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Priority Queue ADT Binary Heaps

Priority Queue ADT Binary Heaps. Gerda Kamberova Department of Computer Science Hofstra University. Overview. Priority Queue (PQ) ADT definition. Applications Implementation - binary heaps (heaps). Complexity analysis. Priority Queue ADT. Priority Queue (PQ) ADT is a dynamic set

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Priority Queue ADT Binary Heaps

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  1. Priority Queue ADTBinary Heaps Gerda Kamberova Department of Computer Science Hofstra University G.Kamberova, Algorithms

  2. Overview • Priority Queue (PQ) ADT definition. • Applications • Implementation - binary heaps (heaps). • Complexity analysis G.Kamberova, Algorithms

  3. Priority Queue ADT • Priority Queue (PQ) ADT is a dynamic set • with keys viewed as priorities • The keys could be compared, so we can identify the element with highest priority • supports • insert • delete element with highest priority • retrieve element with highest priority (without delete) • Depending on what, min or max, is selected as highest priority, we distinguish between min- and max-prioirity queues Historically, PQ means min-PQ. • If x is a reference to a record, the operations on priority queue S are: • For max-priority queue: Max(S), DeleteMax(S), Insert(S,x) • For min-priority queue: Min(S), DeleteMin(S), Insert(S,x) G.Kamberova, Algorithms

  4. Applications • Job scheduling on a multiuser system • Queue (FIFO). Not a good choice. • PQ: • greedy algorithm: let the smallest job run first. • Keep a PQ of incoming jobs and efficiently retrieve the highest priority job to run. • Minimizes avg time to wait • Event-driven discrete-event system simulation: most prevalent application of PQ is in G.Kamberova, Algorithms

  5. PQ ADT Implementation • Linked list (sorted or unsorted)? • Array sorted or unsorted? • Binary search tree (BST)? • O(log n), • overkill, supports all dynamic set operations • Heap • no linked lists • better suited than BST for PQ • Implementation, internally the structure is an array • O(log n) in worst case. • Insert is O(1) on average. • Building PQ is in linear time, O(n) G.Kamberova, Algorithms

  6. Binary Heap (Heap) • Heap (binary heap) is a binary tree (BT) in which the following properties are satisfied: • HS [the heap structure property or heap shape]: the BT is completely filled at all levels except possibly on the very last, where it is filled from left to right • PODR[ the partial order heap order property]: the key value stored at any non-leaf node is >= the key values of its children. • no particular • Note: • order between key values on the same level. • Max is at the root • when you go from a leaf, up a direct path, to the root, the keys are encountered in non-decreasing order. • the operations insert and delete may destroy the heap properties, so these operations should not terminate before restoring the heap properties G.Kamberova, Algorithms

  7. BT Representation of a Max Heap • Check HS and PORD properties 16 10 14 9 8 7 3 2 4 1 G.Kamberova, Algorithms

  8. Implementation of a Heap • since a heap is almost a complete BT, it is easily stored in an array (no need to store pointers): 16 10 14 9 8 7 3 2 4 1 16 14 10 8 7 9 3 2 4 1 … G.Kamberova, Algorithms

  9. Implementation of a Heap Nodes are labeled left-to-right in level order (root first, the right-most leaf last), and this order defines the array indices. 16 10 14 9 8 7 3 2 4 1 Read top to bottom, left to right, fill left to right, start at index 1 Read left to right, fill top to bottom left to right 16 14 10 8 7 9 3 2 4 1 … 1 2 3 4 5 6 7 8 9 10 G.Kamberova, Algorithms

  10. Implementation of the PQ operations with heaps • Retrieve the maximum • Easy, just return the root • Insert/DeleteMax • more complicated. • Must preserve HS and PORD properties. • Insert: • Input: a heap A of size n and an element referenced by x with key k to be inserted. • Output: A heap A of size n+1. • Where to place the new node? • not much choice, must preserve HS. • create a new node at index n+1 • If we place the new key at that place the PROD may be destroyed. • restore PORD by sifting up the new node until it falls in place i.e. where the new key will be in place G.Kamberova, Algorithms

  11. Insertion in a max heap: Example • Insert element with key 15 in the heap • Create a new node as most right leaf in tree • Propagate “hole” up until it falls in place 16 10 14 14 9 8 7 3 7 2 4 1 15 G.Kamberova, Algorithms

  12. Insertion in a heap • Implementation • Create the (n+1)-st node, and sift the key upwards Heap_Insert(A, key, n) n = n+1 i = n // find the spot for the new key while (i > 1 && A(Parent[i]) < key ) A[i] = A[Parent(i)] i = Parent(i) A[i] = key • Complexity: G.Kamberova, Algorithms

  13. Delete in a heap • Input: Heap A of size n; • Output: Heap A of size n-1, the root key value is removed from the heap (may be returned as well) • Since the heap HS must be preserved, the output heap is missing the right-most leaf of the input heap. • The root value must be extracted, and the “hole” at the root must be filled in • This suggest the following procedure: • copy the key from right-most leaf to root • delete right-most leaf • Restore PORD: heapify G.Kamberova, Algorithms

  14. Delete in a heap Heap_Delete(A, n) A[1] = A[n] //copy last leaf in root n = n - 1 Heapify(A,1) G.Kamberova, Algorithms

  15. Deletion from the heap 1 Heap_Delete(A, n) A[1] = A[n] //copy last leaf in root n = n - 1 Heapify(A,1) • Complexity: O(1)+ time to heapify 16 10 14 9 8 7 3 2 4 1 G.Kamberova, Algorithms

  16. Heapify • Input: A rooted tree, such that A[i] is at the root, and the left and the right subtrees (if exist) are heaps. • Output: The tree is made into a heap. • Procedure: • Sift down the key value of the root until it falls in place. • a recursive procedure: • compare key at the root with the keys of its children. • If the root key >= of the children keys, stop, • Otherwise, swap root key with the key of larger child, and proceed in hipifying now the subtree rooted at the node for which swap occured G.Kamberova, Algorithms

  17. Heapify • Hipify(A,i) : makes the subtree rooted at A[i] into a heap, provided that the trees rooted at A[Left(i)] and A[Right(i)] are heaps. Hipify(A,i) if A[i] is smaller than its children m = index of larger child swap A[i] and A[m] Hipify(A,m) G.Kamberova, Algorithms

  18. Deletion from the heap: Example 1 16 10 14 9 8 7 3 2 4 1 G.Kamberova, Algorithms

  19. Deletion from the heap: Example • Complexity: if we swap, iteratively, bounded by the heigh • Complexity using recursive heapify? 1 10 14 9 8 7 3 2 4 G.Kamberova, Algorithms

  20. Complexity of Heapify • Worst case: at each recursive call the subtree on which the call is made on a complete BT (all leaves at last level exist); the height of that tree must be one more than the height of the other subtree • In a complete BT the number of leaves is equal half of the nodes • If the worst-case tree has n nodes, they are distributed as follows • Recurrence for Heapify n/3 n/3 n/3 G.Kamberova, Algorithms

  21. Constructing a heap • Top-down: using HeapInsert, • successively insert nodes staring with an empty heap. • Time complexity is O(nlog n) • Bottom-up: using Heapify • Construct an almost complete binary tree with n vertices and store the keys arbitrarily there. This is Theta(n) • Now going bottom up, from the most right node in the last internal level, heapify the trees rooted at the internal nodes. • This time can be shown to be linear O(n) G.Kamberova, Algorithms

  22. Heapsort • Build a heap: O(n) or O(n log n) • Do n times extract_max n times: O(n log n) • Total time O(n log n) G.Kamberova, Algorithms

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