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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Singly-linked Lists
 A list in which there is a preferred direction.
 A minimally linked list.
 The item before has a pointer to the item after.
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Singly-linked List
 Implement this structure using objects and references.
3

1
Singly-linked List
4
7 head
11
4

1
Singly-linked List
4
7 head
11
5

Singly-linked List class ListElement
{
Object datum ;
ListElement nextElement ;
. . .
} datum nextElement
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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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1 p
Singly-linked List newElement
4 c
7 head
11
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Analysing Singly-linked List
 Accessing a given location is O( n ).
 Setting a given location is O( n ).
 Inserting a new item is O( n ).
 Deleting an item is O( n )
 Assuming both a head at a tail pointer, accessing, inserting or deleting can be O(1).
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Doubly-linked Lists
 A list without a preferred direction.
 The links are bidirectional: implement this with a link in both directions.
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head
Doubly-linked List tail
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Doubly-linked List head tail
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Doubly-linked List class ListElement
{
Object datum ;
ListElement nextElement ;
ListElement previousElement ; previousElement
. . .
} datum nextElement
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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 ;
Spot the deliberate mistake. What needs to be done to correct this?
newItem.nextElement = c.nextElement ; newItem.previousElement = c ; c.nextElement.previousElement = newItem ; c.nextElement = newItem ;
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head c
Doubly-linked List newItem tail
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Doubly-linked List
 Performance of doubly-linked list is formally similar to singly linked list.
 The complexity of managing two pointers makes things very much easier since we only ever need a single pointer into the list.
 Iterators and editing are made easy.
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Doubly-linked List
 Usually find the List type in a package is a doubly-linked list.
 Singly-linked list are used in other data structures.
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Stack and Queue
 Familiar with the abstractions of stack and queue.
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Stack push isEmpty pop top
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Queue isEmpty insert remove
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Implementing Stack tos
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Implementing Queue head tail
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Multi-lists
 Multi-lists are essentially the technique of embedding multiple lists into a single data structure.
 A multi-list has more than one next pointer, like a doubly linked list, but the pointers create separate lists.
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head
Multi-lists
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head
Multi-lists
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head
Multi-lists (Not Required) head
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Linked Structures
 A doubly-linked list or multi-list is a data structure with multiple pointers in each node.
 In a doubly-linked list the two pointers create bi-directional links
 In a multi-list the pointers used to make multiple link routes through the data.
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Linked Structures
 What else can we do with multiple links?
 Make them point at different data: create
Trees (and Graphs).
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Level 1
Level 2
Trees root
Level 3 height = depth = 3 node leaf node degree children parent
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Trees
 Crucial properties of Trees:
 Links only go down from parent to child.
 Each node has one and only one parent (except root which has no parent).
 There are no links up the data structure; no child to parent links.
 There are no sibling links; no links between nodes at the same level.
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Trees
 If we relax the restrictions, it is not a Tree, it is a Graph.
 A Tree is a directed, acyclic Graph that is single parent.
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Trees
 Binary Trees have degree 2.
 Red–Black Trees and AVL Trees are Binary
Trees with special extra properties; they are balanced.
 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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Binary Trees
 Trees are immensely useful for sorting and searching.
 Look at Binary Trees as they are the simplest.
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Binary Trees
This is a complete binary tree.
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Binary Trees
 How to insert something in the list?
 Need a metric, there must be an order relation defined on the nodes.
 The elements are in the tree in a given order; assume ascending order.
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Binary Trees
 Inserting an element in the Binary Tree involves:
 If the tree is empty, insert the element as the root.
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Binary Trees
 If the tree is not empty:
 Start at the root.
 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.
 When the child is a null pointer insert the element.
37

Binary Tree root
37
37
38

Binary Tree root
37
9
3
37, 9, 3
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root
Binary Trees
37
9 68
37, 9, 3, 68, 14, 54
3 14 54
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Binary Trees
Delete this one
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Binary Trees
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Binary Trees
Delete this one
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Binary Trees
Assume ascending order.
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Binary Trees
Delete this one
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Binary Trees
Assume ascending order.
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Binary Tree
 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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Binary Tree
 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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Binary Tree
 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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Binary Tree
 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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Binary Tree
 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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Traversing a Binary Tree
 Four sorts of route through a tree:
 In-order.
 Pre-order.
 Post-order.
 Level-order.
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Traversing a Binary Tree
 Pre-order, post-order and in-order are related since they just rearrange order of behaviour. Depth-first searches.
 Level-order is different. Breadth-first search.
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root
Traversing a Binary Tree
37 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
9 68
This is a complete binary tree.
3 14 54
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Searching and Sorting
 A Tree is an inherently sorted data structure.
 A Tree can be an index to data rather than holding data.
 Searching using a Tree is much better than linear search, in fact it is a sort of binary chop search.
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Binary Trees
 Balance is important when working with
Binary Trees:
 Height is O(log
2 n) in the best case but O(n) in the worst case (tree becomes a linear list).
 Worst case occurs when data is fed in in order.
 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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Problem with Binary Tree
 If data is entered in sorted order, the tree becomes a list.
 This degeneration loses the O(log
2 behaviour.
n)
 How can we get around this?
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Problem with Binary Tree
 Make the tree self-balancing.
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AVL Tree
 A binary tree that is self-modifying.
 Is nearly balanced at all times.
 No sub-tree is more than one level deeper than its sibling.
 Adelson-Velskii and Landis were the progenitors.
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AVL Tree
 AVL trees insert data by inserting as any normal binary tree.
 The tree may become unbalanced.
 Thus, there is then a second stage, the tree re-balances itself if it needs to.
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AVL Tree
 When removal occurs, the tree may become unbalanced.
 There is, therefore, a second stage, the tree re-balances itself if it needs to.
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AVL Tree
 AVL trees are now considered inefficient and are therefore rarely used.
 Trees are, however, so important that efficiency is necessary.
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Red-Black Tree
 These trees have a different algorithm for handling the modifications.
 Instead of measuring the unbalancedness of the tree, each node is coloured.
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Red-Black Tree
 Insertion does not require two phases since the tree can be re-balanced as the position of the insertion point is found.
 This makes it far more efficient than the
AVL tree.
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B-Tree
 Used in database systems.
 Not used in memory bound systems.
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End of this Session
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