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Queues, Stacks and Heaps. Queue. List structure using the FIFO process Nodes are removed form the front and added to the back. Front. A. B. C. D. Back. Queue. Removing a node (popping) Then adding a node (pushing) Uses include Breadth First Search and other graph-related algorithms.

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Queue
Queue

List structure using the FIFO process

Nodes are removed form the front and added to the back

Front

A

B

C

D

Back


Queue1
Queue

Removing a node (popping)

Then adding a node (pushing)

Uses include Breadth First Search and other graph-related algorithms

Front

B

C

D

Back

Front

B

C

D

A

Back


Stack
Stack

List structure using the FILO process

Nodes added to and removed from the top

Top

D

C

B

A

Bottom


Stack1
Stack

Removing a node Then adding a node

popping pushing

Top

Top

C

E

B

C

A

B

A

Bottom

Bottom


Stack2
Stack

Used in Depth First Search and other recursive algorithms


Tree basics
Tree Basics

A tree is a connected graph with no cycles

Nodes can have multiple children and at most one parent

Nodes with no children are called leaves

Topmost node called the root

Root

Parent of node

4

2

6

Child of node

8

5

7

3

A leaf


Heap

A heap is a binary tree - no more than 2 children per parent

The binary heap is complete – all levels are full with the possible exception of the last

The value of each node is greater than or equal to the values of each of its children


Heap

Properties of a heap of size n:

Height of the heap is trunc(log2n)

Root of the heap contains

the largest value

10

8

3

1

5

2


Heap

A heap can be conveniently stored in an array as such:

  • The root is stored at index 1

  • The children of node i are stored at indices 2i and 2i+1

  • The parent of node i is stored at index trunc(i/2)


Heap

A simple heap with array representation

9

8

3

6

5

2

1

4

5

Index

0

1

2

3

4

5

6

7

8

9

Value

0

9

8

3

6

5

2

1

4

5


Heap

Heap construction:

  • Read values into array

  • For each node from the last parent down to the root:

    If the node value is less than either of the children, switch the node with the greater child

    Continue until the node value is greater than or equal to both children (automatically true if it is a leaf)

  • Construction is in O(n)


Heap

Inserting a value:

  • Increment the size and add the value as the last node

  • Sift the node up the heap if it is larger than its parent until its parent is greater than it or it has become the root

  • Insertion is in O(log2n)

8

8

9

6

5

9

5

8

5

4

9

4

6

4

6


Heap

Deleting the root (when popping):

  • Change the value of the root to the value of the last node in the heap and decrement the size of the heap

  • If the node is less than either child, swap it with the larger child, repeat until it is greater than both children

  • Deletion is in O(log2n)

9

3

6

6

5

6

5

3

5

2

3

2

2


Heap

A heap can be used to sort a list of values (heapsort):

  • Heapify the list of values

  • Pop the root off and reheap

  • Repeat until the heap is empty

  • Deletion of a node is O(log2n) and this is repeated n times, so heapsort is in O(nlog2n) (this is also the worst case)

  • Heapsort can be done in-place, but it is not a stable sort


Priority queue
Priority Queue

Priority queues are queues which pop the minimum or maximum value in the queue. As the root of a heap is always the largest or smallest value in the heap, priority queues can use a heap structure.

Priority queues have important uses in:

  • Dijkstra’s Algorithm (shortest path)

  • Prim’s Algorithm (a faster alternative to Kruskal’s for a minimum spanning tree)

  • Simply finding the minimum/maximum value of a dynamic list efficiently


Problem examples
Problem Examples

Shortest path is a fairly common problem, with The Cheese Universe from the first training camp being a straight-forward example. A heap priority queue converts Dijkstra’s to O((E+V)log2n) from O(n2).

An example of a minimum spanning tree problem for which Prim’s Algorithm might be used is the Caves of Caerbannog problem from last years SACO.


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