A Priority Queue: The Binary Heap

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218 A complete binary tree of N elements uses array positions I through N Determine how large the array must be for a a binary tree that has two extra levels (ie, is slightly unbalanced) b a binary tree that has a deepest node at depth 2 log N c a binary tree that has a deepest node at depth 41 log N d the worst-case binary tree 219 Show the following regarding the maximum item in the heap a It must be at one of the leaves b There are exactly N / 2 1 leaves c Every leaf must be examined to find it 2110 Prove Theorem 211 by using a direct summation Do the following a Show that there are 2' nodes of height H - i b Write the equation for the sum of the heights using part (a) c Evaluate the sum in part (b) 2111 Verify that the sum of the heights of all the nodes in a perfect binary tree satisfies N - t>(IV), where v(N) is the number of 1 c in N's binary representation 2112 Prove the bound in Exercise 211 1 by using an induction argument 2113 For heapsort, O(N log IV) comparisons are used in the worst case Derive the leading term (ie, decide whether it is N log N , 2N log N , 3N log N, and so on) 2114 Show that there are inputs that force every percDown in heapsort to Work backward) go all the way to a leaf (Hint: 2115 Suppose that the binary heap is stored with the root at position r: Give formulas for the locations of the children and parent of the node in position i 2116 Suppose that binary heaps are represented by explicit links Give a simple algorithm to find the tree node that is at implicit position i 2117 Suppose that binary heaps are represented by explicit links Consider the problem of merging binary heap l h s with rhsAssume that both heaps are full complete binary trees, containing 2'- 1 and 2' - I nodes, respectively a Give an O(1og N) algorithm to merge the two heaps if I = r

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b Give an O(1ogN) algorithm to merge the two heaps if 11-rl = 1 c Give an O( log2 N) algorithm to merge the two heaps regardless of 1 and r

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2118 A d-heap is an implicit data structure that is like a binary heap, except that nodes have d children A d-heap is thus shallower than a binary heap, but finding the minimum child requires examining d children instead of two children Determine the running time (in terms of d and N) of the insert and deleteMin operations for a d-heap 2119 A min-max heap is a data structure that supports both deleteMin and deleteMax at logarithmic cost The structure is identical to the binary heap The min-max heap-order property is that for any node X at even depth, the key stored at X is the smallest in its subtree, whereas for any node X at odd depth, the key stored at X is the largest in its subtree The root is at even depth Do the following a Draw a possible min-max heap for the items l , 2 , 3, 4, 5 , 6 , 7, 8, 9, and 10 Note that there are many possible heaps b Determine how to find the minimum and maximum elements c Give an algorithm to insert a new node into the min-max heap d Give an algorithm to perform deleteMin and deleteMax e Give an algorithm to perform buildHeap in linear time 2120 The 2 - 0 heap is a data structure that allows each item to have two individual keys The deleteMin operation can be performed with respect to either of these keys The 2-D heap-order property is that for any node X at even depth, the item stored at X has the smallest key #1 in its subtree, and for any node X at odd depth, the item stored at X has the smallest key #2 in its subtree Do the following a Draw a possible 2-D heap for the items (1, lo), (2, 9), (3, 8), (4,7), and (5,6) b Explain how to find the item with minimum key #I c Explain how to find the item with minimum key #2 d Give an algorithm to insert a new item in the 2-D heap e Give an algorithm to perform deleteMin with respect to either key f Give an algorithm to perform buildHeap in linear time

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