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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.

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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.

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

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4 Singly-linked List head

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5 Singly-linked List head

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

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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 ;

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8 Singly-linked List head p c newElement

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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).

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

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11 Doubly-linked List tail head

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12 Doubly-linked List tail head

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

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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?

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15 Doubly-linked List tail head c newItem

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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.

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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.

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

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19 Stack pushpop isEmpty top

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20 Queue insert remove isEmpty

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21 Implementing Stack tos

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22 Implementing Queue tail head

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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.

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24 Multi-lists head

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25 Multi-lists head

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26 Multi-lists (Not Required) head

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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.

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

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

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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.

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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.

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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.

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

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34 Binary Trees This is a complete binary tree.

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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.

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

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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.

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38 Binary Tree root 37

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39 Binary Tree root , 9, 3

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

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41 Binary Trees Delete this one

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42 Binary Trees

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43 Binary Trees Delete this one

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44 Binary Trees Assume ascending order.

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45 Binary Trees Delete this one

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46 Binary Trees Assume ascending order.

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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 ; }

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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 ) ; }

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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); } }

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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); } }

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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) ; }

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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.

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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.

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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.

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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.

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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).

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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?

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

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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.

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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.

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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.

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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.

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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.

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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.

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

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66 End of this Session

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