Computer-Science

Priority Queues and Heaps

• 1. Priority Queues
  • Selection Sort
  • Insertion Sort
  • Priority Queue Sort
• 2. Heaps
• 3. Priority Queues and Heaps
  • Heap Sort

1. Priority Queues

Priority Queue ADT

• A priority queue stores a collection of entries
  • Each element is associated with a “key” value representing rank or weigh
• Functions
  • insert(e): inserts an element e
  • object min(): returns (but does not remove) an element with smallest key; error if empty - return O
  • removeMin(): removes the element referenced by min(); error if empty - return X
  • integer size()
  • boolean empty()
• Applications: Standby flyers, auctions, stock market

Example

Operation	Output	Priority Queue
insert(5)	-	{5}
insert(9)	-	{5, 9}
insert(2)	-	{2, 5, 9}
insert(7)	-	{2, 5, 7, 9}
min()	[2]	{2, 5, 7, 9}
removeMin()	-	{5, 7, 9}
size()	3	{5, 7, 9}
min()	[5]	{5, 7, 9}
removeMin()	-	{7, 9}
removeMin()	-	{9}
removeMin()	-	{}
empty()	true	{}
removeMin()	"error"	{}

• Each call to min() returns a reference to an entry in the queue, not the actual value
  • 값을 주지 않고, position(reference)를 준다.
• Although the “Priority Queue” column shows the items in sorted order, they are not necessarily stored in this order
  • key 값만 알면 min()과 removeMin()을 사용할 수 있기 때문에 Priority Queue는 정렬이 되어있을 필요는 없다.
    • 정렬 O: insert()에서 탐색
    • 정렬 X: min()에서 탐색

List-based Priority Queue

Implementation with an unsorted list	Implementation with a sorted list
Performance	Performance
insert takes O(1) time since we can insert the item at the beginning or end of the sequence	insert takes O(n) time since we have to find the place where to insert the item
removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key	removeMin and min take O(1) time, since the smallest key is at the beginning

Priority Queue Sort

• We can use a priority queue to sort a set of comparable elements
• Insert the elements one by one with a series of insert(e) operations
• Remove the elements in sorted order with a series of removeMin() operations

Algorithm PQ-Sort(L, P)
  Input List L, priority queue P
  Output Sorted list L
  while not L.empty()
    e ← L.front()
    L.eraseFront()
    P.insert(e)
  while not P.empty()
    e ← P.min()
    P.removeMin()
    L.insertBack(e)

Selection-Sort

• Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted list

• Running time of Selection-sort:

  • 1. Inserting the elements into the priority queue with n insert operations takes O(n) time
  • 1. Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to
       • 
• Selection-sort runs in O(n^2) time
  • 

Selection-Sort Example

	List L	Priority Queue P (unsorted list)
Input:	(7, 4, 8, 2, 5, 3, 9)	()
Phase 1
(a)	(4, 8, 2, 5, 3, 9)	(7)
(b)	(8, 2, 5, 3, 9)	(7, 4)
…	…	..
(g)	()	(7, 4, 8, 2, 5, 3, 9)
Phase 2
(a)	(2)	(7, 4, 8, 5, 3, 9)
(b)	(2, 3)	(7, 4, 8, 5, 9)
(c)	(2, 3, 4)	(7, 8, 5, 9)
(d)	(2, 3, 4, 5)	(7, 8, 9)
(e)	(2, 3, 4, 5, 7)	(8, 9)
(f)	(2, 3, 4, 5, 7, 8)	(9)
(g)	(2, 3, 4, 5, 7, 8, 9)	()

Exercise

• Run the selection-sort on the following input sequence:
  • (22, 15, 36, 44, 10, 3, 9)

Result :

	List L	Priority Queue P (unsorted list)
Input:	(22, 15, 36, 44, 10, 3, 9)	()
Phase 1
(a)	(15, 36, 44, 10, 3, 9)	(22)
(b)	(36, 44, 10, 3, 9)	(22, 15)
…	…	..
(g)	()	(22, 15, 36, 44, 10, 3, 9)
Phase 2
(a)	(3)	(22, 15, 36, 44, 10, 9)
(b)	(3, 9)	(22, 15, 36, 44, 10)
(c)	(3, 9, 10)	(22, 15, 36, 44)
(d)	(3, 9, 10, 15)	(22, 36, 44)
(e)	(3, 9, 10, 15, 22)	(36, 44)
(f)	(3, 9, 10, 15, 22, 36)	(44)
(g)	(3, 9, 10, 15, 22, 36, 44)	()

Insertion-sort

• Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted list

• Running time of Insertion-sort:

  • 1. Inserting the elements into the priority queue with n insert operations takes time proportional to
       • 
  • 1. Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time
• Insertion-sort runs in O(n^2) time
  • 

Insertion-sort Example

	List L	Priority Queue P (sorted list)
Input:	(7, 4, 8, 2, 5, 3, 9)	()
Phase 1
(a)	(4, 8, 2, 5, 3, 9)	(7)
(b)	(8, 2, 5, 3, 9)	(4, 7)
…	…	..
(g)	()	(2, 3, 4, 5, 7, 8, 9)
Phase 2
(a)	(2)	(7, 4, 8, 5, 3, 9)
(b)	(2, 3)	(7, 4, 8, 5, 9)
(c)	(2, 3, 4)	(7, 8, 5, 9)
(d)	(2, 3, 4, 5)	(7, 8, 9)
(e)	(2, 3, 4, 5, 7)	(8, 9)
(f)	(2, 3, 4, 5, 7, 8)	(9)
(g)	(2, 3, 4, 5, 7, 8, 9)	()

Exercise

• Run the insertion-sort on the following input sequence:
  • (22, 15, 36, 44, 10, 3, 9)

Result :

	List L	Priority Queue P (unsorted list)
Input:	(22, 15, 36, 44, 10, 3, 9)	()
Phase 1
(a)	(15, 36, 44, 10, 3, 9)	(22)
(b)	(36, 44, 10, 3, 9)	(15, 22)
…	…	..
(g)	()	(3, 9, 10, 15, 22, 36, 44)
Phase 2
(a)	(3)	(22, 15, 36, 44, 10, 9)
(b)	(3, 9)	(22, 15, 36, 44, 10)
(c)	(3, 9, 10)	(22, 15, 36, 44)
(d)	(3, 9, 10, 15)	(22, 36, 44)
(e)	(3, 9, 10, 15, 22)	(36, 44)
(f)	(3, 9, 10, 15, 22, 36)	(44)
(g)	(3, 9, 10, 15, 22, 36, 44)	()

2. Heaps

• A heap T is a binary tree storing keys at its nodes and satisfying the following properties:
  • Heap-Order: for every internal node v other than the root, key(v)>=key(parent(v))
    • A minimum key is always stored at the root of T
    • 부모가 우선순위가 더 높다. (형제는 상관없다.)
  • Complete Binary Tree: let h be the height of the heap
    • The height is the smallest if T is complete
    • For i = 0, ..., h-1,there are 2^i nodes at depth i
  • The last node of a heap is the right most node at the maximum depth
    • 

Height of a Heap

• Theorem: A heap storing n keys has height O(logn)
  • Proof: (we apply the complete binary tree property)
    • Let h be the height of a heap storing n keys
    • Since there are 2^i keys at depth i = 0, ..., h-1 and at least one key at depth h,we have
    • Level h has at most 2^h nodes:
    • Thus,

Vector-based Heap Implementation

• We can represent a heap with n keys by means of a vector of length n + 1
• For the node at rank i
  • the left child is at rank 2i
  • the right child is at rank 2i+1
• Links between nodes are not explicitly stored
• The cell of at rank 0 is not used
• Operation insert corresponds to inserting at rank n+1
• Operation removeMin corresponds to removing at rank n

3. Priority Queues and Heaps

• We can use a heap to implement a priority queue
• We store a (key,element) item at each internal node
• We keep track of the position of the last node

1) Insertion into a Heap

• Function insert of the priority queue ADT corresponds to the insertion of a key k to the heap

• The insertion algorithm consists of three steps

  • Find the insertion node z (the new last node)
  • Store k at z
  • Restore the heap-order property (discussed next)

2) Upheap

• After the insertion of a new key k, the heap-order property may be violated
• Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node
• Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k
• Since a heap has height O(logn), upheap runs in O(logn) time

3) Removal from a Heap

• Function removeMin of the priority queue ADT corresponds to the removal of the root key from the heap
• The removal algorithm consists of three steps
  • Replace the root key with the key of the last node w
  • Remove w
  • Restore the heap-order property (discussed next)

4) Downheap

• After replacing the root key with the key k of the last node, the heap-order property may be violated
• Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root
• Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k
• Since a heap has height O(logn), down heap runs in O(logn) time

5) Merging Two Heaps

• We are given two heaps and a key k
• We create a new heap with the root node storing k and with the two heaps as subtrees
• We perform down heap to restore the heap-order property

Heap-Sort

• Consider a priority queue with n items implemented by means of a heap
  • The space used is O(n)
  • Functions insert and removeMin take O(logn) time
  • Functions size, empty, and min take time O(1) time
• Using a heap-based priority queue, we can sort a sequence of n elements in O(nlogn) time
  • The resulting algorithm is called heap-sort
  • Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort

Priority Queue Summary

Bottom-up Heap Construction

• We can construct a heap storing n given keys in using a bottom-up construction with log n phases
• In phase i, pairs of heaps with 2^i-1 keys are merged into heaps with 2^(i+1)-1 keys

Example

Analysis

• We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path)
• Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n)
• Thus, bottom-up heap construction runs in O(n) time
• Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort

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