Do you remember what is the most powerful tool for managing complexity?

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2 Do you remember what is the most powerful tool for managing complexity? 2

3 Abstractions and more abstractions You are here 3

4 8.1 Three views of data Application or user level Logical or abstract level Implementation level Abstract Data Types (ADTs) are here! Data Structures are here! 4

5 Abstract Data Type (ADT) Abstract data type = A data type whose properties (data and operations) are specified independently of any particular implementation. The terms container and black-box are used to describe ADTs. Python strings and lists have their own methods attached to them, e.g. my_list.sort()! 5

6 8.2 Stacks Stack = An ADT in which accesses are made at only one end LIFO = Last In First Out The insert is called Push and the removal is called Pop Name three everyday structures that are stacks 6

7 Stack example: Towers of Hanoi Tower of Hanoi - Wikipedia, the free encyclopedia 7

8 Stack algorithms Placing elements in the stack: WHILE (more data) Read value push(mystack, value) Removing elements from the stack: 8 WHILE (NOT IsEmpty(myStack)) value = pop(mystack) Write value

9 Example A stack is initially empty. Draw the stack after each of these operations: push(42) push(15) push(10) pop() push(21) pop() pop() 9

10 QUIZ: Stacks A stack contains initially only the number 42. Draw the stack at the end of this sequence operations: push(10) push(11) pop() push(12) pop() push(101) push(102) pop() 10

11 Solution This is the stack at the end: push(10) push(11) pop() push(12) pop() push(101) push(102) pop()

8.3 Queues Queue = An ADT in which items are inserted at one end and removed from the other end FIFO = First In First Out No standard queue terminology Enqueue, Enque, Enq,

12 8.3 Queues Queue = An ADT in which items are inserted at one end and removed from the other end FIFO = First In First Out No standard queue terminology Enqueue, Enque, Enq, Enter, and Insert are used for the insertion operation Dequeue, Deque, Deq, Delete, and Remove are used for the removal operation. Name three everyday structures that are queues 12

13 Interesting trivia: queueing is one of only two words in the English language that has 5 vowels in a row! E.g. Queueing Theory The other one is miaouing (miaoued?) 4 vowels in a row is more common: queue (!), aqueous, onomatopoeia, sequoia, archaeoastronomy, etc. 13

14 Queue algorithms Placing elements in the queue: WHILE (more data) Read value enque(myqueue, value) Removing elements from the queue: WHILE (NOT IsEmpty(myQueue)) value = deque(myqueue) Write value

15 QUIZ A queue is initially empty. Draw the queue after each of these operations: enque(42) enque(15) enque(10) deque() enque(21) deque() deque() 15

16 ADTs are also called containers. What exactly does the stack container contain? The data stored on that stack The main functions push( ), pop( ) Other functions isfull( ), isempty( ), peek( ),

17 Two implementations for ADTs Array-based implementation Objects in the container are kept in an array, i.e. physically next to each other in memory. Linked-based implementation Objects in the container are not kept physically together, but each item tells you where to go to get the next one in the structure Did you ever play treasure hunt, a game in which each clue tells you where to go to get the next clue? 17

18 Array implementation Memory maps for Linked implementation 18

19 Stacks and Queues in the linked implementation 19

20 Why access elements only at the ends? 20

21 8.4 Lists List are more general ADTs than stacks and queues There is an index for the current item (crt) that can move through the list. Access takes place in position crt 21

22 Operations that can be applied to lists: Insert item New item goes before old crt crt points to new item Delete item New item into list at position crt Remove the crt item from the list Move crt left (except when at tail of list, in which case crt points to the new tail) Get Next item Get the crt item Move crt right (if at end of list set crt = NULL) More items Reset Are there more items at crt? Brings crt back to head of list 22

23 Linked-list implementation crt 23

24 Array implementation of list crt Big problem: insertion and deletion require shifting the tail of the array timeconsuming! Details in CS Data Structures!

25 QUIZ: Show the list and the outputs (if any) after each of the following operations: getnext() getnext() delete() getnext() insert(42) moreitems() getnext() getnext() getnext() reset() 25

26 Algorithm for creating a list and printing its elements WHILE (more data) Read value Insert(myList, value) Reset(myList) Write "Items in the list are " WHILE (moreitems(mylist)) GetNext(myList, nextitem) Write nextitem, ' ' movenext(mylist) Brings crt back to the head of the list Trick question: Which implementation is being used (array or linked)? 26

27 Logical Level The algorithm that uses the list does not need to know how the data in the list is stored (array or linked), or how the various operations (Insert(), Reset(), moreitems()) are implemented! We have written algorithms using a stack, a queue, and a list without ever knowing the internal workings, i.e. the implementation of these containers. 27

28 8.5 Trees Structure such as lists, stacks, and queues are linear in nature; only one relationship is being modeled Other (more or less complex) relationships require different structures 28

29 Binary Tree (BT) Root node Node with two children Node with right child Leaf node Node with left child 29

30 To do for next time: Read Sections 8.1, 8.2, 8.3 End-of-chapter 1 5, 12, 13, 14 EOL 2 30

31 It is possible to implement BTs with arrays, but Binary tree = A linked container with a unique starting node called the root, in which each node can have up to two child nodes A node can have 0, 1, or 2 children A unique path (series of nodes) exists from the root to every other node. There are no loops! 31

32 QUIZ BT Write the (unique) path from the root to the node containing:

33 Operations on lists: Insert item New item goes before old crt crt points to new item Delete item New item into list at position crt Remove the crt item from the list Move crt left (except when at head of list) Get next item Get the crt item Move crt right (if at end of list set crt = NULL) More items Reset Are there more items at crt? Brings crt back to head of list 33

34 QUIZ: Show the list (and crt) at the end of the following sequence: insert(42) delete() moreitems() getnext() getnext( delete() delete() delete() getnext() getnext() 34

35 QUIZ BT 0 children 1 child 2 children List all the nodes having: 35

36 QUIZ: Are these Trees? BTs? 36

37 QUIZ: Are these Binary Trees?

38 QUIZ CONCLUSIONS: Trees Have one (and only one) root Have directed edges There is one path from the root to any other node The path above is unique In addition, Binary Trees have to satisfy: Any node can have only 0, 1, or 2 children

39 Putting a BT to work: Binary Search Tree (BST) Do you notice a pattern in this BT? 39

40 Binary Search Tree = A tree with the following properties: 1. Shape property It is a binary tree 2. Semantic property For every node, the value stored in the node is greater than the value in any node in its left subtree and smaller than the value in any node in its right subtree x <X >X 40

41 QUIZ: Are these Binary Search Trees?

42 Algorithm for searching an item in a BST IsThere(tree, item) IF (tree is null) RETURN FALSE ELSE IF (item equals info(tree)) RETURN TRUE ELSE IF (item < info(tree)) IsThere(left(tree), item) ELSE IsThere(right(tree), item) Recursive! 42

43 QUIZ BST: Search for item 18 IsThere(tree, item) IF (tree is null) RETURN FALSE ELSE IF (item equals info(tree)) RETURN TRUE ELSE IF (item < info(tree)) IsThere(left(tree), item) ELSE IsThere(right(tree), item) 43

44 QUIZ: What s the difference between a BT and a BST? 44

45 Inserting an item in a BST: first we have to search for it! Search for Kyrsten in this tree 45

46 Inserting an item in a BST: first we have to search for it! Search for kyrsten in this tree and insert the item where the search ended! 46

47 Insert(tree, item) IF (tree is null) Put item in a new tree ELSE IF (item < info(tree)) Insert (left(tree), item) ELSE Insert (right(tree), item) Note how similar this algorithm is to the search algorithm! 47

48 QUIZ on BST insertion Problem 47/280: The following elements are inserted in an initially empty BST: 50, 72, 96, 107, 26, 12, 11, 9, 2, 10, 25, 51, 16, 17, 95 Find the final tree. 48

49 Extra-credit QUIZ: 49

50 QUIZ What do the acronyms BT and BST mean in Computer Science? How is a BT similar to a BST? How is a BT different from a BST? 50

51 Binary Search Tree = A tree with the following properties: 1. Shape property It is a binary tree 2. Semantic property For every node, the value stored in the node is greater than the value in any node in its left subtree and smaller than the value in any node in its right subtree x <X >X 51

52 QUIZ on BST insertion The following elements are inserted in an initially empty BST: Hello, world!, COSC 1302, MATH 2412, PHYS 2425, ENGL 1301, ENGL 1302 Show the final tree. 52

53 Printing/traversing a BST Print(tree) If (tree is not null) Print (left(tree)) Write info(tree) Print (right(tree)) Is that all there is to it? Yes! Remember that recursive algorithms can be very powerful 53

54 Printing/traversing a BST Print(tree) If (tree is not null) Print (left(tree)) Write info(tree) Print (right(tree)) This is called in-order traversal 54

55 QUIZ: Printing/traversing a BST Print(tree) If (tree is not null) Print (left(tree)) Write info(tree) Print (right(tree)) Apply the algorithm to this BST. Show all output. 55

56 FYI: Other tree algorithms Finding the depth of a tree: Depth(tree) IF (tree is null) RETURN 0 ELSE RETURN max(depth(left(tree), Depth(right(tree))) Finding the length of a tree: Length(tree) IF (tree is null) RETURN 0 ELSE RETURN 1 + Length(left(tree))+ Length(right(tree))

57 8.6 Graphs Graph = A data structure that consists of a set of nodes (called vertices) and a set of edges that connect nodes to each other. Note: The unique path condition from trees has been removed; graphs are more general than trees! 57

58 Graphs: Unlike trees, there can be cycles and unconnected parts 58

59 QUIZ: Find 5 different cycles in this graph 59

60 QUIZ: In each case, decide if the data structure is a tree or just a graph 60

61 Graphs: directed and undirected 61

62 62 Graphs

63 Representing a Graph: adjacency matrix Our text simply calls it table See Table 8.3/262 63

64 QUIZ: Draw the adjacency matrix for this graph EOL 3 64

65 65 Graph Algorithms

66 Depth-First Searching Algorithm--Given a starting vertex and an ending vertex, we can develop an algorithm that finds a path from startvertex to endvertex This is called a depth-first search because we start at a given vertex, we go to the deepest branch and explore as far down one path before taking alternative choices at earlier branches. 66

67 DFS Stack to the rescue! startvertex endvertex Push the start vertex 67

68 DFS Stack to the rescue! startvertex endvertex Pop the start vertex, then push its children (in alphabetical order) 68

69 DFS Stack to the rescue! startvertex endvertex Unleash recursion! Figure 8.11 Using a stack to store the routes 69

70 DFS Nitty gritty: How to avoid loops? startvertex endvertex Mark each node when we push it. Figure 8.11 Using a stack to store the routes 70

71 DFS Nitty gritty: When to stop? startvertex endvertex A node is considered visited when popped (not when pushed) Figure 8.11 Using a stack to store the routes 71

72 QUIZ DFS: Find a path from Austin to Chicago Show all the steps! 72

73 Depth First Search(startVertex, endvertex) Set found to FALSE Push(myStack, startvertex) WHILE (NOT IsEmpty(myStack) AND NOT found) Pop(myStack, tempvertex) IF (tempvertex equals endvertex) Write endvertex Set found to TRUE ELSE IF (tempvertex not visited) Write tempvertex Push all unvisited vertices adjacent to tempvertex Mark tempvertex as visited IF (found) Write "Path has been printed" ELSE Write "Path does not exist") And mark pushed vertex as visited 73

74 Graph Algorithms Depth-First Search (DFS) algorithm It looks so easy do we really need an algorithm for it? 74

75 Source: 75

76 76 Is this the only path from Austin to Washington?

77 77 Is this the shortest path from Austin to Washington?

78 What if there were a direct path Dallas Washington?

79 Conclusion on DFS: The algorithm is guaranteed to find a path (if at least one exists) between the given vertices Not all paths Not (in general) the shortest path To find all, or the shortest path, we need to use different algorithms 79

80 Other Graph Algorithms Breadth-First Search (BFS) Single-Source Shortest-Path Search (SSPS) Skip them in our text! All-Pairs Shortest Path Search Flow algorithms Etc. etc. etc. 80

81 SKIP 8.7 Subprogram Statements READ and take notes: Ethical issues: Workplace Monitoring 81

82 Chapter Review Questions Distinguish between an array-based implementation and a linked implementation Distinguish between an array and a list Distinguish between and a unsorted list and a sorted list Distinguish between the behavior of a stack and a queue Distinguish between the binary tree and a binary search tree 82

83 Chapter Review Questions Draw the binary search tree that is built from inserting a series of items Understand the difference between a tree and a graph Use the DFS algorithm to find a path between two nodes of a graph. 83

84 Who am I? In the 1940s, the new computer architecture I developed revolutionized the design of computers. Today s computers are referred to as [my last name] machines because the architectural principles I described have proven very successful. 84

85 Do you know? What are some of the economic stimulus scams uncovered by the IRS in 2007? What is the Open Source Hardware Bank (OSHB)? Explain its connection to improving the function and expanding the capabilities of hardware How does graph theory relate to terrorist detection? 85

86 Homework for Ch.8 31, 32, 34, 35, 37, 38, 47, 48, Table means adjacency matrix. 56 Instead of Sandler, let the destination be Fred. The edges are all bidirectional (they go both ways). Remember that we push nodes on the stack in alphabetical order. 86

QUIZ QuickSort (Ch.7)

QUIZ QuickSort (Ch.7) 42 55 23 1-4 100 43 44 8 7 6 5 20 Do you remember what is the most powerful tool for managing complexity? 3 Remember from Ch.7: Three types of abstraction Data abstraction Separation