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Chapter 8: Problem 10

Write an algorithm that finds the largest key in a binary search tree. Analyze your algorithm, and show the results using order notation.

Short Answer

Expert verified
The largest key in a binary search tree can be found by traversing from the root to the rightmost node. The time complexity for this operation is O(h), where h is the height of the tree.

Step by step solution

01

Initialize

First, check whether the root of the tree is null. If it is, then the tree is empty and it does not contain any keys. Therefore, return null.
02

Traverse the Tree

If the root is not null, start from the root node and while the current node isn't null, update the current node to its right child. This is because in a binary search tree, all the values to the right of a node are greater than the node.
03

Finding the Largest Key

When the current node is null, its parent node is the node with the largest key. This is because this node doesn’t have a right child i.e., there’s no key in the tree which is larger. So, return this key value.
04

Time Complexity Analysis

The time complexity of this algorithm is \( O(h) \) where h is the height of the tree. This is because in the worst-case scenario, the algorithm will need to traverse from the root of the tree to the farthest leaf node, which takes \( O(h) \) time. The space complexity is also \( O(h) \) due to the memory required to store the call stack during recursive calls.

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Most popular questions from this chapter

Suppose a very large sorted list is stored in external storage. Assuming that this list cannot be brought into internal memory, develop a searching algorithm that looks for a key in this list. What major factor(s) should be considered when an external search algorithm is developed? Define the major factor(s), analyze your algorithm, and show the results using order notation.

Write a probabilistic algorithm that determines whether an array of \(n\) elements has a majority element (the element that appears the most). Analyze your algorithm and show the results using order notation.

Write an algorithm that deletes a node from a binary search tree considering all possible cases. Analyze your algorithm and show the results using order notation.

Use induction to show that \(W(n) \in \Omega(n \lg n)\) for the following recurrence. This is Recurrence 8.2 in Section 8.5 .4 where \(m\) (group size) is 3. \\[ W(n)=W\left(\frac{2 n}{3}\right)+W\left(\frac{n}{3}\right)+\frac{5 n}{3} \\]

Show that the average-case time complexity of Interpolation Search is in \(\Theta\) (Ig (Ig \(n\) )), assuming the keys are uniformly distributed and that search key \(x\) is equally probable to be in each of the array slots.
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