A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property

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1 CmSc 250 Intro to Algorithms Chapter 6. Transform and Conquer Binary Heaps 1. Definition A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property Complete binary tree: Each node has two children, except possibly for the last two levels. (The nodes at the last level do not have children.) New nodes are inserted at the last level from left to right. This is called heap-structure property. A Complete Binary Heap Tree: Next node to be inserted - right child of the yellow node. 2. Binary heap implementation with an array Since the tree is complete, we use the following array implementation: Given array A: A(1) - contains the root The node in A(i) has its left child in A(2*i) and its right child in A(2*i+1) Its parent in A([i/2]). The smallest element is always at the root, so the access time to the element with highest priority is constant O(1). 1

2 This is the array that implements the binary heap in the example. The nodes of the tree are written in the array level by level from left to right. The first element in the array is 6 - this is the node at level 0. Then come and - the nodes at level 1. Further we have 15, 17 (children of ) 18, and 23 (children of - all at level 2. Finally we have 20, 19 and 34 - the nodes at the last level. Let's take a node in the array and find its parent and its children. Consider for example 17. Its position in the array is 5. Its parent is at position [5/2] = 2 Its left child is at position 5*2 =, this is the element 34, and its right child is at position 2*5 + 1 = 11 (still empty). 3. Basic operations 3.1. Insert a node - percolate up A hole is created at the bottom of the tree, in the next available position If the new node has a greater value than the parent of the hole - we insert the node in the hole. For example, if the new value to be inserted is 18, the binary heap would be: 2

3 If the new value is less than its parent, we slide down the parent of the hole, so the hole moves up, and check to see if the new node is less than the parent of the new hole. This process is repeated until we find a place for the new node, possibly at the very root of the binary heap. Let's say we want to insert is less than 17, so we slide down 17, and the hole appears at the node where 17 used to be: Since 16 is less than, we can insert it in the hole: 3

4 Complexity of insertion: O(logN) in the worst case To insert a node in the worst case we would need to percolate up to the root of the binary heap tree. As the tree is a complete tree, the longest path from the bottom to the top is not longer than O(logN), where N is the number of nodes in the tree. Hence we would do no more than O(logN) percolations up Deleting a node - percolate down We delete the node at the root - this is the node with highest priority. After deleting there is a hole at the root, which has to be filled. We may think of sliding the "hole" down and moving up the smaller of the children of the "hole", and repeat this until the hole gets to the bottom. The hole may appear anywhere at the bottom, and if it is not at the last node, the tree is not anymore a complete binary tree. (You may try this approach on the tree below and see what happens with the tree.)

5 To make sure that the hole appears at the last node, we render the deletion to an insertion in top-down direction. We can try to insert in the hole the last element that has been inserted, i.e. the rightmost element in the lowest row (the last element in the array). Thus in the array the last element will become empty, and the tree will be complete. The last element in the example is 18. The hole is at the root will not fit there, and we have to percolate the hole down The left child of the hole is with higher priority than the right one. We slide the hole down to the left: does not fit in the new hole, the children are greater, so we again slide the hole down: The left child of the hole is with higher priority than the right one. We slide the hole down to the left: 5

6 Now 18 fits the hole, and we can safely insert it there: Complexity of deletion: O(logN) in the worst case. We always delete from the top, hence the time to access the element with highest priority is a constant. However, after deletion we have to fill in the hole at the root. In the worst case we will percolate down to the bottom. Since the height of the tree is O(logN), where N is the number of nodes, the worst running time is O(logN). 6

7 3.3. Other heap operations DecreaseKey(p,d) This operation lowers the value of the element at position p by a positive amount d. It is used to increase the priority of an element. We have to find a new position of the element according to its new priority by percolating up. IncreaseKey(p,d) This operation increases the value of the element at position p by a positive amount d. It is used to decrease the priority of an element. We have to find a new position of the element according to its new priority by percolating down. Remove(p) With this operation an element is removed from the queue. This is done in two steps: a. Assigning the highest priority to p - percolate p up to the root. b. Deleting the element in the root and filling the hole by percolating down the last element in the array. 4. Building the heap This operation takes as input N elements and places them into an empty heap through successive inserts. The worst case running time is bound by O(NlogN). However, there is another algorithm to build the binary heap, that runs in O(N) time. The idea: Given an array of elements to be inserted in the heap, treat the array as a heap with order property violated, and then do operations to fix the order property. Let the array A be: A[1] is the root, A[1] = 150. Its children are A[2] and A[3], the children of A[2] are A[4] and A[5], etc. 7

8 In order to fix the order property we compare the nodes with their children starting with the rightmost node at the height 1. a. 1 is less that its children - OK b. 70 is not less than its children. 50 is the node to go up one level. c. is OK d. 30 is not less than its children, 20 is the node to go up After processing the nodes at height 1 we have the following tree:

9 Next we examine the nodes at height 2: a. 40 is OK b. 80 needs to be percolated down twice. The new tree will be: Next we examine the node at height 3 - the root 150 needs to be percolated down three times - until it gets to the bottom (Double dotted line indicates two percolates) Now the tree is in order. 9

10 Complexity: The worst case for a node at height k is k percolations. Thus at worst the number of operation is equal to the sum of the height of the nodes in the tree. Theorem: For a perfect binary tree of height h containing N = 2 h+1-1 nodes, the sum S of the heights of the nodes is S = 2 h (h + 1) = O(N) Proof: We can see that the tree has 1 node at height h, 2 nodes at height h-1, 4 nodes at height h -2, 1 (2 0 ) h 2 (2 1 ) h (2 2 ) h (2 3 ) h h 0 S = 2 i (h - i), i = 0 to h S = h + 2(h - 1) + 4(h - 2) + 8 (h - 3) +. 2 (h-2) (h-1). 1 (1) 2S = 2h + 4(h - 1) + 8(h - 2) + 16 (h - 3) +. 2 (h-1) (h). 1 (2) Subtract (1) from (2): S = h -2h h = - h (2 h+1-1) = (2 h+1-1) - (h + 1) Note: h = (2 h+1-1) Hence S = (2 h+1-1) - (h + 1) Hence the complexity of building a heap with N nodes is linear O(N).

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