Heaps

(ヒープ)

Data Structures and Algorithms

5th lecture, October 27, 2016

http://www.sw.it.aoyama.ac.jp/2016/DA/lecture5.html

Martin J. Dürst

Today's Schedule

Summary of last lecture, homework
Priority queue as an ADT
Efficient implementation of priority queue
Complete binary tree
Heap
Heap sort
How to use irb

Summary of Last Lecture

The order (of growth)/(asymptotic) time complexity of an algorithm can be calculated from the number of the most frequent basic operations
Calculation can use a summation or a recurrence (relation)
The big-O notation compactly express the inherent efficiency of an algorithm
An abstract data type (ADT) combines data and the operations on this data
Stack and queue are typical examples of ADTs
Each ADT can be implemented in different ways
Depending on implementation, the time complexity of each operation of an ADT can change

Priority Queue

Example from IT:

Queue for process management, ...

Operations:

Creation: new, init
Check for emptiness: empty?
Insert additional item: add,...
Return and remove item with highest priority: getNext/delMax/...
Return item with highest priority (without removal): findMax/peekAtNext/...

Simple Implementations

Time complexity of each operation
Implementation	Array (ordered)	Array (unordered)	Linked list (ordered)	Linked list (unordered)
`insert`	`O`(`n`)	`O`(1)	`O`(`n`)	`O`(1)
`getNext`	`O`(1)	`O`(`n`)	`O`(1)	`O`(`n`)
`findMax`	`O`(1)	`O`(`n`)	`O`(1)	`O`(`n`)

Time complexity for each operation differs for different implementations.

But there is always an operation that needs O(n).

Is it possible to improve this?

Ideas for Improving Priority Queue Implementation

Tree Structure

Problem: Balance

Complete Binary Tree

Definition based on tree structure:

Allmost all internal nodes (except maybe one node) have have 2 children
All tree layers except the lowermost are full
The lowermost tree layer is filled from the left

Implementing a Complete Binary Tree with an Array

Implementing a tree by allocating individual nodes and connecting them with pointers is complicated
Compared to this, operations on an array are simple
A complete binary tree can be implemented with an array as follows:
(this is also how Knuth defines a complete binary tree)
Give each node in the complete binary tree with n nodes a number so that:
- Number 0 stays unused
- Each node has a number between 1 and n (inclusive)
- The root has number 1
- The number of the parent of node i is ⌊i/2⌋ (i>1)
- The numbers of the children of note i are 2i and 2i+1

Heap

A heap is

A complete binary tree where
Each parent always has higher priority than its children

⇒ The root always has the highest priority

We need the following operations for implementing a heap:

Addition and removal of data items
Restauration of invariants

Invariant

A condition that is always maintained in a data structure or algorithm (especially loop)
Very important for data structures
Can be used in proofs (properties of data structures, correctness of algorithms, ...)
After an operation on (change to) a data structure, it may be necessary to restore invariants

Restoring Heap Invariants

If the priority at a given node is too high: Use heapify_up

Compare priority with parent
If parent priority is lower, exchange with parent
Continue until parent priority is higher

If the priority at a given node is too low: Use heapify_down

Compare priority with both children
If necessary, exchange with child with higher priority
Continue at exchanged child until exchange becomes unnecessary

Implementation: 5heap.rb

Implementing a Priority Queue with a Heap

Insertion of a new element (insert):
1. Insert the new element at the end of the heap (next empty place in lowermost tree layer, or new layer if necessary)
2. Restore heap invariants for newly inserted element using heapify_up
Removal of element with highest priority (getNext):
1. Remove the root element and store it separately
2. Move the last element of the heap (rightmost element in lowermost layer) to the root
3. Restore heap invariants for root element using heapify_down
4. Return the original root element

Time Complexity of Heap Operations

Time complexity of each operation
Implementation	`Heap` (implemented as an `Array`)
`insert`	`O`(log `n`)
`findMax`	`O`(1)
`getNext`	`O`(log `n`)

Heap Sort

Use priority queue to sort by (decreasing) priority
1. Create a heap from all the items to be sorted
2. Remove items from heap one-by-one: They will be ordered by (decreasing) priority
Implementation optimization:
Use space at the end of the array to store removed items
⇒ The items will end up in the array in increasing order
Time complexity: O(n log n)
- Addition and removal of items is O(log n) for each item
- To sort n items, the total complexity is O(n log n)

How to use `irb`

irb: Interactive Ruby, a 'command prompt' for Ruby

Example usage:

C:\Algorithms>irb
irb(main):001:0> load './5heap'
=> true
irb(main):002:0> h = Heap.new
=> #<Heap:0x2833d60 @array=[nil], @size=0>
irb(main):003:0> h.insert 3
=> #<Heap:0x2833d60 @array=[nil, 3], @size=1>
irb(main):004:0> h.insert(5).insert(7)
=> #<Heap:0x2833d60 @array=[nil, 7, 3, 5], @size=3>
 ...

Other Kinds of Heaps

Priority queues can be used as components in many different algorithms
Often, two priority queues need to be joined
With the 'usual' heap, joining is O(n)
With a binomial queue, joining is O(log n)
With a Fibonacci heap, joining can be improved to O(1)

Ideas to Improve Implementation of Priority Queue

Started with two simple implementations
Advantages and disadvantages for each implementation
New idea: Combining both implementations/finding a balance between the two implementations
Not completely ordered, but also not completely unordered
→ Partially ordered, just to the extent necessary to find highest priority item

Conceptual Layers

Application: Heap sort
ADT: Priority queue
Conceptual data structure: Heap
Actual data structure: Complete binary tree
Internal implementation: Array

Summary

A priority queue is an important ADT
Implementing a priority queue with an array or a linked list is not efficient
In a heap, each parent has higher priority than its children
In a heap, the highest priority item is at the root of a complete binary tree
A heap is an efficient implementation of a priority queue
Many data structures are defined using invariants
A heap can be used for sorting, using heap sort

Report: Manual Sorting

Deadline: November 9, 2016 (Wednesday), 19:00.

Where to submit: Box in front of room O-529 (building O, 5th floor)

Format:

A4, double-sided 4 pages (2 sheets of paper, stapled in upper left corner; NO cover page)
Easily readable handwriting (NO printouts)
Name (kanji and kana), student number, course name and report name at the top of the front page

Problem: Propose and describe an algorithm/algorithms for manual sorting, for the following two cases:

One person sorts 5000 pages
20 people together sort 80000 pages

Each page is a sheet of paper of size A4, where a 10-digit number is printed in big letters.

The goal is to sort the pages by increasing number. There is no knowledge about how the distribution of the numbers.

You can use the same algorithm for both cases, or a different algorithm.

Details:

Describe the algorithm(s) in detail, so that e.g. your friends who don't understand computers can execute them.
Describe the equipment/space that you need.
Calculate the overall time needed for each of both cases.
Analyse the time complexity (O()) of the algorithm(s).
Comment on the relationship to other algorithms you know, and on the special needs of manual (as opposed to computer) execution.
If you use any Web pages, books, ..., list them as references at the end of your report
Caution: Use IRIs (e.g. http://ja.wikipedia.org/wiki/情報), not URLs (e.g. http://ja.wikipedia.org/wiki/%E6%83%85%E5%A0%B1)

Additional Homework

(for next week, no need to submit, but bring the sorting cards)

Cut the sorting cards, shuffle them, and try to find a fast way to sort them. Play against others (who is fastest?).
(Example: Two players, one player uses selection sort, one player uses insertion sort, who wins?)
Implement joining two (normal) heaps.
Think about the time complexity of creating a heap:
heapify_down will be called n/2 times and may take up to O(log n) each time.
Therefore, one guess for the overall time complexity is O(n log n).
However, this upper bound can be improved by careful analysis.
Find five different applications of sorting.

Glossary

priority queue: 順位キュー、優先順位キュー、優先順位付き待ち行列
complete binary tree: 完全二分木
heap: ヒープ
internal node: 内部節
restauration: 修復
invariant: 不変条件
sort: 整列、ソート
decreasing (order): 降順
increasing (order): 昇順
join: 合併
binomial queue/heap: 2項キュー、2 項ヒープ
distribution: 分布