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Chapter 5 – Part 3: Heaps
• A heap is a binary tree structure such that:
– The tree is complete or nearly complete.
– The key value of each node is >= the key value of
each of its descendents (max-heap).
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Heap Properties
• Completeness:
Complete tree:
N = 2H - 1 (last level is full)
Nearly complete: H = [log2N] + 1
All nodes in the last level are on the left
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Heap Properties
• Key value order:
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Insert Heap
• Insert the new node into the bottom level,
as far left as possible
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• and then reheap-up
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Basic Heap Operations
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Reheapup
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Delete Heap
• replace the deleted value by the value X of the last leaf
(rightmost in the last level)
• delete the last leaf
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• and then reheap-up, if X is  its parent, or reheap-down,
if X is < either of its children
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Basic Heap Operations
swap with the larger of its
children
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swap with the larger of its
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Reheapdown
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Heap Data Structure
• Node i: left child 2i + 1, right 2i + 2
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• Node i: parent [(i - 1)/2]
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• Left child j: right sibling j + 1
• Right child j: left sibling j - 1
• Heap size n: first leaf [n/2]
• First leaf k: last nonleaf k - 1
[0] [1] [2] [3] [4] [5] [6]
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ReheapUp Algorithm
Algorithm
reheapUp (ref heap <array>, val newNode <index>)
Reestablishes heap by moving data in child up to its correct location
Pre
Post
heap is an array containing an invalid heap
newNode is index location to new data in heap
Heap has been restored
1 if (newNode not 0)
1 parent = (newNode - 1)/2
2 if (heap[newNode].key > heap[parent].key)
1 swap (newNode, parent)
2 reheapup (heap, parent)
2 return
End
reheapUp
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ReheapDown Algorithm
Algorithm
reheapDown (ref heap <array>,
val root <index>, val last <index>)
Reestablishes heap by moving data in root down to its correct location
Pre
heap is an array containing an invalid heap
root is root of heap
last is an index to the last element in heap
Post
Heap has been restored
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ReheapDown Algorithm
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if (root*2 + 1 <= last)
1 leftKey = heap[root*2 + 1].data.key
2 if (root*2 + 2 <= last)
1 rightKey = heap[root*2 + 2].data.key
3 else
1 rightKey = lowKey
4 if (leftKey > rightKey)
1 largeChildKey = leftKey
2 largeChildIndex = root*2 + 1
5 else
1 largeChildKey = rightKey
2 largeChildIndex = root*2 + 2
6 if (heap[root].data.key < heap[largerChildIndex].data.key)
1 swap (root, largerChildIndex)
2 reheapDown (heap, largerChildIndex, last)
return
End
reheapDown
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Build Heap
Algorithm
buildHeap (ref heap <array>, val size <integer>)
Given an array, rearrange data so that it forms a heap
Pre
Post
heap is an array containing an invalid heap
size is number of elements in array
array is now a heap
1 walker = 1
2 loop (walker <= size)
1 reheapUp (heap, walker)
2 walker = walker + 1
3 return
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End
[0] [1] [2] [3] [4] [5] [6]
buildHeap
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Insert Heap
Algorithm
insertHeap (ref heap <array>, ref last <index>
val data <data type>)
Inserts data into heap
Pre
Post
Return
heap is a valid heap
last is index to last node in heap
data have been inserted into heap
true if successful; false if array is full
1 if (heap full)
1 return false
2 last = last + 1
3 heap[last] = data
4 reheapUp (heap, last)
5 return true
End
insertHeap
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Delete Heap
Algorithm
deleteHeap (ref heap <array>, ref last <index>
ref dataOut <data type>)
Deletes root of heap and passes data back to caller
Pre
Post
Return
heap is a valid heap
last is index to last node in heap
dataOut is reference parameter for output data
root has been deleted from heap
true if successful; false if array is empty
1 if (heap empty)
1 return false
2 dataOut = heap[0]
3 heap[0] = heap[last]
4 last = last - 1
5 reheapDown (heap, 0, last)
6 return true
End
deleteHeap
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Complexity of Heap Operations
• ReheapUp: O(log2n)
• ReheapDown: O(log2n)
• BuildHeap: O(nlog2n)
• InsertHeap: O(log2n)
• DeleteHeap: O(log2n)
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Heap Applications
• Selection Algorithms: to determine the kth
largest element in an unsorted list
– Creat a heap
– Delete k - 1 elements from the heap
– The desired element will be at the top
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Heap Applications
• Priority Queues: each element has a
priority to be dequeued
– Use a heap to represent a queue
– Elements are enqueued according to their priorities
– Top element is dequeued first
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