(c) Finally apply critical path scheduling using two processors to the order requirement

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1 Blue Exam: 1. When scheduling tasks using the list-processing algorithm, increasing the number of machines always reduces the completion time. False. We even had an example in class where it made the completion time longer. 2. An Illinois resident with driver s license number N is a female. True. Females have the last three digits of their driver s license number greater than Assume an order-requirement digraph has a critical path with length 40 minutes. With at least two machines we can always complete all tasks in 40 minutes. False. Only if all the other task can be done together in 40 minutes too. 4. The first-fit decreasing (FFD) algorithm for bin packing is guaranteed to always produce an optimal solution. False. It gives good, fast solutions but not always optimal. This problem is also NP-complete. 5. (a) Given the order-requirement digraph below (with time given in minutes) and the priority list T1, T2, T3, T4, T5, T6, apply the list-processing algorithm to construct a schedule using two processors. Processor 1 would have T 1, T 3, T 4 and T 6 with a total run time of 18 minutes. Processor 2 would have T 2 and T 5 with idle time from time 12 on. (b) Repeat the algorithm but this time use three processors. Processor 1 would have T 1 and T 5 followed by 3 minutes of idle time and finally T 6 for a total of 16 minutes. Processor 2 would have T 2 followed by T 4 with idle time from 13 on. Processor 3 would have T 3 and then idle time from 4 on. (c) Finally apply critical path scheduling using two processors to the order requirement digraph above. The priority list would now be T2, T3, T1, T4, T5, T6. Processor 1 would have T 2, T 4, idle time from minutes and then T 6 from minutes. Processor 2 would have T 3, T 1, and T 5 with idle time from 14 on. 1

2 6. The table below represents species of plants which have competing light or water requirements. Draw the graph which would be useful in determining the minimum number of different habitats that would be needed to display all these plants in a garden. A B C D E A X X X B X X C X X D X X X E X X Put a vertex for each of plants A through E and then connect two vertices with an edge if and only if the plants have competing requirements. 7. Advertisements for the TV show Q are played in groups of ads where each group can last up to 2 minutes. If the ads slated for Q last 63, 32, 11, 19, 24, 87, 64, 36, 27, 42, and 63 seconds, determine, using both first fit (FF) and first fit decreasing (FFD), how many groups of ads would need to run during the show to play all the ads. For FF: Bin 1: 63, 32, 11 Bin 2: 19, 24, 64 Bin 3: 87, 27 Bin 4: 36, 42 Bin 5: 63 For FFD we have the order 87, 64, 63, 63, 42, 36, 32, 27, 24, 19, 11: Bin 1: 87, 32 Bin 2: 64, 42, 11 Bin 3: 63, 36, 19 Bin 4: 63, 27, (a) Given independent tasks of the following length: 9 minutes, 2 minutes, 8 minutes, 5 minutes, 4 minutes, 8 minutes, schedule them on two processors. Processor 1 would have 9, 5, 8 for a total time of 22 minutes. Processor 2 would have 2, 8, 4 and then idle time from 14 on. 2

3 (b) Now use the decreasing-time-list algorithm to schedule these tasks on two machines. We get the priority list: 9, 8, 8, 5, 4, 2. So Processor 1 would have 9, 5 and 4. Processor 2 would have 8, 8 and 2. Both are done in 18 minutes. (c) Is either schedule optimal? Why or Why not? The second one is optimal since there is no idle time. 9. (a) Determine the check digit which should be appended to the US Postal Service money order identification number The sum of the digits is 41 and 41/9 has remainder 5 so we add a check digit of 5. (b) Suppose the fourth digit of the ISBN 0-71x is obliterated. What is the missing digit? We use weights 10, 9, 8, and get that the sum should be x. We want 11 to divide this evenly. So we want to know which multiple of 11 can also be written as a multiple of 7. We get that 165 = = and so x = 6. (c) Determine the check character that should be appended to the bank identification number We use weights 7,3,9,7,3,9,7,3 to get a sum of 120 which ends in a zero so the check digit is zero. 10. Which of the algorithms: first fit (FF), next fit (NF), or worst fit (WF) would be most preferable when filling boxes on an assembly line? Why? We would want to use next fit since you can t go back on an assembly line. 11. If the UPC number is incorrectly entered as , will the error be detected by the check digit? No it will not be corrected. Since the weights on a UPC code are 3, 1, 3, 1, we will decrease the sum by 20 (36-16) and so will still have a number that ends in zero which is what makes a UPC code legitimate. 12. What is the minimum time required to complete 12 independent tasks on 3 processors when the sum of all the times of the 12 tasks is 60 minutes? 3

4 It will take a minimum of 60/3 = 20 minutes. This is the best possible if there is no idle time. 13. Give an example of a graph that cannot be colored with two colors. There are LOTS of answers here. For instance, the complete graph on 5 vertices cannot be colored with two colors. See me if you have questions about a particular graph. 14. Does the following have the correct structure for a legitimate Postnet bar code? Why or why not? Could you find the last two digits of the receiver s street address from this code? Circle the check digit. This has the right structure, two guard bars, 12 groupings of 5 lines each, each grouping has 2 large and 3 small lines. Since this bar code has 12 groupings, it consists of the zip code, the extra 4 code, the last two digits of the receiver s street address or PO box and the check digit which is the last 5 bars before the final guard bar on the right side. Bonus: (5 points) Prove that the ISBN code can detect any single digit error. See class notes or the book for the proof of this. Ivory Exam: 1. Assume an order-requirement digraph has a critical path with length 40 minutes. If the digraph is scheduled on four machines, we may sometimes be able to complete all tasks in 35 minutes. False. We can never complete all jobs in less time than the critical path. 2. An Illinois resident with driver s license number containing the last four digits 7825 was born in

5 False. The 7 in 7825 is the last digit of the year the person was born in. We don t know what decade, though. That would be the number before the When scheduling tasks using the list-processing algorithm, decreasing the time of each task always decreases the completion time. False. We had an example in class where the time was actually increased. 4. The first fit (FF) algorithm never uses more boxes (bins) than the next fit (NF) algorithm. True. Piece by piece, for first fit, each piece is at worst in the same bin as it is for next fit so at worst we use the same number of bins. 5. (a) Given the order-requirement digraph below (with time given in minutes) and the priority list T1, T2, T3, T4, T5, T6, apply the list-processing algorithm to construct a schedule using two processors. Processor 1 would have T1, T3, T6, T4 for a total run time of 17 minutes. Processor 2 would have T2 and T5 with a total run time of 13 minutes and idle time from 13 on. (b) Repeat the algorithm but this time use three processors. Processor 1 would have T1, then idle time from 4-5, then T5 and finally T4 for a total run time of 16 minutes. Processor 2 would have T2 and then idle time from time 5 on. Processor 3 would have T3 and T6 and then idle time from 10 on. (c) Finally apply critical path scheduling using two processors to the order requirement digraph above. The priority list would be T2,T1,T3,T5,T6,T4. Processor 1 would have T2, T5, idle time from 13 to 14 and then T4 for a total run time of 16. Processor 2 would have T1, T3, T6 and idle time from 14 on. 6. (a) Given independent tasks of the following length: 3 minutes, 5 minutes, 8 minutes, 4 minutes, 6 minutes, 2 minutes, schedule them on two processors. Processor 1 has 3, 8, 2 and idle time from Processor 1 has 5, 4, 6 for a total run time of 15 minutes. (b) Now use the decreasing-time-list algorithm to schedule these tasks on two 5

6 machines. Processor 1 has 8, 4, 2 for a total run time of 14 minutes. Processor 2 has 6, 5, 3 for a total run time of 14 minutes also. (c) Is either schedule optimal? Why or Why not? The second schedule is optimal since it has no idle time. 7. Advertisements for the TV show Q are played in groups of ads where each group can last up to 2 minutes. If the ads slated for Q last 63, 32, 11, 19, 24, 87, 64, 36, 27, 42, and 63 seconds, determine, using both first fit (FF) and first fit decreasing (FFD), how many groups of ads would need to run during the show to play all the ads. See Blue The table below shows chemical compounds which cannot be mixed without causing dangerous reactions. Draw the graph which would be used to facilitate scheduling of disposal containers for the compounds. A B C D E F A X X X B X X X X C X X D X X X E X X F X X Put a vertex for each of the chemicals A through F and then connect two vertices with an edge if and only if the chemicals are incompatible. 9. (a) Determine the check digit which should be appended to the US Postal Service money order identification number We sum the digits and get 46. The remainder on division by 9 is 1 so the check digit is 1. (b) Is the number a legitimate bank identification number? We use the weights 7,3,9,7,3,9,7,3 and get 93 so the last digit should be a 3 (last digit of 93) but in this case it is a 5 so the number is not legitimate. 6

7 (c) Determine the check digit that should be appended to the Master Card number This is Codabar. The sum of the odd place digits is 30 and so multiplied by 2 gives us 60. Also, there are 3 odd place digits greater than 4. Finally the even places sum to 25 so we get = 88. So to get a number that 10 evenly divides, we append a check digit of If the UPC number is incorrectly entered as , will the error be detected by the check digit? We are subtracting 2 from the sum and adding 8 which is a net gain of 6 so the sum will no longer end in a zero so we can tell an error has occurred. 11. Which of the algorithms: first fit (FF), next fit (NF), or worst fit (WF) would be most preferable when filling boxes on an assembly line? Why? We would want to use next fit since you can t go back on an assembly line. 12. What is the minimum time to complete 12 independent tasks on 4 processors when the sum of all the times of the 12 tasks is 60 minutes? It will take a minimum of 60/4=15 minutes to complete all the tasks. This is best possible if there is no idle time. 13. Does the following have the correct structure for a legitimate Postnet bar code? Why or why not? Why or why not? Could you find the last two digits of the receiver s street address from this code? Circle the check digit. This has the right structure, two guard bars, 10 groupings of 5 lines each, each grouping has 2 large and 3 small lines. Since this bar code has 10 groupings, it consists of the zip code, the extra 4 code, and the check digit which is the last 5 bars before the final guard bar on the right side. It does not include the last two digits of the receiver s street address or PO box. 7

8 14. Give an example of a graph that cannot be colored with two colors. There are LOTS of answers here. For instance, the complete graph on 5 vertices cannot be colored with two colors. See me if you have questions about a particular graph. Bonus: (5 points) Prove that the ISBN code can detect any single digit error. See class notes or the book for a proof of this. 8