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Assignment 5CSCI 3110 Design and Analysis of Algorithms.docx
- 1. Assignment 5 CSCI 3110:Design and Analysis of Algorithms Due June 16, 2014 Banner ID: Name: Banner ID: Name: Banner ID: Name: Assignments are due on the due date before class and have to include this cover page. Plagiarism in assignment answers will not be tolerated. By submitting their answers to this assignment, the authors named above declare that its content is their original work and that they did not use any sources for its preparation other than the class notes, the textbook, and ones explicitly acknowledged in the answers. Any suspected act of plagiarism will be reported to the Faculty’s Academic Integrity Officer and possibly to the Senate Discipline Committee. The penalty for academic dishonesty may range from failing the course to expulsion from the university, in accordance with Dalhousie University’s regulations regarding academic integrity. This assignment studies a problem with applications in parallel scheduling. You are given a set of jobs J1, J2, . . . , Jn you want to run on your super powerful parallel computer with k processors P1, P2, . . . , Pk. For each job Ji , you are given a prediction ti of the amount of time it takes to run this job on a
- 2. single processor. Ajob cannot be split, that is, once assigned to a given processor it needs to be run to completion by this processor. Let S j be the set of jobs you choose to run on processor Pj . Then processor Pj finishes its work in Tj = ∑ Ji∈ S j ti time. A schedule is a partition of the set S = {J1, J2, . . . , Jn} into subsets S1, S2, . . . , Sk of jobs assigned to processors P1, P2, . . . , Pk (that is, ⋃k j=1 S j = S and, for all j 6= j′, S j ∩ S j′ = ;). The makespan of this schedule is max{T1, T2, . . . , Tk}, that is, the time at which the last processor finishes its assigned work. In order to best utilize the computing power of your parallel computer, you try to find a schedule that minimizes the makespan. This problem turns out to be NP-hard to solve, even with only two processors. For such problems, we need to look for approximate solutions or for special types of inputs for which the problem can be solved exactly in polynomial time. This is the goal of this assignment. Question 1 (10 marks) Describe a simple greedy algorithm that computes a 2-approximation of the optimal schedule. In other words, if T∗ is the smallest possible makespan achievable for the given set of jobs using k processors, then the schedule produced by your algorithm should have a makespan of at most 2T∗ . Your algorithm should run in O(kn) time. Prove that the schedule produced by your algorithm has makespan at most 2T∗ . To do so, you will probably want to use two properties the minimum makespan T∗ must obviously
- 3. satisfy: • T∗ ≥1 k ∑k i=1 ti . (The processor doing the most amount of work must do at least a 1/k fraction of the total amount of work.) • T∗ ≥ max{t1, t2, . . . , tn}. (The longest job needs to run on some processor Pj , so this processor finishes no earlier than the time it takes to complete this job.) You should then prove that the schedule produced by your algorithm has makespan at most 1 k ∑k i=1 ti+max{t1, t2, . . . , tn}. Question 2 (10 marks) Describe a modification of the above greedy algorithm that computes an optimal schedule, that is, one with the smallest possible makespan, provided the given set of jobs satisfies the following condition: Let t1, t2, . . . , tn once again be the predicted amounts of time it takes to run jobs J1, J2, . . . , Jn, and let t ∗ = min{t1, t2, . . . , tn}. You may assume that t∗ = 1. Then, for all 1 ≤ i ≤ n, ti is a power of 2. Your algorithm should run in O(n lg n + kn) time. Prove that it produces an optimal schedule. 1
- 4. Copy of LitwareSales.xlsx Sheet1Litware, Inc.Semiannual Sales ReportSales Representative January Bush, Patricia11850$12,500$14,560$12,215$18,504$16,895Chisholm, Martin1285012,50015,69811,02517,56811,875Cook, Cathan, 210082090021155115222512320357Tippett, John956425800119569870698513789Turner, Olinda109542075421563102541256825845Poe, Deborah118501195615685165212587412456Singh, Paul14856128991552414756189548796Smith-Bates, Lorrin250112387123547205682158723879$ 117,943$ 141,180$ 139,688$ 106,731$ 147,163$ 133,892 Sheet2 Copy of SFA Enrollment Update.xlsx Sheet1School of Fine ArtFine Arts DepartmentEnrollmentCall No.CourseFallFA 100Acrylics145FA104Basic Drawing231FA 107Bookbinding & the Artist's Book87FA 201Ceramics232FA204Figure Drawing187FA205Illustration Techniques243FA206Introduction to Design and Color221FA207Mixed-Media Collage121FA208Oil Painting98FA321Portfolio Development158FA262Water Color206 Sheet2School of Fine ArtsMedia Studies DepartmentEnrollmentCall No.CourseFallMS 100Adobe Illustrator322MS104Adobe Photoshop Illustration233MS 107Digital Photography156MS 201Flash446MS204Graphic Fundamentals126MS205Job Advantage778MS206Photography288MS207Photoshop: Basics & Beyond987MS208Photoshop for Photographers332MS321Using Adobe In Design267MS262Web Design554 Sheet3
- 5. Copy of SFAEnrollment.xlsx Sheet1Fall Quarter EnrollmentFine Arts DepartmentFall Quarter EnrollmentFA100Acrylics145FA104Basic Drawing231FA107Bookbinding & the Artist's Book87FA201Ceramics232FA204Figure Drawing187FA205Illustration Techniques243FA206Introduction to Design and Color221FA207Mixed-Media Collage121FA208Oil Painting98FA321Portfolio Development158FA262Water Color206 Sheet2Fall Quarter EnrollmentMedia Studies DepartmentMS 100Adobe Illustrator322MS104Adobe Photoshop Illustration233MS 107Digital Photography156MS 201Flash446MS204Graphic Fundamentals126MS205Job Advantage778MS206Photography288MS207Photoshop: Basics & Beyond987MS208Photoshop for Photographers332MS321Using Adobe In Design267MS262Web Design554 Sheet3 Copy of Top Ten.xlsx Sheet1Fortune Magazine*Top Ten Companies to Work ForRankCompanyU. S. Employees% Minorities% Women1Genentech181242502Wegmans3189015543Valero Energy1658240404Griffin Hospital104910785W. L. Gore453715406Container Store285729627Vision Service Plan196830688J. M. Smucker293024449REI7443124010S. C. Johnson and Son34041538*Data based on 2006 Rankings Project 1-6, &F Sheet2 Sheet3 Copy of Tailspin Toys.xlsx Sheet1Tailspin ToysSecond Quarter Earnings April May June TOTAL Sales $61,254$69,302$68,521 Cost of Goods Sold
- 6. $38,895$37,568$39,521 Gross MarginMarketing Expense $2,154$2,852$2,822 Administrative Expense $4,456$4,124$4,589Miscellaneous Expense $1,059$1,205$1,502 Total Expenses Net Income Before Taxes Federal Taxes $4,995$8,008$6,830 Net Income After Taxes Sheet2 Sheet3 Copy of Blue Yonder.xlsx Sales HistoryBlue Yonder AirlinesAnnual Sales ( in thousands)FY 2005FY 2006FY 2007FY 2008TotalCorporate Contracts$1,039$1,253$1,427$1,454Sky Diving $76$79$88$89Charter Flights$123$130$133$138Flight School$205$208$207$199Total Sales$1,443$1,670$1,855$1,880Percentage IncreaseFY 2006FY 2007FY 20082005-2008Corporate ContractsSky Diving Charter FlightsFlight School Copy of BY Financials.xlsx Annual ExpensesBlue Yonder AirlinesExpenses (in thousands)Expenses (in thousands)FY 2005FY 2006FY 2007FY 2008TotalCorporate Contracts$723$857$1,059$1,061$3,700Sky Diving $59$62$72$74$267Charter Flights$101$106$109$111$427Flight School$142$147$148$144$581Total Expenses$1,025$1,172$1,388$1,390$4,975 Annual SalesBlue Yonder AirlinesAnnual Sales ( in thousands)FY 2005FY 2006FY 2007FY 2008TotalCorporate Contracts$1,039$1,253$1,427$1,454$5,173Sky Diving $76$79$88$89$332Charter Flights$123$130$133$138$524Flight School$205$208$207$199$819Total Sales$1,443$1,670$1,855$1,880$6,848 IncomeBlue Yonder AirlinesIncome Before Taxes (in thousands)Expenses (in thousands)FY 2005FY 2006FY 2007FY 2008TotalCorporate Contracts$316$396$368$393$1,473Sky
- 7. Diving $17$17$16$15$65Charter Flights$22$24$24$27$97Flight School$63$61$59$55$238Total Expenses$418$498$467$490$1,873 Copyof Financial History.xlsx Sales HistoryFourth CoffeeAnnual Sales (in thousands)20042005200620072008TotalCoffee and Espresso$798$915$1,050$1,204$1,385$5,352Bakery$122$139$ 151$174$196$783Coffee Accessories$95$130$133$138$139$635Packaged Coffee/Tea$101$132$129$137$135$634Deli $205$208$207$199$202$1,021Total Sales$1,321$1,524$1,670$1,852$2,057$8,425 Expense HistoryFourth CoffeeAnnual Expenses (in thousands)20042005200620072008TotalCoffee and Espresso$625$730$830$945$1,060$4,190Bakery$89$99$112$1 29$151$580Coffee Accessories$75$98$96$102$100$471Packaged Coffee/Tea$92$110$118$127$132$579Deli $162$160$153$150$158$783Total Expenses$1,043$1,197$1,309$1,453$1,601$6,603 IncomeFourth CoffeeIncome Before Taxes (in thousands)20042005200620072008TotalCoffee and Espresso$173$185$220$259$325$1,162Bakery$33$40$39$45$4 5$203Coffee Accessories$20$32$37$36$39$164Packaged Coffee/Tea$9$22$11$10$3$55Deli $43$48$54$49$44$238Income Before Taxes$278$327$361$399$456$1,822 Bourne.pptx Stephanie Bourne President, Wide World Importers 2010 Business Person of the Year Director, Twin Cities Chamber of Commerce Governor’s Advisory Council member
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