1 Linked Lists A linked list is a sequence in which there is a defined order as with any sequence but unlike array and Vector there is no property of contiguity of memory.

2 Singly-linked Lists u A list in which there is a preferred direction. u A minimally linked list. u The item before has a pointer to the item after.

3 Singly-linked List u Implement this structure using objects and references.

4 Singly-linked List head

5 Singly-linked List head

6 Singly-linked List class ListElement { Object datum ; ListElement nextElement ;... } datumnextElement

7 Singly-linked List ListElement newItem = new ListElement(new Integer(4)) ; ListElement p = null ; ListElement c = head ; while ((c != null) && !c.datum.lessThan(newItem)) { p = c ; c = c.nextElement ; } newItem.nextElement = c ; p.nextElement = newItem ;

8 Singly-linked List head p c newElement

9 Analysing Singly-linked List u Accessing a given location is O(n). u Setting a given location is O(n). u Inserting a new item is O(n). u Deleting an item is O(n) u Assuming both a head at a tail pointer, accessing, inserting or deleting can be O(1).

10 Doubly-linked Lists u A list without a preferred direction. u The links are bidirectional: implement this with a link in both directions.

11 Doubly-linked List tail head

12 Doubly-linked List tail head

13 Doubly-linked List class ListElement { Object datum ; ListElement nextElement ; ListElement previousElement ;... } datumnextElement previousElement

14 Doubly-linked List ListElement newItem = new ListElement(new Integer(4)) ; ListElement c = head ; while ((c.next != null) && !c.next.datum.lessThan(newItem)) { c = c.nextElement ; } newItem.nextElement = c.nextElement ; newItem.previousElement = c ; c.nextElement.previousElement = newItem ; c.nextElement = newItem ; Spot the deliberate mistake. What needs to be done to correct this?

15 Doubly-linked List tail head c newItem

16 Doubly-linked List u Performance of doubly-linked list is formally similar to singly linked list. u The complexity of managing two pointers makes things very much easier since we only ever need a single pointer into the list. u Iterators and editing are made easy.

17 Doubly-linked List u Usually find the List type in a package is a doubly-linked list. u Singly-linked list are used in other data structures.

18 Stack and Queue u Familiar with the abstractions of stack and queue.

19 Stack pushpop isEmpty top

20 Queue insert remove isEmpty

21 Implementing Stack tos

22 Implementing Queue tail head

23 Multi-lists u Multi-lists are essentially the technique of embedding multiple lists into a single data structure. u A multi-list has more than one next pointer, like a doubly linked list, but the pointers create separate lists.

24 Multi-lists head

25 Multi-lists head

26 Multi-lists (Not Required) head

27 Linked Structures u A doubly-linked list or multi-list is a data structure with multiple pointers in each node. u In a doubly-linked list the two pointers create bi-directional links u In a multi-list the pointers used to make multiple link routes through the data.

28 Linked Structures u What else can we do with multiple links? u Make them point at different data: create Trees (and Graphs).

29 Trees node leaf node degree root children parent Level 1 Level 2 Level 3 height = depth = 3

30 Trees u Crucial properties of Trees: v Links only go down from parent to child. v Each node has one and only one parent (except root which has no parent). v There are no links up the data structure; no child to parent links. v There are no sibling links; no links between nodes at the same level.

31 Trees u If we relax the restrictions, it is not a Tree, it is a Graph. u A Tree is a directed, acyclic Graph that is single parent.

32 Trees u Binary Trees have degree 2. u Red–Black Trees and AVL Trees are Binary Trees with special extra properties; they are balanced. u B-Trees, B+-Trees, B*-Trees are more complicated Trees with flexible branching factor: these are used very extensively in databases.

33 Binary Trees u Trees are immensely useful for sorting and searching. u Look at Binary Trees as they are the simplest.

34 Binary Trees This is a complete binary tree.

35 Binary Trees u How to insert something in the list? u Need a metric, there must be an order relation defined on the nodes. u The elements are in the tree in a given order; assume ascending order.

36 Binary Trees u Inserting an element in the Binary Tree involves: u If the tree is empty, insert the element as the root.

37 Binary Trees u If the tree is not empty: v Start at the root. v For each node decide whether the element is the same as the one at the node or comes before or after it in the defined order. v When the child is a null pointer insert the element.

38 Binary Tree root 37

39 Binary Tree root , 9, 3

40 Binary Trees root , 9, 3, 68, 14, 54 54

41 Binary Trees Delete this one

42 Binary Trees

43 Binary Trees Delete this one

44 Binary Trees Assume ascending order.

45 Binary Trees Delete this one

46 Binary Trees Assume ascending order.

47 Binary Tree u In Java: class Unit { public Unit(Object o, Unit l, Unit r) { datum = o ; left = l ; right = r ; } Object datum ; Unit left ; Unit right ; }

48 Binary Tree u Copying can be done recursively: public Object clone() { return new Unit(datum, (left != null) ? ((Unit)left).clone() : null, (right != null) ? ((Unit)right).clone() : null ) ; }

49 Binary Tree u Can take a tour around the tree, doing something at each stage: void inOrder (Function f) { if (left != null) { left.inOrder(f) ; } f.execute(this) ; if (right != null) { right.inOrder(f); } }

50 Binary Tree u Can take a different tour around the tree, doing something at each stage: void preOrder (Function f) { f.execute(this) ; if (left != null) { left.preOrder(f) ; } if (right != null) { right.preOrder(f); } }

51 Binary Tree u Can take yet another tour around the tree, doing something at each stage: void postOrder (Function f) { if (left != 0) { left.postOrder(f) ; } if (right != 0) { right.postOrder(f); } f.execute(this) ; }

52 Traversing a Binary Tree u Four sorts of route through a tree: v In-order. v Pre-order. v Post-order. v Level-order.

53 Traversing a Binary Tree u Pre-order, post-order and in-order are related since they just rearrange order of behaviour. Depth-first searches. u Level-order is different. Breadth-first search.

54 Traversing a Binary Tree root inorder: 3, 9, 14, 37, 54, 68 preorder: 37, 9, 3, 14, 68, 54 postorder: 3, 14, 9, 54, 68, 37 levelorder: 37, 9, 68, 3, 14, 54 This is a complete binary tree.

55 Searching and Sorting u A Tree is an inherently sorted data structure. u A Tree can be an index to data rather than holding data. u Searching using a Tree is much better than linear search, in fact it is a sort of binary chop search.

56 Binary Trees u Balance is important when working with Binary Trees: v Height is O(log 2 n) in the best case but O(n) in the worst case (tree becomes a linear list). v Worst case occurs when data is fed in in order. v Lookup time, insertion time and removal time are all O(log 2 n) when the tree is balanced and O(n) in the worst case (directly proportional to approximate height).

57 Problem with Binary Tree u If data is entered in sorted order, the tree becomes a list. u This degeneration loses the O(log 2 n) behaviour. u How can we get around this?

58 Problem with Binary Tree u Make the tree self-balancing.

59 AVL Tree u A binary tree that is self-modifying. u Is nearly balanced at all times. u No sub-tree is more than one level deeper than its sibling. u Adelson-Velskii and Landis were the progenitors.

60 AVL Tree u AVL trees insert data by inserting as any normal binary tree. u The tree may become unbalanced. u Thus, there is then a second stage, the tree re-balances itself if it needs to.

61 AVL Tree u When removal occurs, the tree may become unbalanced. u There is, therefore, a second stage, the tree re-balances itself if it needs to.

62 AVL Tree u AVL trees are now considered inefficient and are therefore rarely used. u Trees are, however, so important that efficiency is necessary.

63 Red-Black Tree u These trees have a different algorithm for handling the modifications. u Instead of measuring the unbalancedness of the tree, each node is coloured.

64 Red-Black Tree u Insertion does not require two phases since the tree can be re-balanced as the position of the insertion point is found. u This makes it far more efficient than the AVL tree.

65 B-Tree u Used in database systems. u Not used in memory bound systems.

66 End of this Session