Heaps

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Data Structures and Algorithms

5th lecture, October 19, 2017

http://www.sw.it.aoyama.ac.jp/2017/DA/lecture5.html

Martin J. Dürst

© 2009-17 Martin J. Dürst 青山学院大学

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

Last Week's Homework 1

Last Week's Homework 2

Last Week's Homework 3

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 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

Heap

A heap is a binary tree where each parent always has higher priority than its children

⇒ The root always has the highest priority

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

Heap

A heap is (full definition):

• 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

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

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

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_many [5, 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

Homework

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

1. 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?)
2. Find five different applications of sorting.
3. Implement joining two (normal) heaps.
4. 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.
5. Continue to work on the report (manual 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