Data Structures and Algorithm Analysis - Computer Science at ...

176 Chap. 5 Binary TreesRH1H2Figure 5.21 Final stage in the heap-building algorithm. Both subtrees of node Rare heaps. All that remains is to push R down to its proper level in the heap.177575 15 64 2 6 3 4 2 6 3 4 2 1 3(a) (b) (c)Figure 5.22 The siftdown operation. The subtrees of the root are assumed tobe heaps. (a) The partially completed heap. (b) Values 1 and 7 are swapped.(c) Values 1 and 6 are swapped to form the final heap.One good algorithm stems from induction. Suppose that the left and right subtreesof the root are already heaps, and R is the name of the element at the root.This situation is illustrated by Figure 5.21. In this case there are two possibilities.(1) R has a value greater than or equal to its two children. In this case, constructionis complete. (2) R has a value less than one or both of its children. In this case,R should be exchanged with the child that has greater value. The result will be aheap, except that R might still be less than one or both of its (new) children. Inthis case, we simply continue the process of “pushing down” R until it reaches alevel where it is greater than its children, or is a leaf node. This process is implementedby the private method siftdown. The siftdown operation is illustrated byFigure 5.22.This approach assumes that the subtrees are already heaps, suggesting that acomplete algorithm can be obtained by visiting the nodes in some order such thatthe children of a node are visited before the node itself. One simple way to do thisis simply to work from the high index of the array to the low index. Actually, thebuild process need not visit the leaf nodes (they can never move down because theyare already at the bottom), so the building algorithm can start in the middle of thearray, with the first internal node. The exchanges shown in Figure 5.20(b) resultfrom this process. Method buildHeap implements the building algorithm.What is the cost of buildHeap? Clearly it is the sum of the costs for the callsto siftdown. Each siftdown operation can cost at most the number of levels it