1 Linked Lists A linked list is a sequence in which there is a defined order as with any sequence...

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Linked ListsLinked 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 ListsSingly-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 ListSingly-linked List

Implement this structure using objects and references.

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11

7

4

1

head

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11

7

4

1

head

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class ListElement

{

Object datum ;

ListElement nextElement ;

. . .

}

datum nextElement

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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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11

7

4

1

head

pc

newElement

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Analysing Singly-linked ListAnalysing 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 ListsDoubly-linked Lists

A list without a preferred direction. The links are bidirectional: implement this

with a link in both directions.

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Doubly-linked ListDoubly-linked List

tail

head

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tail

head

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class ListElement

{

Object datum ;

ListElement nextElement ;

ListElement previousElement ;

. . .

}

datum nextElement

previousElement

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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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tail

head

c

newItem

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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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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 QueueStack and Queue

Familiar with the abstractions of stack and queue.

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StackStack

push pop

isEmpty

top

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QueueQueue

insert remove

isEmpty

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Implementing StackImplementing Stack

tos

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Implementing QueueImplementing Queue

tail

head

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Multi-listsMulti-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

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head

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Multi-lists (Not Required)Multi-lists (Not Required)

headhead

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Linked StructuresLinked 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 StructuresLinked Structures

What else can we do with multiple links? Make them point at different data: create

Trees (and Graphs).

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TreesTrees

node

leaf node

degree

root children

parentLevel 1

Level 2

Level 3

height = depth = 3

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TreesTrees

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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TreesTrees

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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TreesTrees

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 TreesBinary Trees

Trees are immensely useful for sorting and searching.

Look at Binary Trees as they are the simplest.

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This is a complete binary tree.

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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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Inserting an element in the Binary Tree involves:

If the tree is empty, insert the element as the root.

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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.

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Binary TreeBinary Tree

root

37

37

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root

3

9

37

37, 9, 3

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root

3 14

9

37

68

37, 9, 3, 68, 14, 54

54

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Delete this one

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Delete this one

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Assume ascendingorder.

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Delete this one

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Assume ascendingorder.

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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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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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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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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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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 TreeTraversing a Binary Tree

Four sorts of route through a tree: In-order. Pre-order. Post-order. Level-order.

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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

3 14

9

37

68

54

inorder: 3, 9, 14, 37, 54, 68preorder: 37, 9, 3, 14, 68, 54postorder: 3, 14, 9, 54, 68, 37levelorder: 37, 9, 68, 3, 14, 54

This is a complete binary tree.

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Searching and SortingSearching 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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Balance is important when working with Binary Trees: Height is O(log2 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(log2 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 TreeProblem with Binary Tree

If data is entered in sorted order, the tree becomes a list.

This degeneration loses the O(log2 n) behaviour.

How can we get around this?

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Problem with Binary TreeProblem with Binary Tree

Make the tree self-balancing.

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AVL TreeAVL 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 TreeAVL 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 TreeAVL 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 TreeAVL 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 TreeRed-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 TreeRed-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-TreeB-Tree

Used in database systems. Not used in memory bound systems.

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1 Linked Lists A linked list is a sequence in which there is a defined order as with any sequence...

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