Binary and Binomial Heaps
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
An in-depth exploration of priority queues and binary heaps. It covers various operations such as make-heap, insert, find-min, delete-min, union, decrease-key, delete, and heap sort. The document also discusses the properties and use of binomial trees and heaps, as well as their representation in a binomial heap.
Typology: Slides
2012/2013
1 / 42
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Priority Queues and Binary Heaps and more Slides Computer Science in PDF only on Docsity!
Binary and Binomial Heaps
Priority Queues
• Supports the following operations.
- Insert element x.
- Return min element.
- Return and delete minimum element.
- Decrease key of element x to k.
• Applications.
- Dijkstra's shortest path algorithm.
- Prim's MST algorithm.
- Event-driven simulation.
- Huffman encoding.
- Heapsort. -...
Priority Queues
Dijkstra/Prim
1 make-heap
|V| insert
|V| delete-min
|E| decrease-key
make-heap
Operation
insert
find-min
delete-min
union
decrease-key
delete
Binary
log N
log N
N
log N
log N
Binomial
log N
log N
log N
log N
log N
log N
Fibonacci *
log N
log N
Relaxed
log N
log N
Linked List
1 N N 1 1 N
is-empty 1 1 1 1 1
Heaps
O(|V| O(|E| log |V|) O(|E| + |V| log |V|)
2
Binary Heap: Definition
• Binary heap.
– Almost complete binary tree.
• filled on all levels, except last, where filled from left to
right
– Min-heap ordered.
• every child greater than (or equal to) parent
Binary Heaps: Array
Implementation
• Implementing binary heaps.
– Use an array: no need for explicit parent or child
pointers.
• Parent(i) = i/2
• Left(i) = 2i
• Right(i) = 2i + 1
Binary Heap: Insertion
• Insert element x into heap.
– Insert into next available slot.
– Bubble up until it's heap ordered.
• Peter principle: nodes rise to level of incompetence
83 84 996442 next free slot
Binary Heap: Insertion
• Insert element x into heap.
– Insert into next available slot.
– Bubble up until it's heap ordered.
• Peter principle: nodes rise to level of incompetence
swap with parent
Binary Heap: Insertion
• Insert element x into heap.
– Insert into next available slot.
– Bubble up until it's heap ordered.
• Peter principle: nodes rise to level of incompetence
– O(log N) operations.
stop: heap ordered
Binary Heap: Delete Min
• Delete minimum element from heap.
– Exchange root with rightmost leaf.
– Bubble root down until it's heap ordered.
• power struggle principle: better subordinate is
promoted
Binary Heap: Delete Min
• Delete minimum element from heap.
– Exchange root with rightmost leaf.
– Bubble root down until it's heap ordered.
• power struggle principle: better subordinate is
promoted
Binary Heap: Delete Min
• Delete minimum element from heap.
– Exchange root with rightmost leaf.
– Bubble root down until it's heap ordered.
• power struggle principle: better subordinate is
promoted
exchange with right child
Binary Heap: Delete Min
• Delete minimum element from heap.
– Exchange root with rightmost leaf.
– Bubble root down until it's heap ordered.
• power struggle principle: better subordinate is
promoted
– O(log N) operations.
stop: heap ordered
Binary Heap: Union
• Union.
– Combine two binary heaps H
1
and H
2