Question

Implement the non-recursive functions nextPreOrder(u), nextInOrder(u), and nextPostOrder(u) that return the node that follows u in a pre-order, in-order, or post-order traversal, respectively. These functions should take amortized constant time; if we start at any node u and repeatedly call one of these functions and assign the return value to u until u = null, then the cost of all these calls should be O(n). java

Answer 1

The question involves implementing nextPreOrder(u), nextInOrder(u), and nextPostOrder(u) non-recursive functions for binary tree traversals in Java. These functions should perform under amortized constant time complexity. The functions respectively return the next node in pre-order, in-order, and post-order traversal paths from a given node.

Step-by-step explanation:

The student has asked about implementing three non-recursive traversal functions in a binary tree: nextPreOrder(u), nextInOrder(u), and nextPostOrder(u) in Java, which should run in an amortized constant time. Implementing such functions require an understanding of binary tree traversals. In pre-order traversal, you visit the node before its children. For in-order traversal, you visit the left child, then the node, and lastly the right child. Lastly, in post-order traversal, you visit the children before the node itself.

Amortized constant time complexity suggests that while a single operation may take more than constant time, the average time per operation over a sequence of operations is constant. This is typically achieved by balancing out costly operations with many less expensive ones.

The nextPreOrder(u) function would first look to the left child if available, then the right child, and ascend to the lowest ancestor which is a left child, moving to its parent if neither child is available. The nextInOrder(u) function looks to the right child, and if it doesn't exist, it goes up to the lowest ancestor of which the node is in the left subtree. The nextPostOrder(u) function looks to ascend to the parent from a left child, and if coming from a right child or there is no right child, to continue ascending to the ancestor until one is found which is a left child of its parent.

Note that pseudocode or actual Java code is not provided, but the student should understand the logic and be able to implement it given these guidelines.