Priority Queues, Binary Heaps, and Heapsort

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1 Priority Queues, Binary eaps, and eapsort Learning Goals: Provide examples of appropriate applications for priority queues and heaps. Implement and manipulate a heap using an array as the underlying data structure. escribe the eapify (Build eap) and eapsort algorithms, and analyze their complexity. A priority queue is an abstract data type (AT) that maintains a bag of items and supports the operations: Insert(item) and RemoveMin (or RemoveMax). What is a bag? Every item has a priority. RemoveMin returns an item of smallest priority and deletes it from the set. Two or more distinct items in a priority queue can have the same priority. If all items have the same priority, you might think a priority queue should behave like a queue, but it may not. We ll see why, shortly. CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page A binary heap can implement a priority queue, efficiently. A minimum binary heap is a nearly complete binary tree such that the data in every node is less than or equal to the data in each of its child nodes and the node s left and right sub-trees are also minimum heaps. (There s also a maximum binary heap.) It is important to realize that two binary heaps can contain the same data but the data may appear in different positions in the heap: Both of the minimum binary heaps above contain the same data:,,, 7, 7,. Even though both heaps satisfy all the properties necessary of a minimum binary heap, the data is stored in different positions in the tree. CPSC Priority Queues, Binary eaps, and eapsort Page 3 CPSC Priority Queues, Binary eaps, and eapsort Page 4 The fact that a binary heap is a nearly complete binary tree allows us to represent the heap using an array. ence we have a contiguous rather than linked structure to represent the heap. K L K L We must now determine a way of navigating the heap. If the heap were represented as a linked structure, each node would have a pointer to its left and right child. Using the array representation, a node with index k in the array has: index of left child: k + index of parent: floor((k-)/) index of right child: k + Inserting an Item into a eap Consider the minimum heap presented below. Let s suppose that we want to insert an E into the heap. We begin by inserting the E at the bottom of the heap (in such a way that the binary tree remains nearly complete) and then performing a ReheapUp operation: we successively compare the new item with its parent. If it is smaller than its parent, we swap it with the parent and continue on up E CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page 6

2 K L M K L M ReheapUp( root, bottom ) // subscripts are passed if ( bottom > root ) set parent to subscript of parent of bottom element if ( data[parent] > data[bottom] ) swap( data[parent], data[bottom] ) ReheapUp( root, parent ) Removing the Item at the Top of a eap Consider the minimum heap presented below. Suppose we want to remove the item at the top of the heap. After doing so, we move the rightmost element on the bottom level of the heap, up to the top. This preserves the shape of the binary tree (it is still nearly complete) but the order property is lost. We regain the ordering by performing a Reheapown operation. We swap the root node with the smaller of its two child nodes and continue down the tree in this fashion until we reach the bottom of the tree. K L K CPSC Priority Queues, Binary eaps, and eapsort Page 7 CPSC Priority Queues, Binary eaps, and eapsort Page K Reheapown( root, bottom ) // subscripts are passed if ( root node is not a leaf ) set minchild to subscript of child s smallest data value if ( data[root] > data[minchild] ) swap( data[root], data[minchild] ) Reheapown( minchild, bottom ) Time Complexity of ReheapUp and Reheapown Operations When performing either a ReheapUp or Reheapown operation, the number of operations depends on the depth of the tree. Notice that we traverse only one path or branch of the tree. A nearly complete binary tree of height h has between h and h+ - nodes: We can now determine the height of a heap in terms of the number of nodes N in the heap: the height is floor(lg N). The time complexity of the ReheapUp and Reheapown operations is therefore O(lg N). CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page Inserting an item into a binary heap (operation Insert for a priority queue) would amount to performing a ReheapUp operation and therefore takes worst case O( ) time. Removing the smallest value from the heap (operation RemoveMin from a priority queue) would amount to performing a Reheapown operation and is O( ). Recall that when a sorted or unsorted linear structure (such as a linked list or an array) is used as the underlying concrete data type, then one or the other of Insert and RemoveMin is O(N). The eapsort Algorithm Motivation: We have seen that Quicksort takes O(N ) time in the worst case, for N elements. In this section, we examine a sorting algorithm that guarantees worst case O( N lg N ) time. We will use a minimum binary heap to sort an array of data in descending order. Similarly, a maximum binary heap can be used to sort data in ascending order. (Or, in each case, you can simply read the resulting array backwards to get the same result, so it doesn t really matter.) Recall that a minimum binary heap is a nearly complete binary tree such that the data in each node is less than or equal to the data in each of its children s nodes. Also, recall that because a binary heap is a nearly complete binary tree, it can conveniently be represented using an array. CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page

3 The eapsort algorithm consists of phases:. [eapify] Build a heap using the elements to be sorted.. [Sort] Use the heap to sort the data. Building a eap from an Arbitrary Array: It is easier to picture this process if we represent the heap using a binary tree rather than an array. Algorithm eapify : Let index be the subscript of the last parent node in the tree. while index : Perform a Reheapown operation starting with the node at index. ecrement index by. Example: Convert the following array to a heap: 7 3 To do so, picture the array as a nearly complete binary tree: 7 3 or an array with elements, the subscript or index of the last parent node in the tree is: CPSC Priority Queues, Binary eaps, and eapsort Page 3 CPSC Priority Queues, Binary eaps, and eapsort Page 4 7 aving built the heap, we now sort the array: Note: In this section, we represent the data in both binary tree and array formats. It is important to understand that in practice the data is stored only as an array. Let swapindex = N. While swapindex is greater than : Swap data at position swapindex with data at position. Reheapown between positions and swapindex. CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page 7 And so the process continues until the entire array is sorted. CPSC Priority Queues, Binary eaps, and eapsort Page 3

4 Implementation: void sort( type* data, int size ) //PRE: The capacity of the array pointed to by data // is at least size. //POST: The first size elements of data have been // sorted in descending order. int swpindx; Buildeap( data, size ); // eapify algorithm for( swpindx = size ; swpindx > ; swpindx-- ) swap( data[], data[swpindx] ); Reheapown( data,, swpindx ); void Buildeap( type* data, int size ) //PRE: data points to an array of data of capacity at // least size. //POST: The first size elements of data are a heap. int index; for( index = (size ) / ; index >= ; index-- ) Reheapown( data, index, size ); CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page void Reheapown( type* data, int top, int size ) // PRE: data between subscript top+ and size // is a heap. // POST: data between subscript top and size is // a heap. int leftchild = * top + ; int rightchild = * top + ; int minchild; if( leftchild < size ) // find subscript of smallest child if( rightchild >= size data[leftchild] < data[rightchild] ) minchild = leftchild; else minchild = rightchild; // if data at top is greater than smallest // child then swap and continue if (data[top] > data[minchild]) swap( data[top], data[minchild] ); Reheapown( data, minchild, size ); Note: This function is tail-recursive i and so it can easily be replaced with an iterative version having O() space requirements. Why might this be important for some implementations? CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page Time Complexity of eapsort We need to determine the time complexity of the Buildeap (i.e., eapify) operation, and the time complexity of the subsequent sorting operation. The time complexity of the sorting operation once the heap has been built is fairly easy to determine. or each element in the heap, we perform a single swap and a Reheapown. If there are N elements in the heap, the Reheapown operation is O( ) and hence the sorting operation is O( ). We must now consider the cost of building the heap. Surprisingly, it is an O( ) operation! If we examine the Buildeap function we see that the Reheapown function is called O( ) times but we have to realize that the Reheapown operation does not always start at the top of the heap and that it is not called at all on any of the leaf nodes (which account for roughly half the nodes in the tree!) CPSC Priority Queues, Binary eaps, and eapsort Page 3 We can determine the time complexity of the Buildeap function by looking at the total number of times the comparison and swap operations occur while building the heap. Let us consider the worst case, which is when the last level in the heap is full. We will colour all the paths from each node, starting with the lowest parent and working up to the root, each going down to a leaf node. The number of edges on the path from each node to a leaf node represents an upper bound on the number of comparison and swap operations that will occur while applying the Reheapown operation to that node. By summing the total length of these paths, we will determine the time complexity of the Buildeap function. CPSC Priority Queues, Binary eaps, and eapsort Page 4 4

5 Let be the height of the tree, N be the number of elements in the tree, and E be the number of edges in the tree, then: Total number of coloured edges or swaps = Note that no edge is coloured more than once. ence the work done by the Buildeap function can be measured in terms of the number of coloured edges. ence, in the worst case, the overall time complexity of the eapsort algorithm is: O(N) + O(N lg N) = O(N lg N) build heap from unsorted array essentially perform N RemoveMin s CPSC Priority Queues, Binary eaps, and eapsort Page CPSC Priority Queues, Binary eaps, and eapsort Page 6

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