Edexcel. Decision Mathematics 1

Transcription

1 Edexcel Decision Mathematics 1 Sorts and Searches Edited by K V Kumaran kvkumaran@gmail.com Decision Maths 1 SORTS and SEARCHES Page 1

2 SORTS and SEARCHES BUBBLE-SORT ALGORITHM Page 03 QUICK-SORT ALGORITHM Page 08 BINARY SEARCH ALGORITHM Page 11 BIN-PACKING ALGORITHMS Page 14 Decision Maths 1 SORTS and SEARCHES Page 2

3 BUBBLE-SORT ALGORITHM It is so named because numbers (or letters) which are in the wrong place bubble-up to their correct positions (like fizzy lemonade) You can bubble from bottom to top, right to left, top to bottom or left to right as long as you bubble in the same direction. Assuming you are bubbling from right to left: Step 1 Compare the last two numbers on the extreme right. If the smaller number is on the right, swap the two numbers and reorder the list, if not, leave them. Step 2 Move one step back in the list (to the left) and compare the two numbers. If the smaller is on the right swap the two numbers and reorder the list, if not, leave them. Step 3 Repeat Step 2 until the two numbers on the extreme left have been compared, then return to Step 1. Step 4 Repeat Step 3 until all the numbers are in order (i.e. no more swaps are performed in a pass) Decision Maths 1 SORTS and SEARCHES Page 3

4 Example Sort into ascending order: This completes the first pass with 1 now in its correct position. Each pass succeeds in placing at least one number in its correct position. Now apply the algorithm to the sublist with 1 removed This completes the second pass with 2 now in its correct position (note that 16 & 38 also swapped). Now apply the algorithm to the sublist with 1 and 2 removed This completes the third pass with 15 now in its correct position. Now apply the algorithm to the sublist with 1, 2 and 15 removed Decision Maths 1 SORTS and SEARCHES Page 4

5 The final pass does not produce any changes This completes the sort with all numbers in their correct position Note: a pass is also known as an iteration What needs to be shown in an exam? The result of each pass! i.e Stop Decision Maths 1 SORTS and SEARCHES Page 5

6 Example Use a bubble-sort to arrange these names into alphabetical order: Smith Jones Wells Fox Davis S J W F D J S W F D J S W F D J S F W D J S F D W At the end of the first pass, Wells is in the correct position J S F D W J S F D W J F S D W J F D S W At the end of the second pass, Smith is in the correct position J F D S W F J D S W F D J S W At the end of the third pass, Jones is in the correct position F D J S W D F J S W At the end of the fourth pass, Fox is in the correct position. Record S J W F D J S F D W. J F D S. W. F D J. S. W. D F. J. S. W. Stop Decision Maths 1 SORTS and SEARCHES Page 6

7 Common Errors 1) Don t forget to keep bubbling in the same direction 2) Towards the end of the sorting, it is very tempting to just switch over those two that are in the wrong place without going through a tedious pass of the algorithm - don t, you ll lose marks! 3) Write out the list each time it alters, otherwise it is impossible to mark! 4) There are no marks for getting the list in the right order! It is method that is being assessed here! Decision Maths 1 SORTS and SEARCHES Page 7

8 QUICK-SORT ALGORITHM Introduced by Hoare in 1962 The mid-point of a list has position [½(N+1)] where [x] is the smallest integer greater than or equal to x e.g. for 3, 6, 7, 11, 15 [½ (5+1)] = [3] = 3 mid-point = 7 for A, C, Y, B, D, R [½ (6+1)] = [3½] = 4 mid-point = B Step 1 Locate the pivot element (use the element at the mid-point) Step 2 Split the list into two sub-lists. Sublist L 1 contains those numbers less than or equal to the pivot and are written to the left of the pivot. Sublist L 2 contains those numbers greater than the pivot element and are written to the right of the pivot. Do not reorder the numbers in the sub-lists Step 3 Repeat Step 2 on each sub-list and each successive sub-list Step 4 Stop when each sub-list contains only one number. The list is now sorted! Decision Maths 1 SORTS and SEARCHES Page 8

9 Example Use a quick-sort to arrange these numbers in numerical order Mid-point = [½(6+1)] = [3½] = 4 th number = L 1 L Stop Example Use a quick-sort to arrange these letters in alphabetical order. Y T A B F S F L Mid-point = [½(8 + 1)] = [4½] = 5 th letter = F Y T A B F S F L A B F F Y T S L A B F F L S Y T A B F F L S T Y Stop Decision Maths 1 SORTS and SEARCHES Page 9

10 Example Use a quick-sort to arrange these letters in numerical order Mid-point = [½(10 + 1)] = [5½] = 6 th number = STOP Common Errors 1) The choice of pivot must be constant. If you have an even number of items in a list, the middle number is not clear. If you choose the right-hand number the first time, you must continue to select the right-hand number each time you have a choice. 2) Do not be tempted to reorder the items in a sub-list. 3) Remember that it is the method that is being assessed and not the answer. So make sure enough working is being shown. The list should be rewritten each time a new sub-list is created. 4) Remember to indicate the stop step. Decision Maths 1 SORTS and SEARCHES Page 10

11 BINARY SEARCH ALGORITHM This only applies to a list of names in alphabetical order or a list of numbers in increasing order. Unordered lists would have a sorting algorithm applied first. This algorithm concentrates on the midpoint of an ever reducing list. We define the midpoint of a list of N names, numbered N 1, N 1 +1,...,N 2 as [(N 1 +N 2 )/2] where [x] = the smallest integer greater than or equal to x We wish to search the list for a name, Fred, say. Step 1 The algorithm compares Fred with the middle name in the list. Either a) the name Fred is found to be at this position, or b) the name Fred occurs before the middle of the list or c) the name Fred occurs after the middle of the list Step 2 If a) occurs, the search is over. If b) occurs, the search continues with Step 1 on a reduced list consisting of those names before the middle name. If c) occurs, the search continues with Step 1 on a reduced list consisting of those names after the middle name. Stop when Fred has been found or when it has been shown that Fred does not appear on the list. At each stage half of the remaining list is discarded (hence the name of the algorithm). Decision Maths 1 SORTS and SEARCHES Page 11

12 Example a) Find the name Robinson in the list below. b) Find the name Davis in the list below. 1 Bennett 2 Blackstock 3 Brown 4 Ebenezer 5 Fowler 6 Laing 7 Leung 8 Robinson 9 Saludo 10 Scadding Solution a) The middle name is [(1+10)/2] = [5.5] = 6 Laing Robinson is after Laing, so the list reduces to 7 Leung 8 Robinson 9 Saludo 10 Scadding The middle name is [(7+10)/2] = [8.5] = 9 Saludo Robinson is before Saludo, so the list reduces to 7 Leung 8 Robinson The middle name is [(7+8)/2] = [7.5] = 8 Robinson The search is complete and Robinson has been found Decision Maths 1 SORTS and SEARCHES Page 12

13 b) As before, the middle name is Laing Davis is before Laing, so the list reduces to 1 Bennett 2 Blackstock 3 Brown 4 Ebenezer 5 Fowler The middle name is [(1+5)/2] = [3] = 3 Brown Davis is after Brown, so the list reduces to 4 Ebenezer 5 Fowler The middle name is [(4+5)/2] = [4.5] = 5 Fowler Davis is before Fowler, so the list reduces to 4 Ebenezer The list is now only one item and this item is not Davis We conclude that Davis is not on the list. Decision Maths 1 SORTS and SEARCHES Page 13

14 BIN-PACKING ALGORITHMS We are seeking to pack bins (of given size) with various items. To find the lower bound, sum the numbers to be packed and divide this total by the size of the bin. The lower bound is the least integer greater than or equal to the result. Problem: Pack the following items in bins of size The sum is divided by 20 is 4.7 So the lower bound is 5 we need at least 5 bins FIRST-FIT ALGORITHM Take the items in the order given and place each item in the first available bin that can take it, starting from Bin 1 each time Bin Bin Bin Bin 4 15 Bin Bin Decision Maths 1 SORTS and SEARCHES Page 14

15 FIRST-FIT DECREASING ALGORITHM Firstly, reorder the items so that they are in decreasing order. Then apply the first-fit algorithm to the reordered list. The reordered list (using a sorting algorithm if necessary): Bin Bin Bin Bin Bin Decision Maths 1 SORTS and SEARCHES Page 15

16 FULL-BIN PACKING Here we use observations to find combinations of items that will fill a bin. Pack those items first. Any remaining items are placed in the next available bin that can take the item, starting from Bin 1 each time. Scanning the list, = 20 (so put in Bin 1) = 20 (so put in Bin 2) = 20 (so put in Bin 3) Bin Bin Bin Bin Bin Decision Maths 1 SORTS and SEARCHES Page 16

17 ADVANTAGES and DISADVANTAGES First-fit Algorithm Advantages Disadvantages It is quick to do It is not likely to lead to a good solution First-fit Decreasing Algorithm Advantages Disadvantages Usually get a fairly good solution It is easy to do May not get the optimal solution Full-Bin Packing Advantages Disadvantages Usually get a good solution Difficult to do, especially when the numbers are plentiful and awkward Of course, not all bin-packing problems will involve bins! Decision Maths 1 SORTS and SEARCHES Page 17

18 Example Andy wants to record the following twelve TV programmes onto video tape. Each video tape has space for up to three hours of programmes. Programme A B C D E F G H I J K L Length (hr) ½ ½ ¾ ½ 1½ 1¾ 2 2 i) Suppose that Andy records the programmes in the order A to L using the first-fit algorithm. Find the number of tapes needed, and show which programmes are recorded onto which tape. ii) Suppose instead that Andy is transferring the programmes from previously recorded tapes, so that they can be copied in any order, and that Andy uses the firstfit decreasing algorithm. Find the number of tapes needed, and show which programmes are recorded onto which tape. i) Tape 1 ½ ½ ¾ 1 (ABCD) Tape (EFG) Tape 3 1½ 1½ (HI) Tape 4 1¾ (J) Tape 5 2 (K) Tape 6 2 (L) 6 Tapes needed ii) Reordered list: 2 2 1¾ 1½ 1½ ¾ ½ ½ Tape (LG) Tape (KF) Tape 3 1¾ 1 (JE) Tape 4 1½ 1½ (IH) Tape 5 1 ¾ ½ ½ (DCBA) 5 Tapes needed Decision Maths 1 SORTS and SEARCHES Page 18

19 1. (a) Use the binary search algorithm to locate the name HUSSAIN in the following alphabetical list. Explain each step of the algorithm. 1. ALLEN 2. BALL 3. COOPER 4. EVANS 5. HUSSAIN 6. JONES 7. MICHAEL 8. PATEL 9. RICHARDS 10. TINDALL 11. WU (6 marks) (b) State the maximum number of comparisons that need to be made to locate a name in an alphabetical list of 11 names. (1 mark) (D1, Jan 2001, Q2) 2. 90, 50, 55, 40, 20, 35, 30, 25, 45 (a) Use the bubble sort algorithm to sort the list of numbers above into descending order showing the rearranged order after each pass. (5) Jessica wants to record a number of television programmes onto video tapes. Each tape is 2 hours long. The lengths, in minutes, of the programmes she wishes to record are: 55, 45, 20, 30, 30, 40, 20, 90, 25, 50, 35 and 35. (b) Find the total length of programmes to be recorded and hence determine a lower bound for the number of tapes required. (2) (c) Use the first fit decreasing algorithm to fit the programmes onto her 2-hour tapes. Jessica s friend Amy says she can fit all the programmes onto 4 tapes. (3) (d) Show how this is possible. (2) (D1, June 2001, Q5) Decision Maths 1 SORTS and SEARCHES Page 19

20 3. (i) Use the binary search algorithm to try to locate the name SABINE in the following alphabetical list. Explain each step of the algorithm. 1. ABLE 2. BROWN 3. COOKE 4. DANIEL 5. DOUBLE 6. FEW 7. OSBORNE 8. PAUL 9. SWIFT 10. TURNER (5) (ii) Find the maximum number of iterations of the binary search algorithm needed to locate a name in a list of 1000 names. (2) (D1, Jan 2002, Q2) 4. Ashford 6 Colnbrook 1 Datchet 18 Feltham 12 Halliford 9 Laleham 0 Poyle 5 Staines 13 Wraysbury 14 The table above shows the points obtained by each of the teams in a football league after they had each played 6 games. The teams are listed in alphabetical order. Carry out a quick sort to produce a list of teams in descending order of points obtained. (5) (D1, May 2002, Q1) Decision Maths 1 SORTS and SEARCHES Page 20

21 The list of numbers above is to be sorted into descending order. (a) (i) Perform the first pass of a bubble sort, giving the state of the list after each exchange. (ii) Perform further passes, giving the state of the list after each pass, until the algorithm terminates. (5) The numbers represent the lengths, in cm, of pieces to be cut from rods of length 50 cm. (b) (i) Show the result of applying the first fit decreasing bin packing algorithm to this situation. (ii) Determine whether your solution to (b) (i) has used the minimum number of 50 cm rods. (D1, Jan 2003, Q6) 6. The following list gives the names of some students who have represented Britain in the International Mathematics Olympiad. Roper (R), Palmer (P), Boase (B), Young (Y), Thomas (T), Kenney (K), Morris (M), Halliwell (H), Wicker (W), Garesalingam (G). (a) Use the quick sort algorithm to sort the names above into alphabetical order. (b) Use the binary search algorithm to locate the name Kenney. (D1, June 2003, Q4) 7. Nine pieces of wood are required to build a small cabinet. The lengths, in cm, of the pieces of wood are listed below. 20, 20, 20, 35, 40, 50, 60, 70, 75 Planks, one metre in length, can be purchased at a cost of 3 each. (a) The first fit decreasing algorithm is used to determine how many of these planks are to be purchased to make this cabinet. Find the total cost and the amount of wood wasted. (5) Planks of wood can also be bought in 1.5 m lengths, at a cost of 4 each. The cabinet can be built using a mixture of 1 m and 1.5 m planks. (b) Find the minimum cost of making this cabinet. Justify your answer. (D1, Nov2003, Q5) (5) Decision Maths 1 SORTS and SEARCHES Page 21

22 8. 1. Glasgow 2. Newcastle 3. Manchester 4. York 5. Leicester 6. Birmingham 7. Cardiff 8. Exeter 9. Southampton 10. Plymouth A binary search is to be performed on the names in the list above to locate the name Newcastle. (a) Explain why a binary search cannot be performed with the list in its present form. (1) (b) Using an appropriate algorithm, alter the list so that a binary search can be performed. State the name of the algorithm you use. (c) Use the binary search algorithm on your new list to locate the name Newcastle. (D1, June 2004, Q4) 9. 45, 56, 37, 79, 46, 18, 90, 81, 51 (a) Using the quick sort algorithm, perform one complete iteration towards sorting these numbers into ascending order. (2) (b) Using the bubble sort algorithm, perform one complete pass towards sorting the original list into descending order. (2) Another list of numbers, in ascending order, is 7, 23, 31, 37, 41, 44, 50, 62, 71, 73, 94 (c) Use the binary search algorithm to locate the number 73 in this list. (Total 8 marks) (D1, Nov 2004, Q4) Decision Maths 1 SORTS and SEARCHES Page 22

23 (a) The list of numbers above is to be sorted into descending order. Perform a Quick Sort to obtain the sorted list, giving the state of the list after each pass, indicating the pivot elements. (5) The numbers in the list represent the lengths, in mm, of some pieces of wood. The wood is sold in one metre lengths. (b) Use the first-fit decreasing bin packing algorithm to determine how these pieces could be cut from the minimum number of one metre lengths. (You should ignore wastage due to cutting.) (c) Determine whether your solution to part (b) is optimal. Give a reason for your answer. (2) (D1, Jan 2005, Q4) 11. Ali 74 Bobby 28 Eun-Jung 63 Katie 54 Marciana 54 Peter 49 Rory 37 Sophie 68 The table shows the marks obtained by students in a test. The students are listed in alphabetical order. Carry out a quick sort to produce a list of students in descending order of marks. You should show the result of each pass and identify your pivots clearly. (Total 5 marks) (D1, June 2005, Q1) The list of numbers above is to be sorted into descending order. Perform a bubble sort to obtain the sorted list, giving the state of the list after each completed pass. (D1, May2006, Q1) Decision Maths 1 SORTS and SEARCHES Page 23

24 13. (D1, Jan 2007, Q1) The list of numbers shown above is to be sorted into ascending order. Apply quick sort to obtain the sorted list. You must make your pivots clear. (5) (D1, Jan 2008, Q2) The numbers in the list represent the lengths in minutes of nine radio programmes. They are to be recorded onto tapes which each store up to 100 minutes of programmes. (a) Obtain a lower bound for the number of tapes needed to store the nine programmes. (2) (b) Use the first-fit bin packing algorithm to fit the programmes onto the tapes. (c) Use the first-fit decreasing bin packing algorithm to fit the programmes onto the tapes. (3) (3) (D1, May2008, Q1) Decision Maths 1 SORTS and SEARCHES Page 24

25 16. Max Lauren John Hannah Kieran Tara Richard Imogen (a) Use a quick sort to produce a list of these names in ascending alphabetical order. You must make your pivots clear. (5) (b) Use the binary search algorithm on your list from part (a) to try to locate the name Hugo. (D1, Jan 2009, Q1) The numbers in the list above represent the lengths, in metres, of ten lengths of fabric. They are to be cut from rolls of fabric of length 60m. (a) Calculate a lower bound for the number of rolls needed. (b) Use the first-fit bin packing algorithm to determine how these ten lengths can be cut from rolls of length 60m. (c) Use full bins to find an optimal solution that uses the minimum number of rolls. (3) 18. Miri Jessie Edward Katie Hegg Beth Louis Philip Natsuko Dylan (a) Use the quick sort algorithm to sort the above list into alphabetical order. (b) Use the binary search algorithm to locate the name Louis. (2) (Total 9 marks) (D1, May 2009, Q2) (5) (Total 9 marks) (D1, May 2009, Q4) Decision Maths 1 SORTS and SEARCHES Page 25

26 19. A builder is asked to replace the guttering on a house. The lengths needed, in metres, are Guttering is sold in 4 m lengths. 0.6, 4.0, 2.5, 3.2, 0.5, 2.6, 0.4, 0.3, 4.0 and 1.0 (a) Carry out a quick sort to produce a list of the lengths needed in descending order. You should show the result of each pass and identify your pivots clearly. (5) (b) Apply the first-fit decreasing bin-packing algorithm to your ordered list to determine the total number of 4 m lengths needed. (c) Does the answer to part (b) use the minimum number of 4 m lengths? You must justify your answer. (2) (D1, Jan 2010, Q4) 20. Hajra (H) Vicky (V) Leisham (L) Alice (A) Nicky (N) June (J) Sharon (S) Tom (T) Paul (P) The table shows the names of nine people. 21. (a) Use a quick sort to produce the list of names in ascending alphabetical order. You must make your pivots clear. (b) Use the binary search algorithm on your list to locate the name Paul (D1, May 2010, Q1) The numbers in the list represent the weights, in kilograms, of seven statues. They are to be transported in crates that will each hold a maximum weight of 60 kilograms. (a) Calculate a lower bound for the number of crates that will be needed to transport the statues. (2) (b) Use the first-fit bin packing algorithm to allocate the statues to the crates. (c) Use the full bin algorithm to allocate the statues to the crates. (d) Explain why it is not possible to transport the statues using fewer crates than the number needed for part (c). (2) (D1, May 2010, Q3) (3) (2) Decision Maths 1 SORTS and SEARCHES Page 26

27 The numbers represent the sizes, in megabytes (MB), of eight files. The files are to be stored on 50 MB discs. (a) Calculate a lower bound for the number of discs needed to store all eight files. (b) Use the first-fit bin packing algorithm to fit the files onto the discs. (c) Perform a bubble sort on the numbers in the list to sort them into descending order. You need only write down the final result of each pass. (d) Use the first-fit decreasing bin packing algorithm to fit the files onto the discs Jenny 2. Merry 3. Charles 4. Ben 5. Toby 6. Hyo 7. Kim 8. Richard 9. Greg 10. Freya (2) (3) (3) (D1, Jan2011, Q2) A binary search is to be performed on the names in the list above to locate the name Kim. (a) Explain why a binary search cannot be performed with the list in its present form. (1) (b) Using an appropriate algorithm, alter the list so that a binary search can be performed, showing the state of the list after each complete iteration. State the name of the algorithm you have used. (c) Use the binary search algorithm to locate the name Kim in the list you obtained in part (b). You must make your method clear. (D1, May 2011, Q1) Decision Maths 1 SORTS and SEARCHES Page 27

28 (a) Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 20. (3) (b) The list of numbers is to be sorted into descending order. Use a bubble sort to obtain the sorted list, giving the state of the list after each complete pass. (5) (c) Apply the first-fit decreasing bin packing algorithm to your ordered list to pack the numbers into bins of size 20. (3) (d) Determine whether your answer to (c) uses the minimum number of bins. You must justify your answer. (2) (D1, Jan 2012, Q5) 25. A carpet fitter needs the following lengths, in metres, of carpet He cuts them from rolls of length 50 m. (a) Calculate a lower bound for the number of rolls he needs. You must make your method clear. (b) Use the first-fit bin packing algorithm to determine how these lengths can be cut from rolls of length 50 m. (3) (c) Carry out a bubble sort to produce a list of the lengths needed in descending order. You need only give the state of the list after each pass. (d) Apply the first-fit decreasing bin packing algorithm to show how these lengths may be cut from the rolls. (2) (3) (D1, May 2012, Q1) Decision Maths 1 SORTS and SEARCHES Page 28

29 (a) Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 2. (3) (b) The list of numbers is to be sorted into descending order. Use a quick sort to obtain the sorted list. You must make your pivots clear. (c) Apply the first-fit decreasing bin packing algorithm to your ordered list to pack the numbers into bins of size 2. (3) (d) Determine whether your answer to part (c) uses the minimum number of bins. You must justify your answer. (1) (D1, May 2013, Q2) Sam (S) 2. Janelle (J) 3. Haoyu (H) 4. Alfie (A) 5. Cyrus (C) 6. Komal (K) 7. Polly (P) 8. David (D) 9. Tom (T) 10. Lydia (L) A binary search is to be performed on the names in the list above to locate the name Lydia. (a) Using an appropriate algorithm, rearrange the list so that a binary search can be performed, showing the state of the list after each complete iteration. State the name of the algorithm you have used. (b) Use the binary search algorithm to locate the name Lydia in the list you obtained in (a). You must make your method clear. (D1, May 2013_R, Q4) Decision Maths 1 SORTS and SEARCHES Page 29

30 x The numbers in the list represent the exact weights, in kilograms, of 9 suitcases. One suitcase is weighed inaccurately and the only information known about the unknown weight, x kg, of this suitcase is that 19 < x 23. The suitcases are to be transported in containers that can hold a maximum of 50 kilograms. (a) Use the first-fit bin packing algorithm, on the list provided, to allocate the suitcases to containers. (3) (b) Using the list provided, carry out a quick sort to produce a list of the weights in descending order. Show the result of each pass and identify your pivots clearly. (c) Apply the first-fit decreasing bin packing algorithm to the ordered list to determine the 2 possible allocations of suitcases to containers. After the first-fit decreasing bin packing algorithm has been applied to the ordered list, one of the containers is full. (d) Calculate the possible integer values of x. You must show your working. (D1, May 2014, Q6) The numbers above are Alan s batting scores for the first 10 cricket matches of the season. (a) Use a quick sort to sort this list of numbers into ascending order. You must make your pivots clear. Alan s batting scores for the final 10 cricket matches of the same season were (b) Carry out a bubble sort on this second list of numbers to produce a list of these scores in ascending order. You need only give the state of the list after each pass. Alan s combined batting scores for the entire season were (c) Use the binary search algorithm to locate 68 in the combined list of 20 scores. You must make your method clear. (3) (D1, May 2015, Q2) (2) Decision Maths 1 SORTS and SEARCHES Page 30