cisc 235: topic 7 priority queues and binary heaps

CISC 235: Topic 7

Priority Queues and Binary Heaps

CISC 235 Topic 7 2

Outline

• Priority Queues• Binary Heaps

– Ordering Properties– Structural Property

• Array Representation• Algorithms and Analysis of Complexity

• insert• minimin• extractMin• buildHeap

CISC 235 Topic 7 3

Priority Queues

A queue that is ordered according to some priority value

Standard Queue Priority Queue

enqueue dequeue enqueue dequeueAdd at Remove Add Remove end from front according from

frontto priorityvalue

CISC 235 Topic 7 4

Example Applications

Line-up of Incoming Planes at AirportPossible Criteria for Priority?

Operating Systems Priority Queues?

Several criteria could be mapped to a priority status

CISC 235 Topic 7 5

Min-Priority Queue Operations

insert( S, x ) – Inserts element x into set S, according to its priority

minimum( S ) – Returns, but does not remove, element of S with the smallest key

extractMin( S ) – Removes and returns the element of S with the smallest key

CISC 235 Topic 7 6

Possible Implementations?

CISC 235 Topic 7 7

Binary Heaps

A binary tree with an ordering property and a structural property

Ordering Property: Min Heap– The element in the root is less than or equal to all elements in

both its sub-trees– Both of its sub-trees are Min Heaps

Ordering Property: Max Heap– The element in the root is greater than or equal to all elements in

both its sub-trees– Both of its sub-trees are Max Heaps

CISC 235 Topic 7 8

Binary Heap Structural Property

A binary heap is a binary tree that is completely filled, with the possible exception of the bottom level, which is filled from left to right with no missing nodes.

CISC 235 Topic 7 9

Which Tree is a Min Heap?

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CISC 235 Topic 7 10

A Min Heap and its Array Representation

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Note: Only the array exists in memory

heap

CISC 235 Topic 7 11

Locating Parents & Children

parent( i )

return i / 2

left( i )

return 2i

right( i )

return 2i + 1

CISC 235 Topic 7 12

Min Heap Operations: Insert

1. Find the next position at which to insert – after the “last” element in the heap – and create a “hole” at that postion

2. “Bubble” the hole up the tree by moving its parent element into it until the hole is at a position where the new element ≤ its children and ≥ its parent:

the percolateUp() operation3. Insert the new element in the hole

CISC 235 Topic 7 13

Insert Step 1: Create hole at next insert position

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Insert 14

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heap

CISC 235 Topic 7 14

Insert Step 2: Bubble hole up to where element ≤ its children and ≥ its parent

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Insert 14

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heap

CISC 235 Topic 7 15

Insert Step 2: Bubble hole up to where element ≤ its children and ≥ its parent

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Insert 14

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heap

CISC 235 Topic 7 16

Insert Step 3: Insert the new element in the hole

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Insert 14

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CISC 235 Topic 7 17

Analysis of Insert?

Worst-case complexity?

CISC 235 Topic 7 18

Min Heap Operations: MinimumReturn minimum at top of heap: O(1)

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heap

return heap[1];

CISC 235 Topic 7 19

Min Heap Operations: ExtractMin

1. Copy the root (the minimum) to a temporary variable, thus creating a “hole” at the root.

2. Delete the last item in the heap by copying its element to a temporary location.

3. Find a place for that element by bubbling the hole down by moving its smallest child up until the element ≤ its children and ≥ its parent:

the percolateDown() operation.

4. Place the former last element in that hole and return the former root.

CISC 235 Topic 7 20

ExtractMin Step 1:Copy the root to a temporary variable

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minimum

heap

minimum = heap[1];

CISC 235 Topic 7 21

ExtractMin Step 1:thus creating a “hole” at the root

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minimum

heap

minimum = heap[1];

CISC 235 Topic 7 22

ExtractMin Step 2: Delete last item in heap by copying it to a temporary location

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minimum

heap

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temp

temp = heap[10];

CISC 235 Topic 7 23

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minimum

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temp

ExtractMin Step 2: Delete last item in heap by copying it to a temporary location

CISC 235 Topic 7 24

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minimum

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temp

ExtractMin Step 3: Bubble hole down until temp ≤ its children and ≥ its parent

CISC 235 Topic 7 25

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minimum

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temp

ExtractMin Step 3: Bubble hole down until temp ≤ its children and ≥ its parent

CISC 235 Topic 7 26

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minimum

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temp

ExtractMin Step 4: Copy temp to the hole and return the minimum

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heap[6] = temp;

return minimum;

CISC 235 Topic 7 27

Analysis of ExtractMin?

Worst-case complexity?

CISC 235 Topic 7 28

Insert n Elements into an Initially Empty Heap

Worst-case complexity?

CISC 235 Topic 7 29

Bottom-Up Heap Construction: BuildHeap

If all the keys are given in advance and stored in an array, we can build a heap in worst-case O(n) time.

Note: For simplicity, we’ll describe bottom-up heap construction for a full tree

Algorithm: Call percolateDown() on nodes in non-heap tree in reverse level-order traversal to convert tree to a heap.

CISC 235 Topic 7 30

BuildHeap Algorithm

1. Build (n+1)/2 elementary heaps, with one key each, of the leaves of the tree (no work)

2. Build (n+1)/4 heaps, each with 3 keys, by joining pairs of elementary heaps with their parent as the root. Call percolateDown()

3. Build (n+1)/8 heaps, each with 7 keys, by joining pairs of 3-key heaps with their parent as the root. Call percolateDown()

4. Continue until reach root of entire tree.

CISC 235 Topic 7 31

BuildHeap at Start: Elements are Unordered: Not in Heap Order

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CISC 235 Topic 7 32

BuildHeap Step 1: Build (n+1)/2 elementary heaps, with one key each

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CISC 235 Topic 7 33

BuildHeap Step 2: Build (n+1)/4 heaps, with 3 keys each

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CISC 235 Topic 7 34

BuildHeap Step 2: Call percolateDown( ) for each 3-key heap

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CISC 235 Topic 7 35

BuildHeap Step 2: Call percolateDown( ) for each 3-key heap

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CISC 235 Topic 7 36

BuildHeap Step 2: Call percolateDown( ) for each 3-key heap

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CISC 235 Topic 7 37

BuildHeap Step 2: Call percolateDown( ) for each 3-key heap

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CISC 235 Topic 7 38

BuildHeap Step 2: Now we have (n+1)/4 heaps, with 3 keys each

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CISC 235 Topic 7 39

BuildHeap Step 3: Build (n+1)/8 heaps, with 7 keys each

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CISC 235 Topic 7 40

BuildHeap Step 3: Call percolateDown( ) for each 7-key heap

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CISC 235 Topic 7 41

BuildHeap Step 3: Call percolateDown( ) for each 7-key heap

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CISC 235 Topic 7 42

BuildHeap Step 3: Call percolateDown( ) for each 7-key heap

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CISC 235 Topic 7 43

BuildHeap Step 3: Call percolateDown( ) for each 7-key heap

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CISC 235 Topic 7 44

BuildHeap Step 3: Call percolateDown( ) for each 7-key heap

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CISC 235 Topic 7 45

BuildHeap Step 3: Now we have (n+1)/8 heaps, with 7 keys each

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CISC 235 Topic 7 46

BuildHeap Step 4: Build (n+1)/16 heaps, with 15 keys each (only one here)

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CISC 235 Topic 7 47

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 48

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 49

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 50

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 51

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 52

BuildHeap Step 4: Call percolateDown( ) for each 15-key heap

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CISC 235 Topic 7 53

BuildHeap Step 4: The Entire Tree is Now a Heap!

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CISC 235 Topic 7 54

Heapsort

How could we use a heap to sort an array?