2. CHAPTER 5 PRIORITY QUEUES (HEAPS) §1 ADT Model Objects: A finite ordered list with zero or more elements. Operations:  PriorityQueue Initialize( int MaxElements );  void Insert( ElementType X, PriorityQueue H );  ElementType DeleteMin( PriorityQueue H );  ElementType FindMin( PriorityQueue H ); —— delete the element with the highest \ lowest priority 1/15

3. §2 Simple Implementations  Array : Insertion — add one item at the end ~  ( 1 ) Deletion — find the largest \ smallest key ~  ( n ) remove the item and shift array ~ O( n )  Linked List : Insertion — add to the front of the chain ~  ( 1 ) Deletion — find the largest \ smallest key ~  ( n ) remove the item ~  ( 1 )  Ordered Array : Insertion — find the proper position ~ O( n ) shift array and add the item ~ O( n ) Deletion — remove the first \ last item ~  ( 1 )  Ordered Linked List : Insertion — find the proper position ~ O( n ) add the item ~  ( 1 ) Deletion — remove the first \ last item ~  ( 1 ) Better since there are never more deletions than insertions 2/15

4.  Binary Search Tree : §2 Simple Implementations Ah! That’s a good idea! Both insertion and deletion will take O(log N) only. Well, insertions are random, but deletions are NOT. We are supposed to delete The minimum element only. Oh, right, then we must always delete from the left subtrees…. But hey, what if we keep a balanced tree? Hey you are getting smarter! Yes a balanced tree such as AVL tree is not a bad idea since only a constant factor will be added to the run time. However… Oh no… what’s wrong? There are many operations related to AVL tree that we don’t really need for a priority queue. Besides, pointers are always dangerous. I bet you have a better option? Now you begin to know me 3/15

5. §3 Binary Heap 1. Structure Property: 【 Definition 】 A binary tree with n nodes and height h is complete iff its nodes correspond to the nodes numbered from 1 to n in the perfect binary tree of height h. 4 89 5 1011 6 1213 7 1415 23 1 A complete binary tree of height h has between andnodes. 2h2h 2 h+1  1h =  log N   Array Representation : BT [ n + 1 ] ( BT [ 0 ] is not used) D HI E J FG BC A BT 01 A 2 B 3 C 4 D 5 E 6 F 7 G 8 H 9 I 10 J 111213 4/15

6. §3 Binary Heap 【 Lemma 】 If a complete binary tree with n nodes is represented sequentially, then for any node with index i, 1  i  n, we have: 5/15

7. §3 Binary Heap PriorityQueue Initialize( int MaxElements ) { PriorityQueue H; if ( MaxElements < MinPQSize ) return Error( "Priority queue size is too small" ); H = malloc( sizeof ( struct HeapStruct ) ); if ( H ==NULL ) return FatalError( "Out of space!!!" ); /* Allocate the array plus one extra for sentinel */ H->Elements = malloc(( MaxElements + 1 ) * sizeof( ElementType )); if ( H->Elements == NULL ) return FatalError( "Out of space!!!" ); H->Capacity = MaxElements; H->Size = 0; H->Elements[ 0 ] = MinData; /* set the sentinel */ return H; } 6/15

8. §3 Binary Heap 2. Heap Order Property: 【 Definition 】 A min tree is a tree in which the key value in each node is no larger than the key values in its children (if any). A min heap is a complete binary tree that is also a min tree. Note: Analogously, we can declare a max heap by changing the heap order property. 9 6 5 3 [1] [2][3] [4] A max heap 10 20 50 83 [1] [2][3] [4] A min heap The largest keyThe smallest key 7/15

9. §3 Binary Heap 3. Basic Heap Operations:  insertion 10 12 15 20 [1] [2][3] [4] 18 [5][6]  Sketch of the idea: The only possible position for a new node since a heap must be a complete binary tree. Case 1 : new_item = 21 21 2021 < Case 2 : new_item = 17 17 2017 > 20 1017 < Case 3 : new_item = 9 209 > 9 109 > 9 8/15

10. §3 Binary Heap /* H->Element[ 0 ] is a sentinel */ void Insert( ElementType X, PriorityQueue H ) { int i; if ( IsFull( H ) ) { Error( "Priority queue is full" ); return; } for ( i = ++H->Size; H->Elements[ i / 2 ] > X; i /= 2 ) H->Elements[ i ] = H->Elements[ i / 2 ]; H->Elements[ i ] = X; } Percolate up Faster than swap H->Element[ 0 ] is a sentinel that is no larger than the minimum element in the heap. T (N) = O ( log N ) 9/15

11. §3 Binary Heap  DeleteMin  Sketch of the idea: 10 12 15 20 [1] [2][3] [4] 18 [5] The node which must be removed to keep a complete binary tree.  move 18 up to the root 18  find the smaller child of 18 1218 < 12 1518 < 15 Ah! That’s simple -- we only have to delete the root node... And re-arrange the rest of the tree so that it’s still a min heap. T (N) = O ( log N ) 10/15

12. §3 Binary Heap ElementType DeleteMin( PriorityQueue H ) { int i, Child; ElementType MinElement, LastElement; if ( IsEmpty( H ) ) { Error( "Priority queue is empty" ); return H->Elements[ 0 ]; } MinElement = H->Elements[ 1 ]; /* save the min element */ LastElement = H->Elements[ H->Size-- ]; /* take last and reset size */ for ( i = 1; i * 2 Size; i = Child ) { /* Find smaller child */ Child = i * 2; if (Child != H->Size && H->Elements[Child+1] Elements[Child]) Child++; if ( LastElement > H->Elements[ Child ] ) /* Percolate one level */ H->Elements[ i ] = H->Elements[ Child ]; else break; /* find the proper position */ } H->Elements[ i ] = LastElement; return MinElement; } Percolate down What if this condition is omitted? Can we remove it by adding another sentinel? 11/15

13. §3 Binary Heap 4. Other Heap Operations: Note: Finding any key except the minimum one will have to take a linear scan through the entire heap.  DecreaseKey ( P, , H ) Lower the value of the key in the heap H at position P by a positive amount of  ……so my programs can run with highest priority. sys. admin.  IncreaseKey ( P, , H ) Percolate up sys. admin. Increases the value of the key in the heap H at position P by a positive amount of  ……drop the priority of a process that is consuming excessive CPU time. Percolate down 12/15

14. §3 Binary Heap  Delete ( P, H ) Remove the node at position P from the heap H …… delete the process that is terminated (abnormally) by a user. sys. admin. DecreaseKey(P, , H); DeleteMin(H)  BuildHeap ( H ) Place N input keys into an empty heap H. sys. admin. N successive Insertions ? Nehhhhh that would be toooo slow ! 30 10020 10 9060 70 50120 110 140130 8040 150 150, 80, 40, 30, 10, 70, 110, 100, 20, 90, 60, 50, 120, 140, 130 PercolateDown (7) PercolateDown (6) 50 70 PercolateDown (5) PercolateDown (4) 20 30 PercolateDown (3) PercolateDown (2) 80 10 80 60 PercolateDown (1) 150 10 150 20 150 30 T (N) = ? 13/15

15. §3 Binary Heap 【 Theorem 】 For the perfect binary tree of height h containing 2 h+1  1 nodes, the sum of the heights of the nodes is 2 h+1  1  (h + 1). T ( N ) = O ( N ) §4 Applications of Priority Queues 〖 Example 〗 Given a list of N elements and an integer k. Find the kth largest element. How many methods can you think of to solve this problem? What are their complexities? Home work: p.180 5.2, 5.3, 5.4 A test case for operations on a heap 14/15

16. §5 d-Heaps ---- All nodes have d children 3 15136 5 1789 2 7410 911 1 3-heap Question: Shall we make d as large as possible? Note:  DeleteMin will take d  1 comparisons to find the smallest child. Hence the total time complexity would be O(d log d N).  *2 or /2 is merely a bit shift, but *d or /d is not.  When the priority queue is too large to fit entirely in main memory, a d-heap will become interesting. Note:  DeleteMin will take d  1 comparisons to find the smallest child. Hence the total time complexity would be O(d log d N).  *2 or /2 is merely a bit shift, but *d or /d is not.  When the priority queue is too large to fit entirely in main memory, a d-heap will become interesting. Home work: p.181 5.13 Index computations for parents and children in a d-heap 15/15