Sorting

1
Sorting

• Dr. Yingwu Zhu

2
Heaps

• A heap is a binary tree with properties
• It is complete
• Each level of tree completely filled
• Except possibly bottom level (nodes in left most
  positions)
• It satisfies heap-order property
• Data in each node gt data in children

3
Heaps

• Which of the following are heaps?

C
B
A
4
Heaps

• Maxheap? by default
• Minheap?

5
Implementing a Heap

• What data structure is good for its
  implementation?

6
Implementing a Heap

• Use an array or vector, why?
• Number the nodes from top to bottom
• Number nodes on each row from left to right
• Store data in ith node in ith location of array
  (vector)

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Implementing a Heap

• Note the placement of the nodes in the array

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Implementing a Heap

• In an array implementation children of ith node
  are at myArray2i and myArray2i1
• Parent of the ith node is at mayArrayi/2
• How about in C, position starting from 0?

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Basic Heap Operations

• Constructor
• Set mySize to 0, allocate array
• Empty
• Check value of mySize
• Retrieve max item
• Return root of the binary tree, myArray1

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Basic Heap Operations

• Delete max item
• Max item is the root, replace with last node in
  tree
• Then interchange root with larger of two children
• Continue this with the resulting sub-tree(s)
• Semiheap 1 complete 2 both subtrees are heaps

11
Implementing Heap

• define CAP 10000
• template lttypename Tgt
• class Heap private T myArrayCAP
• int mySize
• private
• void percolate_down(int pos) //percolate
  down
• void percolate_up(int pos) //percolate up
  
• public Heap()mySize(0) Heap()
• void deleteMax() //remove the max element
• void insert(const T item) //insert an
  item

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Percolate Down Algorithm

• 1. Set c 2 r 1
• 2. While c lt n do following //what does this
  mean?
• a. If c lt n-1 and myArrayc lt myArrayc
  1 Increment c by 1b. If myArrayr lt
  myArrayc i. Swap myArrayr and
  myArrayc ii. set r c iii. Set c 2 c
  1else Terminate repetitionEnd while

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Basic Heap Operations

• Insert an item
• Amounts to a percolate up routine
• Place new item at end of array
• Interchange with parent so long as it is greater
  than its parent

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Percolate Up Algorithm

• Why percolate up?
• When to terminate the up process?
• void HeapltTgtpercolate_up()
• void HeapltTgtinsert(const T item)

15
How to do remove?

• Remove an item from the heap?
• Question A leaf node must be less or equal than
  any internal node in a heap?

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Heapsort

• Given a list of numbers in an array
• Stored in a complete binary tree
• Convert to a heap
• Begin at last node not a leaf pos (size-2)/2?
• Apply percolated down to this subtree
• Continue

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Heapsort

• Algorithm for converting a complete binary tree
  to a heap called "heapify"For r (n-1-1)/2
  down to 0
• apply percolate_down to the subtree in
  myArrayr , myArrayn-1End for
• Puts largest element at root
• Do you understand it? Think why?

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Heapsort

• Why?
• Heap is a recursive ADT
• Semiheap ? heap from bottom to up
• Percolate down for this conversion

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Heapsort

• Now swap element 1 (root of tree) with last
  element
• This puts largest element in correct location
• Use percolate down on remaining sublist
• Converts from semi-heap to heap

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Heapsort

• Again swap root with rightmost leaf
• Continue this process with shrinking sublist

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Summary of HeapSort

• Step 1 put the data items into an array
• Step 2 Heapify this array into a heap
• Step 3 Exchange the root node with the last
  element and shrink the list by pruning the last
  element.
• Now it is a semi-heap
• Apply percolate-down algorithm
• Go back step 3

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Heapsort Algorithm

• 1. Consider x as a complete binary tree, use
  heapify to convert this tree to a heap
• 2. for i n-1 down to 1a. Interchange x0 and
  xi (puts largest element at end)b. Apply
  percolate_down to convert binary tree
  corresponding to sublist in x0 .. xi-1

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Heapsort

• Fully understand how heapsort works!
• T(n) O(nlogn)
• Why?

24
Heap Algorithms in STL

• Found in the ltalgorithmgt library
• make_heap() heapify
• push_heap() insert
• pop_heap() delete
• sort_heap() heapsort
• Note program which illustrates these operations,
  Fig. 13.1

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Priority Queue

• A collection of data elements
• Items stored in order by priority
• Higher priority items removed ahead of lower
• Operations
• Constructor
• Insert
• Find, remove smallest/largest (priority) element
• Replace
• Change priority
• Delete an item
• Join two priority queues into a larger one

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Priority Queue

• Implementation possibilities
• As a list (array, vector, linked list)
• T(n) for search, removeMax, insert operations
• As an ordered list
• T(n) for search, removeMax, insert operations
• Best is to use a heap
• T(n) for basic operations
• Why?

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Priority Queue

• STL priority queue adapter uses heap
• Note operations in table of Fig. 13.2 in text,
  page 751