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binary heap The classic method used to implement priority queues The binary heap has two properties: structure and ordering (p 756) buildHeap operation The process of reinstating heap order in a complete tree, which can be done in linear time by applying a percolate down routine to nodes in reverse level order (p 766)

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complete binary tree A tree that is completely filled and has no missing nodes The heap is a complete binary tree, which allows representation by a simple array and guarantees logarithmic depth (p 756) external sorting A form of sorting used when the amount of data is too large to fit in main memory (p 778) heap-order property States that in a (min) heap, the item in the parent is never larger than the item in a node (p 758) heapsort An algorithm based on the idea that a priority queue can be used to sort in O(N log N ) time (p 773) implicit representation Using an array to store a tree (p 758) max heap Supports access of the maximum instead of the minimum ( P 759) rnultiway merge K-way merging that reduces the number of passes The obvious implementation uses 2K tapes (p 781) percolate down Deletion of the minimum involves placing the former last item in a hole that is created at the root The hole is pushed down the tree through minimum children until the item can be placed without violating the heap-order property (p 763) percolate up Implements insertion by creating a hole at the next available location and then bubbling it up until the new item can be placed in it without introducing a heap-order violation with the hole's parent (p 762) polyphase merge Implements a K-way merge with K + 1 tapes (P 782) replacement selection The length of the runs initially constructed can be larger than the amount of available main memory If we can store M objects in main memory, then we can expect runs of length 2M (p 784) run A sorted group in the external sort At the end of the sort, a single run remains (p 779)

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1 The trickiest part of the binary heap is the percolate down case in which only one child is present This case occurs rarely, so spotting an incorrect implementation is difficult 2 For heapsort, the data begins in position 0, so the children of node i are in positions 2i + 1 and 2i + 2 3 The STL priority queue is a max heap, not a min heap

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The code to implement the BinaryHeap is available in two files

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BinaryHeaph BinaryHeapcpp TestBinaryHeapcpp queueh queuecpp TestQueuecpp

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Contains the interface for the BinaryHeap class Contains the implementation of the BinaryHeap class Contains a test program for the BinaryHeap class Contains the interface for the p r i o r i ty-queue class Contains the implementation of the p r i o r i ty-queue class Contains a test program for the p r i o r i t y - q u e u e class

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211 Describe the structure and ordering properties of the binary heap 212 In a binary heap, for an item in position i where are the parent, left child, and right child located 213, Show the result of inserting 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, and 2, one at a time, in an initially empty heap Then show the result of using the linear-time buildHeap algorithm instead 214 Where could the 11th dashed line in Figures 2 117-2120 have been 215 A max heap supports i n s e r t , deleteMax, and findMax (but not d e l e t e M i n or fi n m i n ) Describe in detail how max heaps can be implemented 216 Show the result of the heapsort algorithm after the initial construction and then two d e l e t eMax operations on the input in Exercise 2 13 217 Is heapsort a stable sort (ie, if there are duplicates, do the duplicate items retain their initial ordering among themselves)

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