Chapter 9. Heaps

Internet Computing Laboratory @ KUT

Youn-Hee Han

1. Basic ConceptsWhere We Are?

Tree Binary Tree Binary Search Tree AVL Search Tree

Heap

힙 (Heap): 수북하게 쌓아

올린 더미

1. Basic ConceptsHeap: a binary tree structure with the properties …

The tree is complete or nearly complete The key value of each node is greater than or equal to the key

value in each of its descendents Root 의 키가 왼쪽자식과 오른쪽 자식의 키보다 크거나 같다 .

The subtrees are in turn heaps

주의할 특징 왼쪽 자식과 오른쪽 자식 사이에는 어느 쪽 키가 크던 상관이 없다 .

1. Basic ConceptsHeap 의 속성

A heap is a special kind of tree. It has two properties that are not generally true for other trees:[Completeness] - The tree is (nearly) complete, which means that nodes are added from top to bottom, left to right, without leaving any spaces. [Heapness] - The item in the tree with the highest priority is at the top of the tree, and the same is true for every subtree.

(nearly)

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1. Basic Concepts

Max Heap ( 일반적 의미의 Heap) 키 값이 큰 레코드가 우선순위가 높은 것으로 간주 루트노드의 키가 가장 크다 .

Min Heap Max Heap 과 정 반대 키 값이 작을수록 우선순위가 높다 . 예 ] 내신등급

1. Basic ConceptsExamples of Heaps

Invalid heaps (Why?)

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1. Basic Concepts

정렬 관점에서… BST 는 약한 의미로 정렬 .

왼쪽 자식 키보다 오른쪽 자식 키가 크다 . Heap 도 약한 의미로 정렬 .

부모의 키가 자식의 키보다 우선 순위가 높다 왼쪽 자식 키와 오른쪽 자식 키는 무관하다 .

Heap Binary search tree

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2. Maintenance Operations (p. 391)

Although Heap is a tree, it is meaningless to traverse it, search it, or print it out.

Important operations Insertion Deletion

Problem: preserving conditions of max heap 1st condition: Completeness 2nd condition: Heapness

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2. Maintenance Operations

Inserting a node into heap1. Insert the node to heap

To preserve 1st condition, the new node should be added at the last leaf level at the first empty position

2nd condition could be broken

2. Repair the structure so that 2nd condition is satisfied Reheap Up

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2. Maintenance Operations

Inserting a node into heap 효율

O(lgN) 삭제보다 비교 횟수가 적음

[ 힙의 삽입 ]“ 신입사원이 오면 일단 말단 자리에 앉힌 다음 , 능력껏 위로 진급시키는 것 ( 進級 , Reheap Up, Promotion)”

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2. Maintenance Operations

Deleting a node from heap1. Deletion always occurs at the root

To preserve 1st condition, the last node should move to the root

2nd condition could be broken

2. Repair the structure so that 2nd condition is satisfied Reheap Down

[ 힙의 삭제 ]“ 사장자리가 비면 말단 사원을 그 자리에 앉힌 다음 , 바로 아래 부하직원보다는 능력이 좋다고 판단될 때까지 강등시키는 것 ( 降等 , Reheap Down, Demotion)”

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2. Maintenance Operations

Deleting a node from heap 우선순위가 가장 큰

루트노드를 삭제 Completeness 를 유지

하기 위해서는 루트의 삭제와 동시에 마지막 요소를 루트노드 위치로 옮김 .

Reheap Down 힙 모습의 복원 (Heap Rebuilding) Reheap ! 루트로부터 시작해서 제자리를 찾기까지 굴러 떨어짐 왼쪽 , 오른쪽 자식 모두를 비교해서 더 큰 것과 스와핑 (Swaping)

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2. Maintenance Operations

Deleting a node from heap 효율

마지막 원소를 루트로 복사하는 데 O(1) 최악의 경우 루트로부터 리프까지 굴러 떨어짐 .. 비교의 횟수는 총 2lgN 이 된다 . 삭제 효율은 O(1) + O(2lgN) ≈ O(lgN)

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2. Maintenance Operations

BST vs. Heap 삽입 , 삭제 , 탐색의 관점에서 Heap 이 BST 보다 더 좋은

구조임 . BST 는 최악의 경우 연결 리스트와 유사 O(N) 의 효율

Skewed Binary Tree ( 편향된 이진 트리 ) 힙은 완전 이진트리로서 균형 트리 균형 트리의 높이는 항상 lg(N) 에 가까움

탐색은 주로 가장 큰 (or 작은 ) 값의노드를 찾는 작업

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3. Heap Implementation

Heap 구현 배열로 표시하는 것이 효율적

Why? Heap is complete or nearly complete!!! Parent 와 Child 사이 , Siblings 사이에 배열에서의 그 위치에 대한 규칙

형성 배열에 위치하는 순서

루트부터 시작해서 위에서 아래로 , 왼쪽에서 오른쪽으로 진행 Root 노드는 항상 배열 인덱스 0.

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3. Heap Implementation

Heap 구현 (n: 노드의 총 개수 ) LeftChild(i) is at 2i + 1 RightChild(i) is at 2i + 2 Parent(i) is at (i-1)/2

If i = 0, i is at the root and has no parent.

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3. Heap Implementation

Heap 구현 (n: 노드의 총 개수 ) Right Sibling(j) is at j + 1 Left Sibling(k) is at k - 1 The location of the first leaf is at n/2 The location of the last non-leaf is at n/2 - 1

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3. Heap Implementation

Insert Heap

Algorithm insertHeap (heap, last, data)// heap: array of valid heap// last: index to the last node in heap// data: data to be inserted1. if (heap full) 1. return false2. end if3. last++4. move data to last node5. reheapUp (heap, last)6. Return true

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3. Heap Implementation

Reheap Up

Algorithm reheapUp (heap, newNode)// heap: array containing an invalid heap// newNode: index location to new data in heap1. if (newNode not the root) 1. set parent to parent of newNode 2. if (newNode Key > parent Key) 1. exchange newNode and parent 2. reheapUp (heap, parent) 3. end if2. end if

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3. Heap Implementation

Heap Build

Algorithm buildHeap (heap, size)// heap: array containing an invalid heap// size: number of elements in array1. Set walker to 12. loop (walker < size) 1. reheapUp (heap, walker) 2. walker++3. end loop

Building a heap from unsorted array

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3. Heap Implementation

Delete Heap Node

Algorithm deleteHeap (heap, last, dataOut)// heap: array of valid heap// last: index to the last node in heap// dataOut: reference for output area1. if (heap empty) 1. return false2. end if3. set dataOut to root data4. move last data to root5. reheapDown (heap, 0)6. Return true

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3. Heap Implementation

Reheap Down

Algorithm reheapDown (heap, root)// heap: array of data// root: root of heap or subheap1. if (there is a left subtree) 1. set leftkey to left subtree key 2. if (there is a right subtree) 1. set rightKey to tight subtree key 2. if (leftKey > rightKey) 1. set largeSubtree to left subtree 3. else 1. set largeSubtree to right subtree 3. else 1. set largeSubtree to left subtree 4. if (root key < largeSubtree key) 1. exchange root and largeSubtree 2. reheapDown (heap, largeSubtree) 5. end if2. end if

4545

6767

5656

3232

88 2323 1919

45 67 32 56 8 32 23 19

reheapDown

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4. Heap Application - Selecting Kth Element

Selecting Kth Element in an unsorted list First solution

Sort the list & Select the element at location k Second solution

USE HEAP!!!

How to execute the second solution? 1. Heap Creation from an unordered list

크기 순으로 4 번째 엘리먼트는 ?

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How to execute the second solution? 2. delete k – 1 elements from heap 3. place the deleted element at the end of the heap and

reduce the heap size by 1 4. After k – 1 elements have been deleted and moved, the top

element (index 0 in array) is the kth element

5. reheap to restore the heap so that we are ready for another selection

찾고자 하는 4 번째 엘리먼트

4. Heap Application - Selecting Kth Element

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What is Priority Queue? 응급실에서 환자 치료의 예

큐 : 먼저 온 사람을 먼저 치료 스택 : 나중에 온 사람을 먼저 치료 우선순위 큐 (Priority Queue) : 우선순위가 높은 사람 ( 예 : 위급한 사람 ) 을 먼저 치료

Priority Queue

5. Heap Application – Priority Queue

an abstract data type supporting the following three operations:

- add an element to the queue with an associated priority - Remove and return the element from the queue that has the highest priority- (optionally) peek at the element with highest priority without removing it

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Priority Queue vs. Stack/Queue 시간 : 스택 , 큐

스택 , 큐에서는 시간에 따라 자료구조를 조직화 시간을 포함한 여러 가지 가치를 우선순위로 가지는 자료구조

우선순위 큐 우선 순위 큐는 각 노드에 “ 우선순위 값” 필드가 필요 .

우선순위 큐는 스택이나 큐 보다 일반적인 구조큐나 스택은 우선순위 큐의 특수한 형태로서 시간에 그 우선순위를 부여한 것임

따라서 키 필드가 불필요

5. Heap Application – Priority Queue

Priority Queue

Stack Queue

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Heap is an excellent structure to implement priority queue

5. Heap Application – Priority Queue

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Priority Queue Example An event is represented by

Priority number (1 ~ 5) Serial number (0 ~ 999)

5. Heap Application – Priority Queue

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Priority Queue Example

5. Heap Application – Priority Queue

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6. Heap Efficiency

Heap Efficiency Heap is efficient for extraction of the largest (or

smallest) element. Heap is efficient for both insertion and deletion Heap is a compromise of sorted and unsorted structures

Sorted (array or linked-list) structures Very efficient in extraction of largest (smallest)

element Inefficient in insertion

Unsorted (array or linked-list) structures Inefficient in extraction of largest (smallest) element Very efficient in insertion

Heap Efficiency

Representation Insertion Deletion

Unordered array O(1) O(n)

Unordered linked list O(1) O(n)

Sorted array O(n) O(1)

Sorted linked list O(n) O(1)

Heap O(log n) O(log n)

6. Heap Efficiency

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