 27ECh. 3.3 - Prob. 28ECh. 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. 32ECh. 3.3 - Prob. 33ECh. 3.3 - Prob. 34ECh. 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. 38ECh. 3.3 - Prob. 39ECh. 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. 43ECh. 3.3 - Prob. 44ECh. 3.3 - Prob. 45ECh. 3.3 - Prob. 46ECh. 3.3 - Prob. 47ECh. 3.3 - Prob. 48ECh. 3.3 - Prob. 49ECh. 3 - Define the termalgorithm. What are the different...Ch. 3 - Describe, using English, an algorithm for finding...Ch. 3 - Prob. 3RQCh. 3 - Prob. 4RQCh. 3 - Prob. 5RQCh. 3 - Define what the worst-case time complexity,...Ch. 3 - Prob. 7RQCh. 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. 11RQCh. 3 - Describe an algorithm for locating the last...Ch. 3 - Prob. 2SECh. 3 - Give an algorithm to determine whether a bit...Ch. 3 - Suppose that a list contains integers that are in...Ch. 3 - Prob. 5SECh. 3 - Prob. 6SECh. 3 - Prob. 7SECh. 3 - Prob. 8SECh. 3 - Prob. 9SECh. 3 - Prob. 10SECh. 3 - Show the steps used by the shaker sort to sort the...Ch. 3 - Express the shaker sort in pseudocode.Ch. 3 - Prob. 13SECh. 3 - Prob. 14SECh. 3 - Prob. 15SECh. 3 - w that8x3+12x+100logxisO(x3).Ch. 3 - Prob. 17SECh. 3 - Prob. 18SECh. 3 - Prob. 19SECh. 3 - w thatnnis notO(n!).Ch. 3 - Prob. 21SECh. 3 - Prob. 22SECh. 3 - Prob. 23SECh. 3 - Prob. 24SECh. 3 - Arrange the...Ch. 3 - Prob. 26SECh. 3 - Prob. 27SECh. 3 - Show that if the denominations of coins arec0,c1,...Ch. 3 - Prob. 29SECh. 3 - Prob. 30SECh. 3 - Prob. 31SECh. 3 - Show that the deferred acceptance algorithm given...Ch. 3 - Prob. 33SECh. 3 - Show that when woman do the proposing in the...Ch. 3 - Prob. 35SECh. 3 - Prob. 36SECh. 3 - Prob. 37SECh. 3 - Prob. 38SECh. 3 - Prob. 39SECh. 3 - Prob. 40SECh. 3 - Prob. 41SECh. 3 - Exercises 4246 we will study the problem of load...Ch. 3 - Prob. 43SECh. 3 - Prob. 44SECh. 3 - Prob. 45SECh. 3 - Prove that the algorithm from Exercise 44 is a...Ch. 3 - Prob. 1CPCh. 3 - Prob. 2CPCh. 3 - Prob. 3CPCh. 3 - Prob. 4CPCh. 3 - Prob. 5CPCh. 3 - Prob. 6CPCh. 3 - Prob. 7CPCh. 3 - Given an integern, use the cashier’s algorithm to...Ch. 3 - Prob. 9CPCh. 3 - Prob. 10CPCh. 3 - Prob. 11CPCh. 3 - Prob. 1CAECh. 3 - Prob. 2CAECh. 3 - Using a generator of random orderings of the...Ch. 3 - Prob. 4CAECh. 3 - Write a program that animates the progress of all...Ch. 3 - Examine the history of the wordalgorithmand...Ch. 3 - Prob. 2WPCh. 3 - Explain how sorting algorithms can be classified...Ch. 3 - Prob. 4WPCh. 3 - Prob. 5WPCh. 3 - Prob. 6WPCh. 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. 10WPCh. 3 - Prob. 11WPCh. 3 - Describe six different NP-complete problems.Ch. 3 - Prob. 13WP
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Consider the following mathematical model:Max z = X1 – X2 + 3X3S.TX1 + X2 <= 20X1 + X3 = 5X2 + X3 >= 10X1, X2, X3 >= 0a) Solve the above mathematical model using Simplex Algorithm. Note that you need to show all the steps and calculations in order to receive a good mark.b) Solve the above mathematical model using LINDO software and print out the first page (mathematical) and the second page of the solutions.c) Compare the solutions from part (a) with part (b). What is your conclusion?pls type solutionarrow_forwardConsider a region of the memory address space that is divided into 5 sequential portions. Assume that five files of various sizes are randomly inserted into this region (one file in one section). How likely is it that files will be written in the parts in order of their sizes? (the largest in the first section, the second largest in the second section, and so on).arrow_forwardThe Infinite Monkey Theorem states that a monkey hitting keys completely at random on a typewriter for an infinite amount of time will, eventually, type any given text. Even the complete works of William Shakespeare. Some quotes are truly amazing. Consider the famous quote by Joey Tribbiani, "HOW YOU DOIN?" That's 13 amazing characters. Suppose a monkey is seated in front of a KEYBOARD WITH 28 CHARACTERS (one for each of the 26 letters in the alphabet, plus the space bar, and a comma). Suppose the monkey types only 13 characters completely at random. If X is the number of different 13 character strings it could have typed, then the chances that it typed the quote is one divided by X. What is X? Group of answer choices 8.29351e+15 2.48115e+18 6.50211e+18 1.55029e+31arrow_forward
- Help me fast so that I will give Upvote.arrow_forwardThere is a central fact table and 4 dimensions – Time, A, B, and C. The ratio m: n between 2 attributes of the hierarchy represents that for every ‘m’ occurrence of a parent element there will be a maximum of ‘n’ children. As per the current state, this fac5*t table contains 3 Years 4 A1s 5 B1s 5 C1s Q1: What is the maximum possible fact data volume in terms of record/row count? Q3: The m:n ratio from attribute B3 to attribute B4 is changed from 1:3 to 1:1 due to some organizational changes. By how many times the maximum possible fact data volume (by record count) decreases?arrow_forwardNewyork University Building 8 has a 15-story structure. Mr. Raju works on the 11th floor of this building. Except for stopping on a floor where the button has been pressed, the single elevator travels endlessly between floors 1,2,3,..., 14,15,14,...,3,2,1. Assume that the amount of time required to load and unload passengers is minimal compared to the actual trip time.Mr. Raju always wonders when he is about to leave his office at 6 PM, most of the time, the lift goes up before stopping at his floor. One day, Raju encounters you in the lift and shared his thoughts with you. Since you are taking Discrete Mathematics course, he expects you to solve the mystery for him. Provide a valid explanation in terms of probability to Mr. Raju.arrow_forward
- The Königsberg bridge problem asks if the seven bridges of the city of Königsberg (picture attached), over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. Trace over each bridge ONLY ONCE WITH ONE CONTINUOUS STROKE. What is the area of mathematics used to solve the problem?arrow_forwardConsider the following graph which shows the distances (in miles) between towns Abbyville (A), Barryton (B), Charlotte (C), and Derry (D). Use the Brute Force method to find the length of the shortest route from Abbyville to all the other towns and back to Abbyville. PLEASE HELParrow_forwardThe Infinite Monkey Theorem states that a monkey hitting keys completely at random on a typewriter for an infinite amount of time will, eventually, type any given text. Even the complete works of William Shakespeare. Some quotes are truly amazing. Consider the famous quote, "NOT TODAY" --Arya Stark That's 9 amazing characters. Suppose a monkey is seated in front of a KEYBOARD WITH 28 CHARACTERS (one for each of the 26 letters in the alphabet, plus the space bar, and a comma). Suppose the monkey types only 9 characters completely at random. If X is the number of different 9 character strings it could have typed, then the chances that it typed the quote is one divided by X. What is X?arrow_forward
- Five workers are available to perform four jobs. The time it takes each worker to perform each job is given in the following table. The goal is to assign workers to jobs so as to minimize the total time required to perform the five jobs: Time Worker Job 1 Job 2 Job 3 Job 4 Job 5 1 10 15 10 15 16 2 12 8 20 16 14 3 12 9 12 18 22 4 6 12 15 18 8 5 16 12 8 12 8 Which worker should be assigned to which job? What is the total time required? (Hint: Use the Assignment model). If there are only 4 jobs (assume that job 5 is not required), how would the workers be assigned to jobs? (Hint: Change the constraints in the Assignment model).arrow_forwardA company produces three types of bicycles, i.e., mountain bike, road bike, and touring bike. For each mountain bike (x) requires 8 worker hours to assemble the frame, 3 worker hours to assemble the components, and 2 worker hours to test and complete the bicycle. For each road bike (y) requires 4 worker hours to assemble the frame, 9 worker hours to assemble the components, and 1 worker hour to test and complete the bicycle. And for each touring bike (z) requires 2 worker hours to assemble the frame, 3 worker hours to assemble the components, and 1 worker hour to test and complete the bicycle. On a daily basis, the company has 296 worker hours for assembling the frame, 456 worker hours for assembling the components, and 84 worker hours for testing and completing the bicycle. Calculate how many bicycles of each type can be produced such that all work power used is equivalent by using Cramer's rule.arrow_forwardHere's a problem that was used as the basis of a television show to illustrate the difference between the way a human mind approaches such a problem and the brute-force approach of a computer that finds the solution by trying all possible 40,320 different arrangements of the digits. Place the digits from 1 though 8 in the eight circles shown in the diagram, but with this restriction: no two digits next to each other in serial order may go in circles that are connected by a direct line. (For example: if 2 is placed in the top circle, neither 1 nor 3 may be placed in any of the three circles in the horizontal row beneath it because each of these circles is connected to the top circle by a direct line.) There is only one solution.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Solve ANY Optimization Problem in 5 Steps w/ Examples. What are they and How do you solve them?; Author: Ace Tutors;https://www.youtube.com/watch?v=BfOSKc_sncg;License: Standard YouTube License, CC-BY
Types of solution in LPP|Basic|Multiple solution|Unbounded|Infeasible|GTU|Special case of LP problem; Author: Mechanical Engineering Management;https://www.youtube.com/watch?v=F-D2WICq8Sk;License: Standard YouTube License, CC-BY
Optimization Problems in Calculus; Author: Professor Dave Explains;https://www.youtube.com/watch?v=q1U6AmIa_uQ;License: Standard YouTube License, CC-BY
Introduction to Optimization; Author: Math with Dr. Claire;https://www.youtube.com/watch?v=YLzgYm2tN8E;License: Standard YouTube License, CC-BY