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CS 225
Data Structures
Oc October 31 – He Heaps and Priority Qu Queues
G G Carl Evans
Ru Running T Times
Hash Table AVL Linked List
Find
SUHA: Worst Case:
Insert
SUHA: Worst Case:
Storage Space
CS 225 Data Structures Oc October 31 He Heaps and Priority Qu - - PowerPoint PPT Presentation
CS 225 Data Structures Oc October 31 He Heaps and Priority Qu - - PowerPoint PPT Presentation
CS 225 Data Structures Oc October 31 He Heaps and Priority Qu Queues G G Carl Evans Ru Running T Times Hash Table AVL Linked List SUHA: Find Worst Case: SUHA: Insert Worst Case: Storage Space Se Secr cret, M Mystery D
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Se Secr cret, M Mystery D Data St Stru ruct cture
ADT: insert remove isEmpty
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Pr Priority Queue Implementat ation
insert removeMin unsorted sorted sorted unsorted
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An Another p r possibly s stru ructure…
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(m (min)He n)Heap
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A complete binary tree T is a min-heap if:
- T = {} or
- T = {r, TL, TR}, where r is
less than the roots of {TL, TR} and {TL, TR} are min-heaps.
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(m (min)He n)Heap
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in insert
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template <class T> void Heap<T>::_insert(const T & key) { // Check to ensure there’s space to insert an element // ...if not, grow the array if ( size_ == capacity_ ) { _growArray(); } // Insert the new element at the end of the array item_[++size] = key; // Restore the heap property _heapifyUp(size); } 1 2 3 4 5 6 7 8 9 10 11 12
in insert
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gr growA wArray
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template <class T> void Heap<T>::_heapifyUp( _________________ ) { if ( index > _________ ) { if ( item_[index] < item_[ parent(index) ] ) { std::swap( item_[index], item_[ parent(index) ] ); _heapifyUp( ________________ ); } } } 1 2 3 4 5 6 7 8 9
in insert t - he heapi pifyUp yUp
template <class T> void Heap<T>::_insert(const T & key) { // Check to ensure there’s space to insert an element // ...if not, grow the array if ( size_ == capacity_ ) { _growArray(); } // Insert the new element at the end of the array item_[++size] = key; // Restore the heap property _heapifyUp(size); } 1 2 3 4 5 6 7 8 9 10 11 12
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template <class T> void Heap<T>::_heapifyUp( _________________ ) { if ( index > _________ ) { if ( item_[index] < item_[ parent(index) ] ) { std::swap( item_[index], item_[ parent(index) ] ); _heapifyUp( ________________ ); } } } 1 2 3 4 5 6 7 8 9
he heapi pifyUp yUp
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re removeMin
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template <class T> void Heap<T>::_removeMin() { // Swap with the last value T minValue = item_[1]; item_[1] = item_[size_]; size--; // Restore the heap property heapifyDown(); // Return the minimum value return minValue; } 1 2 3 4 5 6 7 8 9 10 11 12 13
re removeMin
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template <class T> void Heap<T>::_removeMin() { // Swap with the last value T minValue = item_[1]; item_[1] = item_[size_]; size--; // Restore the heap property _heapifyDown(); // Return the minimum value return minValue; } template <class T> void Heap<T>::_heapifyDown(int index) { if ( !_isLeaf(index) ) { T minChildIndex = _minChild(index); if ( item_[index] ___ item_[minChildIndex] ) { std::swap( item_[index], item_[minChildIndex] ); _heapifyDown( ________________ ); } } } 1 2 3 4 5 6 7 8 9 10
re removeMin - he heapi pifyD yDown wn
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template <class T> void Heap<T>::_heapifyDown(int index) { if ( !_isLeaf(index) ) { T minChildIndex = _minChild(index); if ( item_[index] ___ item_[minChildIndex] ) { std::swap( item_[index], item_[minChildIndex] ); _heapifyDown( ________________ ); } } } 1 2 3 4 5 6 7 8 9 10
re removeMin
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bui buildHe dHeap
U L D P B I H E W A O N
B U I L D H E A P N O W
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bui buildHe dHeap – so sorted d array
B E H O A D I L W N U P
B U I L D H E A P N O W A B D E H I L N O P U W
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bui buildHe dHeap - he heapi pifyUp yUp
U L D P B I H E W A O N
B U I L D H E A P N O W
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bui buildHe dHeap - he heapi pifyD yDown wn
U L D P B I H E W A O N
B U I L D H E A P N O W
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bui buildHe dHeap
U L D P B I H E W A O N
B U I L D H E A P N O W
- 1. Sort the array – it’s a heap!
2. 3.
template <class T> void Heap<T>::buildHeap() { for (unsigned i = 2; i <= size_; i++) { heapifyUp(i); } } 1 2 3 4 5 6 template <class T> void Heap<T>::buildHeap() { for (unsigned i = parent(size); i > 0; i--) { heapifyDown(i); } } 1 2 3 4 5 6
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Pr Proving bui buildHe dHeap Ru Running T Time
Theorem: The running time of buildHeap on array of size n is: _________. Strategy:
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Pr Proving bui buildHe dHeap Ru Running T Time
S(h): Sum of the heights of all nodes in a complete tree of height h. S(0) = S(1) = S(h) =
U L D P B I H E W A O N
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Pr Proving bui buildHe dHeap Ru Running T Time
Proof the recurrence: Base Case: General Case:
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Pr Proving bui buildHe dHeap Ru Running T Time
From S(h) to RunningTime(n): S(h): Since h ≤ lg(n): RunningTime(n) ≤
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Hea Heap p Sort
Running Time? Why do we care about another sort?
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1. 2. 3.
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A( A(no nothe her) ) thr hrowba wback k to CS 173…
Let R be an equivalence relation on us where (s, t) ∈ R if s and t have the same favorite among: { ___, ___, ____, ___, ____, }