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Chapter 8: Problem 31

Discuss the advantages of using each of the following instead of the other: a. A binary search tree with a balancing mechanism b. \(A 3-2\) tree

Short Answer

Expert verified
The binary search tree with balancing mechanism is simpler to implement and requires less memory, but the 3-2 tree requires fewer rotations in operations, which might be more optimal in large and dense datasets.

Step by step solution

01

Understanding Binary Search Trees with Balancing Mechanism

Binary search trees (BSTs) maintain their elements in sorted order, allowing for efficient search, insertion, and deletion operations. The balancing mechanism ensures that the tree's height is logarithmic and thus, the aforementioned operations can be completed in \(O(\log n)\) time in the worst case.
02

Understanding 3-2 Trees

A 3-2 tree is a type of balanced search tree. It's also known as a B-tree of order 2, where every node in the tree can hold 1-2 keys and has 2-3 children, respectively. The tree adjusts its structure as it grows, ensuring that it remains balanced, leading to efficient operations in the worst-case scenario since the height of the tree is always logarithmic in the number of keys.
03

Comparing Advantages

The advantage of using a BST with a balancing mechanism over a 3-2 tree lies primarily in its simplicity: it is easier to code a BST and it generally requires less memory. On the other hand, the advantage of the 3-2 tree over the BST with balancing mechanism is that it requires fewer rotations when adding and removing elements, which can potentially lead to more efficient operations.

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Most popular questions from this chapter

Write an algorithm to delete an element from a hash table that uses linear probing as its clash resolution strategy. Analyze your algorithm and show the results using order notation.

Show that the worst-case time complexity of Interpolation Search is in \(\Theta\left((\lg n)^{2}\right)\), assuming the keys are uniformly distributed and that search key \(x\) is equally probable to be in each of the array slots.

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} \\]

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.

Give at least two examples of situations in which hashing is not appropriate.
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