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Lecture 28 Heaps Chapter 12 of textbook Concepts of heaps Binary heaps Binomial heaps

Lecture 28 Heaps Chapter 12 of textbook Concepts of heaps Binary heaps Binomial heaps Fibonacci heaps Heap summary. 1. Concept of Heaps. A heap (or heap-ordered tree ) is a tree-based data structure that satisfies the heap property : each node has a key value

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Lecture 28 Heaps Chapter 12 of textbook Concepts of heaps Binary heaps Binomial heaps

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  1. Lecture 28 Heaps Chapter 12 of textbook • Concepts of heaps • Binary heaps • Binomial heaps • Fibonacci heaps • Heap summary

  2. 1. Concept of Heaps • A heap (or heap-ordered tree) is a tree-based data structure that satisfies the heap property: • each node has a key value • if A has child B, then key(A) ≥ key(B). • This implies that the root node has the highest key value in the heap. Such a heap is called a max-heap. • min-heap: if A has a child B, then key(A) ≤ key(B)

  3. Example of Heaps 4 99 min-heap max-heap 6 83 12 99 9 69 21 45 7 36 54 10 27 13

  4. Heap operations find-max : find the node with maximum key value from a max-heap, O(1) time find-min : find the node with maximum key value from a min-heap, O(1) time Insert - insert a node into heap, maintain the property of heap delete - delete min/max key node from min-heap / m-heap. maintain the property of heap delete-min/extract-min, delete-max / extract-max decrease-key – decrease key value of a node, maintain the heap order

  5. Applications of Heaps Heap-sort: insert elements into heap, find-min/find-min delete get one after another, return a sorted list of elements. Used to implement priority queue. Heap Implemented priority queues are used in Graph algorithms like Prim's Algorithm and Dijkstra's algorithm for shortest path

  6. 2. Binary Heaps Binary heap is heap of complete binary tree structure Complete binary tree • all levels of the tree, except possibly the last one (deepest) are fully filled • if the last level of the tree is not complete, the nodes of that level are filled from left to right.

  7. Binary Heaps 4 99 Binary min-heap Binary-max-heap 6 83 12 99 9 21 39 69 21 39 45 7 36 19 54 10 27 13

  8. Array Representation of Binary Heaps • A complete binary tree of n element can be represented by an array of n elements • Order the complete binary tree node data elements in breadth-first and left-first order, and put the elements into array in the same order. The root node has index 0. • The node of index i has children indices at 2i + 1 and 2i + 2, and parent at index (i − 1) ∕ 2 • The height of the complete binary tree is log2n • In case binary heap, the element at index 0 is the max/min element

  9. Insertion in a Binary Heap Algorithm: Insert element to binary max-heap H Step 1. add the new element node to the bottom left H Step 2. If the new node value is bigger than its parent, swap their values. Set the parent node as new node, go to step 2. Step 3. else exit. Step 2 is called heapify

  10. Example of Insertion in a Binary Heap 54 45 36 27 21 18 21 11 54 54 54 99 99 36 45 45 36 36 54 36 45 21 18 21 27 99 21 21 18 18 21 21 45 21 18 21 11 27 11 11 99 27 11 27 • Consider the heap given below and insert 99 in it

  11. Insertion in a Binary Heap Algorithm to insert a value in the heap Step 1: [Add the new value and set its POS] SET N = N + 1, POS = N Step 2: SET HEAP[N] = VAL Step 3: [Find appropriate location of VAL] Repeat Steps 4 and 5 while POS < 0 Step 4: SET PAR = POS/2 Step 5: IF HEAP[POS] <= HEAP[PAR], then Goto Step 6. ELSE SWAP HEAP[POS], HEAP[PAR] POS = PAR [END OF IF] [END OF LOOP] Step 6: ReturnTime complexity: ? Space complexity: ?

  12. Deletion in a Binary Heap • An element is always deleted from the root of the heap. So, deleting an element from the heap is done in two major steps. Step 1. Replace the root node’s value with the last node’s value so that H is still a complete binary tree but not necessarily a heap. Step 2. Delete the last node. Step 3. Sink down the new root node’s value so that H satisfies the heap property. In this step, interchange the root node’s value with its child node’s value (whichever is largest among its children).

  13. Deletion in a Binary Heap 45 45 11 36 29 36 27 29 18 21 27 11 18 21 54 11 45 36 45 36 27 29 18 21 27 29 18 21 11 • Consider the heap H given below and delete the root node’s value.

  14. Deletion in a Binary Heap Algorithm to delete the root element from the heap DELETE_HEAP(HEAP, N, VAL) Step 1: [ Remove the last node from the heap] SET LAST = HEAP[N], SET N = N – 1 Step 2: [Initialization] SET PTR = 0, LEFT = 1, RIGHT = 2 Step 3: SET HEAP[PTR] = LAST Step 4: Repeat Steps 5 to 7 while LEFT <= N Step 5: IF HEAP{PTR] >= HEAP[LEFT] AND HEAP[PTR] >= HEAP[RIGHT], then Go to Step 8 [END OF IF] Step 6: IF HEAP[RIGHT] <= HEAP[LEFT], then SWAP HEAP[PTR], HEAP[LEFT] SET PTR = LEFT ELSE SWAP HEAP[PTR], HEAP[RIGHT] SET PTR = RIGHT [END OF IF] Step 7: SET LEFT = 2 * PTR and RIGHT = LEFT + 1 [END OF LOOP] Step 8: Return Time complexity: ? Space complexity: ?

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