A Practical Introduction to Data Structures and Algorithm Analysis

182 Chap. 5 Binary TreesMin-heaps and max-heaps both have their uses. For example, the Heapsortof Section 7.6 uses the max-heap, while the Replacement Selection algorithm ofSection 8.5.2 uses a min-heap. The examples in the rest of this section will use amax-heap.Be careful not to confuse the logical representation of a heap with its physicalimplementation by means of the array-based complete binary tree. The two are notsynonymous because the logical view of the heap is actually a tree structure, whilethe typical physical implementation uses an array.Figure 5.19 shows an implementation for heaps. The class is a generic with onetype parameter. E defines the type for the data elements stored in the heap. E mustextend the Comparable interface, and we use the compareTo for comparingrecords in the heap.This class definition makes two concessions to the fact that an array-based implementationis used. First, heap nodes are indicated by their logical position withinthe heap rather than by a pointer to the node. In practice, the logical heap positioncorresponds to the identically numbered physical position in the array. Second, theconstructor takes as input a pointer to the array to be used. This approach providesthe greatest flexibility for using the heap because all data values can be loaded intothe array directly by the client. The advantage of this comes during the heap constructionphase, as explained below. The constructor also takes an integer parameterindicating the initial size of the heap (based on the number of elements initiallyloaded into the array) and a second integer parameter indicating the maximum sizeallowed for the heap (the size of the array).Method heapsize returns the current size of the heap. H.isLeaf(pos)will return true if position pos is a leaf in heap H, and false otherwise. Membersleftchild, rightchild, and parent return the position (actually, thearray index) for the left child, right child, and parent of the position passed, respectively.One way to build a heap is to insert the elements one at a time. Method insertwill insert a new element V into the heap. You might expect the heap insertion processto be similar to the insert function for a BST, starting at the root and workingdown through the heap. However, this approach is not likely to work because theheap must maintain the shape of a complete binary tree. Equivalently, if the heaptakes up the first n positions of its array prior to the call to insert, it must takeup the first n + 1 positions after. To accomplish this, insert first places V at positionn of the array. Of course, V is unlikely to be in the correct position. To moveV to the right place, it is compared to its parent’s value. If the value of V is lessthan or equal to the value of its parent, then it is in the correct place and the insert