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- 1. Data structures & algorithms Basics: Part I By Daniel Gomez-Prado Sept 2012 Disclaimer: This tutorial may contain errors, use it at your own discretion. The slides were prepared for a class review on basic data structures at University of Massachusetts, Amherst. http://www.dgomezpr.com
- 2. Outline• Analysis of complexity• Classes, objects and containers• Array• Stack• Queue• (Single) Linked and Double Linked List• Iterators (overview)• Linear and Binary search• Merge sort• Quick sort 2
- 3. Analysis big-Oh Random access memory Your program Memory Import java.util.* class review { Static public main() { // this is a review // for ECE242 exam } }• Assumptions: o Unlimited memory o All memory accesses takes 1 unit time 3
- 4. Analysis big-Oh Random access memory Your program Memory Import java.util.* class review { Static public main() { // this is a review // for ECE242 exam } }• Assumptions: o Unlimited memory We have what we need: 1 Gb or 1,000 Tb o All memory accesses takes 1 unit time 4
- 5. Analysis big-Oh Random access memory Your program Memory Import java.util.* class review { Static public main() { // this is a review // for ECE242 exam } }• Assumptions: o Unlimited memory We have what we need: 1 Gb or 1,000 Tb No hierarchical memory, o All memory accesses takes 1 unit time Cache, L1, L2, hard drive 5
- 6. Analysis big-Oh Running timePROGRAM OPERATIONS STEPSint sum = 0; 1 assignment 1for (i=0;i<128;i=++) i =1, 2, 3, 4 … 128 128 for (j = 128; j>0; j=j/2) j = 128,64,32,16,8,4,2,1 log2128+1 sum = sum + a[i][j]; 1 addition, 1 assignment 2 6
- 7. Analysis big-Oh Running timePROGRAM OPERATIONS STEPSint sum = 0; 1 assignment 1for (i=0;i<128;i=++) i =1, 2, 3, 4 … 128 128 for (j = 128; j>0; j=j/2) j = 128,64,32,16,8,4,2,1 log2128+1 sum = sum + a[i][j]; 1 addition, 1 assignment 2 1+128*(log2128+1)*2 7
- 8. Analysis big-Oh Running timePROGRAM OPERATIONS STEPSint sum = 0; 1 assignment 1for (i=0;i<128;i=++) i =1, 2, 3, 4 … 128 128 for (j = 128; j>0; j=j/2) j = 128,64,32,16,8,4,2,1 log2128+1 sum = sum + a[i][j]; 1 addition, 1 assignment 2 1+128*(log2128+1)*2 In general we have an arbitrary number “n” instead of 128, in that case: 1+n*(log2n+1)*2 1+2n+2n*log2n 8
- 9. Analysis big-Oh Running timePROGRAM OPERATIONS STEPSint sum = 0; 1 assignment 1for (i=0;i<128;i=++) i =1, 2, 3, 4 … 128 128 for (j = 128; j>0; j=j/2) j = 128,64,32,16,8,4,2,1 log2128+1 sum = sum + a[i][j]; 1 addition, 1 assignment 2 1+128*(log2128+1)*2 n could be the size of the stack, queue, list or the dimension of a matrix, etc. In general we have an arbitrary number “n” instead of 128, in that case: 1+n*(log2n+1)*2 can we simplify the expression? 1+2n+2n*log2n YES!, By using big-Oh notation we can specify the asymptotic complexity of the algorithm 9
- 10. Analysis big-Oh Definition• Given functions f(n) and g(n): o f(n) is said to be O(g(n)) o if and only if • there are (exist) 2 positive constants, C>0 and N>0 o such that • f(n) ≤ Cg(n) for every n>N 10
- 11. Analysis big-OhExample of definition usage 11
- 12. Analysis big-OhExample of definition usage Relationship between C & n 12
- 13. Analysis big-OhExample of definition usage Relationship between C & n 13
- 14. Analysis big-OhExample of definition usage Relationship between C & n O(n*log2n) is true for C=3 and n≥32 14
- 15. Analysis big-OhExample of definition usage keywords Relationship between C & n O(n*log2n) is true for C=3 and n≥32 15
- 16. Analysis big-OhExample of definition usage keywords f(n) is given g(n) is given Relationship between C & n O(n*log2n) is true for C=3 and n≥32 16
- 17. Analysis big-Oh Example 1State the asymptotic complexity of:i. Print out middle element of an array of size n 17
- 18. Analysis big-Oh Example 1State the asymptotic complexity of: big-Ohi. Print out middle element of an array of size n 18
- 19. Analysis big-Oh Example 1State the asymptotic complexity of: big-Ohi. Print out middle element of an array of size n arrays allow access to any position randomly 19
- 20. Analysis big-Oh Example 1State the asymptotic complexity of: big-Ohi. Print out middle element of an array of size n arrays allow access to any position randomly 20
- 21. Analysis big-Oh Example 1State the asymptotic complexity of: big-Ohi. Print out middle element of an array of size n arrays allow access to any position randomly recall Your program Memory o All memory accesses takes 1 unit time 21
- 22. Analysis big-Oh Example 1State the asymptotic complexity of: big-Ohi. Print out middle element of an array of size n arrays allow access to any position randomly recall Your program Memory Solution is: O(1) o All memory accesses takes 1 unit time 22
- 23. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n 23
- 24. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n recall a linked list 24
- 25. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n head tail next next next next object object object object 25
- 26. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n head tail next next next next object object object object 26
- 27. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n head tail next next next next n/2 object object object memory object locations 27
- 28. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n head tail next next next next n/2 object object object memory object locations 28
- 29. Analysis big-Oh Example 1ii. Print out the middle element of a linked list of size n head tail next next next next n/2 object object object memory object locations f(n) = n/2 Solution is: the asymptotic complexity is O(n) 29
- 30. Analysis big-Oh Example 1iii. Print out the odd elements of an array of size n f(n) = n/2 Solution: the asymptotic complexity is O(n)iv. Pop 10 elements from a stack that is implemented with an array. Assume that the stacks contains n elements and n > 10. When in doubt, ASK! is n = 11 or is n > 1000 ? f(n) = 10 Solution: the asymptotic complexity is O(1) 30
- 31. Classes and objects• The goal of a “class” (in object-oriented language) o Encapsulate state and behavior• A class is a blueprint that has o a constructor to initialize its data members o a destructor to tear down the object o A coherent interface to interact with the object (public methods) o Private methods unreachable from the outside o The possibility to extend and inherit members from other classes• An object is an instant of a class• What are the benefits: o Through inheritance, extensions, packages allows to structure a program o Exposes behavior that could be reused o Alleviates the problem of understanding somebody else code 31
- 32. ADT (Abstract Data Type)• ADTs are containers• ADTs are primarily concern in: o Aggregation of data o Access of data o Efficiency of memory is used o Efficiency of the container access 32
- 33. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 33
- 34. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; int size 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 34
- 35. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 0 511 35
- 36. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; ObjectY size 0 511 36
- 37. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 In Java memory management is o ObjectY[] object_y_array = new ObjectY[512]; taken care for you. ObjectY size 0 511 37
- 38. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 1,934,218ObjectX 0 511 0 ObjectZ … … 38
- 39. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 1,934,218ObjectX 0 511 0 ObjectZ … … 39
- 40. Arrays• Contiguous blocks of memory of a data type o Any position can be randomly access• Example o Int[] integer_array = new int[1024]; 0 1023 o ObjectY[] object_y_array = new ObjectY[512]; 1,934,218ObjectX 0 511 0 ObjectZ … … 512 ObjectsY fixed boundary 40
- 41. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 0 41
- 42. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 peek push push 0 0 push 42
- 43. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 1023 peek push pop pop peek push 0 0 push 0 43
- 44. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false peek push pop pop peek push 0 0 push 0 44
- 45. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false peek push pop pop peek index push 0 0 push 0 45
- 46. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false peek push pop pop peek index push 0 0 push 0 46
- 47. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false peek push pop pop status peek index push 0 0 push 0 47
- 48. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1024 1023Recall what a class encapsulates isEmptyo Status & falseo Behavior pop peek index 0 48 -1
- 49. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1024 1023Recall what a class encapsulates isEmptyo Status & falseo Behavior popDoes it mean we are always safeo index = -1, stack is empty, goodo index = 1024, peek index o refuse to push objects o overflow, runtime exception 0 49 -1
- 50. Stacks• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 1024 1023Recall what a class encapsulates isEmptyo Status & falseo Behavior popDoes it mean we are always safeo index = -1, stack is empty, goodo index = 1024, peek index o refuse to push objects o overflow, runtime exception 0 50 -1
- 51. Stacks … ObjectZ• Enforce a LIFO behavior (last in, first out) o It is based on an array o It overrides the random access of an array by a LIFO access 0 1024 1023Recall what a class encapsulates isEmptyo Status & falseo Behavior popDoes it mean we are always safeo index = -1, stack is empty, goodo index = 1024, peek index o refuse to push objects o overflow, runtime exception 0 51 -1
- 52. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 0 52
- 53. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 1023 peek enqueue enqueue 0 0 enqueue 53
- 54. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 1023 1023 peek enqueue peek dequeue enqueue 0 0 enqueue 0 dequeue 54
- 55. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false peek enqueue peek dequeue enqueue 0 0 enqueue 0 dequeue 55
- 56. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 1023 1023isEmpty isEmpty isEmpty true false false status peek enqueue peek index1 index2 dequeue enqueue 0 0 enqueue 0 dequeue 56
- 57. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023Recall what a class encapsulates isEmptyo Status & false statuso Behavior peek index1 index2 dequeue 0 dequeue 57
- 58. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023Recall what a class encapsulates isEmptyo Status & false statuso Behavior peek index1Does it mean we are always safeo index1 = index2, stack is empty, good index2o index1 or index2 = 1024, dequeue o rewind to 0 o test condition o Increment using mod 1024o What if index2 > index1 0 dequeue 58
- 59. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023Recall what a class encapsulates isEmptyo Status & false statuso Behavior peek index1Does it mean we are always safe >o index1 = index2, stack is empty, good index2o index1 or index2 = 1024, dequeue o rewind to 0 o test condition o Increment using mod 1024o What if index2 > index1 0 dequeue 59
- 60. Queues• Enforce a FIFO behavior (first in, first out) o It is based on an array o It overrides the random access of an array by a FIFO access 1023 1023 status index2 status peek index1 > > index2 peek index1 dequeue 0 0 dequeue 60
- 61. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access• Let’s use another construct o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … 61
- 62. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access• Let’s use another construct o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … 62
- 63. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … 63
- 64. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … 64
- 65. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next 65
- 66. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next next object 66
- 67. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next next head object 67
- 68. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next prev next head object 68
- 69. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next prev next Node head object 69
- 70. Is everything an Array? can we use something else? beginning end• Recall an array o Contiguous memory ObjectX 0 ObjectY N-1 0 ObjectZ … … o Fixed bound size fixed boundary o Random access How do we know in this container?• Let’s use another construct o the beginning, the end o which element is next o Non contiguous memory o Unlimited size ObjectX ObjectY ObjectZ … o Sequential access only … head next edges prev next Node head object 70
- 71. Linked List• Use the prior construct (node and edge) head next next next next … object object object 71
- 72. Linked List• Use the prior construct (node and edge) head next next next next … object object object push next object 72
- 73. Linked List• Use the prior construct (node and edge) head next next next next … object object object push next object 73
- 74. Linked List• Use the prior construct (node and edge) head next next next next … object object object push next object 74
- 75. Linked List• Use the prior construct (node and edge) head next next next next … object object object next pop object 75
- 76. Linked List• Use the prior construct (node and edge) head next next next next … object object object next pop object 76
- 77. Linked List• Use the prior construct (node and edge) head next next next next … object object object next pop object 77
- 78. Linked List• Use the prior construct (node and edge) head next next next next … object object object next pop object 78
- 79. Linked List• Use the prior construct (node and edge) head next next next next … object object object pop 79
- 80. Linked List• Use the prior construct (node and edge) head next next next next … object object object peek 80
- 81. Linked List• Use the prior construct (node and edge) head next next next next … object object object peek 81
- 82. Double linked List • Use the prior construct (node and edge) headprev next prev next prev next … prev next object object object 82
- 83. Quick QuestionsCan we do:• a linked list or double linked list from an array o Yes• a queue with nodes and edges o Why not.• a stack with nodes and edges o Sure 83
- 84. Iterators encapsulate container traversals• we have two implementations of a stack 1023 pop prev next prev next head prev next prev 4 nextpeek index 1 2 3 4 prev 3 next prev 2 next 0 push prev 1 next 84
- 85. Iterators encapsulate container traversals• we have two implementations of a stack 1023 pop prev next prev next head prev next prev 4 nextpeek index 1 2 3 4 prev 3 next prev 2 next 0 push prev 1 next 85
- 86. Iterators encapsulate container traversals• we have two implementations of a stack 1023 peek push pop pop prev next prev next head prev next prev 4 nextpeek index 1 2 3 4 prev 3 next prev 2 next 0 push prev 1 next 86
- 87. Iterators encapsulate container traversals• we have two implementations of a stack 1023 prev next head prev next prev 5 nextpeek index 1 2 3 4 5 prev 4 next prev 3 next prev 2 next 0 push prev 1 next 87
- 88. Iterators encapsulate container traversals• we have two implementations of a stack 1023 update next update prev prev next increment by 1 head decrement by 1 prev next prev 5 nextpeek index 1 2 3 4 5 prev 4 next prev 3 next prev 2 next 0 push prev 1 next 88
- 89. Iterators encapsulate container traversals• we have two implementations of a stack Traverse container 1023 according to container rules update next update prev prev next increment by 1 head decrement by 1 prev next prev 5 nextpeek index 1 2 3 4 5 prev 4 next prev 3 next prev 2 next 0 push prev 1 next 89
- 90. Iterators encapsulate container traversals• we have two implementations of a stack Traverse container behavior 1023 according to container rules state update next update prev prev next increment by 1 head decrement by 1 prev next prev 5 nextpeek index 1 2 3 4 5 prev 4 next prev 3 next prev 2 next 0 push prev 1 next 90
- 91. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element a x z b n m l j i b c u• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process 91
- 92. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process 92
- 93. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process 93
- 94. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process a b b c i j l n m u x z 94
- 95. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process a b b c i j l n m u x z Looking for u 95
- 96. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process a b b c i j l n m u x z Looking for u 96
- 97. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process a b b c i j l n m u x z Looking for u 97
- 98. Searching Linear vs Binary• If you make no assumptions o iterate (traverse) all elements to find an existing element o iterate (traverse) all elements to realize you don’t have an element Looking for u worst case all elements a x z b n m l j i b c u are visited. O(n)• If you assume the container is already order (according to certain rule) o Iterate back/forth skipping some elements to speed up the process worst case there are log(n)+1 a b b c i j l n m u x z element visited. O(log(n)) Looking for u 98
- 99. So binary search is faster but the assumption is…• The container is already order, so o how do we sort a container o how do insert elements in a sorted container o How do we remove elements in a sorted container• What is more expensive (big-Oh) o A linear search o Order a container and then a binary search o Maintain a container sorted and then a binary search 99
- 100. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 100
- 101. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 101
- 102. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 102
- 103. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 103
- 104. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 85 24 104
- 105. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 85 24 63 45 105
- 106. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 24 85 63 45 106
- 107. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 24 85 45 63 107
- 108. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 24 45 63 85 19 37 91 56 24 85 45 63 108
- 109. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 109
- 110. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 110
- 111. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 24 85 45 63 111
- 112. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 take the min 24 85 45 63 112
- 113. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 85 85 24 24 45 63 113
- 114. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 take the min 85 85 24 24 45 63 114
- 115. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 85 85 24 24 45 24 63 63 45 115
- 116. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56Wait, what? 24 45 63 85 19 37 91 56 24 85 45 63 take the min 85 85 24 24 45 24 63 63 45 116
- 117. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 24 45 63 85 19 37 91 56 24 85 45 63 117
- 118. Sorting a container Merge sort• Use the divide and conquer approach o Divide the problem into 2 subsets (unless you have a base case) o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 45 19 37 91 56 DIVIDE 24 45 63 85 19 37 91 56 CONQUER RECUR 24 85 45 63 118
- 119. Sorting a container Merge sort• What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 85 24 63 45 24 85 24 45 45 63 119
- 120. Sorting a container Merge sort• What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 85 24 63 45 19 37 91 56 85 24 63 45 24 85 24 45 45 63 120
- 121. Sorting a container Merge sort• What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 2f(n/2) 85 24 63 45 19 37 91 56 85 24 63 45 24 85 24 45 45 63 121
- 122. Sorting a container Merge sort • What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 2f(n/2) 85 24 63 45 19 37 91 56 4f(n/4) 85 24 63 45log2(n) 24 85 24 45 45 63 122
- 123. Sorting a container Merge sort • What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 2f(n/2) 85 24 63 45 19 37 91 56 4f(n/4) 85 24 63 45log2(n) 24 85 x elements 24 45 45 63 y elements 123
- 124. Sorting a container Merge sort • What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 2f(n/2) 85 24 63 45 19 37 91 56 4f(n/4) 85 24 63 45log2(n) 24 85 x elements O(x+y) 24 45 45 63 y elements 124
- 125. Sorting a container Merge sort • What is the complexity of merge sort o Divide the problem into 2 subsets (2 times half the problem) o Recursively solve each subset (keep subdividing the problem) o Conquer: Solve the sub-problem and merge (the merge running time) f(n) 85 24 63 45 19 37 91 56 2f(n/2) 85 24 63 45 19 37 91 56 4f(n/4) 85 24 63 45log2(n) 24 85 O(x+y) 24 45 45 63 125
- 126. Sorting a container Merge sort O(nlogn) 126
- 127. Sorting a container Merge sort O(nlogn) 127
- 128. Sorting a container Merge sort O(nlogn) 128
- 129. Sorting a container Merge sort O(nlogn) 129
- 130. Sorting a container Merge sort• Drawback of merge sort algorithm o The merge is not in place take the min Additional memory 85 24 45 63• The merge could be modified to be in place, but the overhead will slow down the running time. 130
- 131. Sorting a container Quick sort• Use the divide and conquer approach o Divide the problem into 3 subsets (unless you have a base case) • A (random) pivot x • A subset with numbers lower than x • A subset with numbers greater than x o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 19 37 91 56 45 24 37 19 85 63 91 56 24 37 131
- 132. Sorting a container Quick sort• Use the divide and conquer approach o Divide the problem into 3 subsets (unless you have a base case) • A (random) pivot x • A subset with numbers lower than x • A subset with numbers greater than x o Recursively solve each subset o Conquer: Solve the sub-problem and merge them into a solution 85 24 63 19 37 91 56 45 24 37 19 85 63 91 56 In place sorting, 24 37 it does not require additional memory 132
- 133. Sorting a container Quick sort• Complexity o Quick sort (with random pivot) is O(nlogn)• Drawback o No replicated element allowed in the container * The pivot is randomly chosen, * if an element occurs twice, in different divisions * then the merging mechanism won’t work 133
- 134. Arrays Example 2Accept 2 integer Arrays: A and B. And find the number ofcommon elements in both assuming no duplicates in each array.o Brute force A, n elements O(nm) B, m elementso Merge-sort modified C, n+m elements Instead of merging compare and increment count when equal O( (n+m)log(n+m) ) 134
- 135. Arrays Example 2Accept 2 integer Arrays: A and B. And find the number ofcommon elements in both assuming no duplicates in each array.o Brute force A, n elements O(nm) B, m elementso Merge-sort modified C, n+m elements Instead of merging compare and increment count when equal O( (n+m)log(n+m) ) 135
- 136. Stacks and Queues Example 3Write a reverse_Queue method using only stacks andqueues in a b c d e f g h out Queue FIFO 136
- 137. Stacks and Queues Example 3Write a reverse_Queue method using only stacks andqueues in in out in a b c d e f g h a b c d e f g h h g f e d c b a O(n) out out Queue Stack Queue FIFO LIFO FIFO 137
- 138. Stacks and Queues Example 4Write a method cut_Queue that adds an element to the head ofthe queue using only stacks and queues N in a b c d e f g h out Queue FIFO 138
- 139. Stacks and Queues Example 4Write a method cut_Queue that adds an element to the head ofthe queue using only stacks and queues N O(n) in in out in out in h g f e d c b a N a b c d e f g h N a b c d e f g h N a b c d e f g h out out Queue FIFO Stack Stack Queue LIFO LIFO FIFO 139
- 140. List Example 5Write a method is_Sorted_Ascedent to check if asingle linked list is sorted in non-decreasing order head next next next … next Java pseudo code while ( node.next ) { if (node.next.key < node.key) return false; node = node.next; } return true; 140
- 141. List Example 6Write a method compress to remove duplicatedelements in a single linked list head next next next … next Java pseudo code while ( node.next ) { if ( node.key == node.next.key) ) { node.next = node.next.next; } else { node = node.next; } } 141

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