(ヒープ)

http://www.sw.it.aoyama.ac.jp/2019/DA/lecture5.html

© 2009-19 Martin J. Dürst 青山学院大学

- 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`

- 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)* - 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- Most ADTs can be implemented in different ways
- Depending on implementation, the time complexity of each operation of an ADT can change

Order the following orders of growth, and explain the reason for your order:

`O`(`n`^{2}), `O`(`n`!),
`O`(`n` log log `n`), `O`(`n` log
`n`), `O`(20` ^{n}`)

Solution: `O`(`n` log log `n`) ⊂
`O`(`n` log `n`) ⊂
`O`(`n`^{2}) ⊂
`O`(20` ^{n}`) ⊂

f(n) |
g(n) |
n_{0} (example solution) |
c (example solution) |

n log log n |
n log n |
2 | 1 |

n log n |
n^{2} |
2 | 1 |

n^{2} |
20^{n} |
1 | 1 |

20^{n} |
n! |
52 | 1 |

20^{n} |
n! |
20 | 20^{20}/20! (=43099804) |

Write a simple program that uses the classes in 4ADTs.rb.

Use this program to compare the implementations.

Hint: Use the second part of 2search.rb as an
example.

Implement the *priority queue* ADT (Ruby or any other programming
language is okay)

A priority queue keeps a priority for each data item.

In the simplest case, the only data is the priority.

The items with the highest priority leave the queue first.

Your implementation can use an array or a linked list or any other data
structure.

Example solution: 5prioQ.rb

- Example from IT:
- Queue for process management, ...
- Operations:
- Creation: new, init
- Check for emptiness: empty?
- Insert additional item: add/enqueue,...
- Return and remove item with highest priority: getNext/delMax/dequeue...
- Return item with highest priority (without removal): peekAtNext/findMax/...

Implementation | Ordered array or linked list | Unordered array or linked list |

`insert` |
O(n) |
O(1) |

`getNext` /`findMax` |
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?

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

⇒ The root always has the highest priority

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

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
- Restoration of
*invariants*

- 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 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 2`i`and 2`i`+1

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 the child that has higher priority
- Continue at exchanged child until exchange becomes unnecessary

Implementation: 5heap.rb

- Insertion of a new element (
`insert`

):- Insert the new element at the end of the heap (next empty place in lowermost tree layer, or new layer if necessary)
- Restore heap invariants for newly inserted element using
`heapify_up`

- Removal of element with highest priority (
`getNext`

):- Remove the root element and store it separately
- Move the last element of the heap (rightmost element in lowermost layer) to the root
- Restore heap invariants for root element using
`heapify_down`

- Return the original root element

Implementation | `Heap` (implemented as an `Array` ) |

`insert` |
O(log n) |

`findMax` |
O(1) |

`getNext` |
O(log n) |

- Use priority queue to sort by (decreasing) priority
- Create a heap from all the items to be sorted
- 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`)

- Addition and removal of items is

`irb`

`irb`

: Interactive Ruby, a 'command prompt' for Ruby

Example usage:

C:\Algorithms>irb irb(main):001:0> load '5heap.rb' => 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> ...

Alternative to `load '5heap.rb'`

: `irb -r./5heap`

- 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)

- Started with two simple implementations:

completely ordered, completely unordered - 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

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

Caution: Implementation of heap sort directly uses the array. This is a
*layer violation*.

- 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*

- Cut the sorting cards, and bring them with you to the next lecture
- Shuffle the sorting cards, and try to find a fast way to sort them. Play against others (who is fastest?).
- Find five different applications of sorting (no need to submit)
- Implement joining two (normal) heaps (no need to submit)
- 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.

(no need to submit)

- 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
- restoration
- 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
- layer violation
- 階層侵害