Question

2-3 tree T contains m keys and the LL red-black tree R contains n keys. You need to generate a 2-3 tree containing the combined n + m keys. You can assume that all keys are distinct and you know m > n. Explain how to efficiently generate a combined 2-3 or LL RB tree size n + m. What is the time bound to do so?

Answer 1

To combine a 2-3 tree containing m keys and an LL red-black tree containing n keys into one, perform in-order traversals of both trees, merge the resulting sorted lists, and construct a new 2-3 tree or red-black tree from the combined list. The time complexity for this operation is O(m+n).

Step-by-step explanation:

To efficiently generate a combined 2-3 or LL red-black (RB) tree with n + m keys, where a 2-3 tree T contains m keys and the LL red-black tree R contains n keys, and given that all keys are distinct and m > n, you can proceed as follows:

• Since 2-3 trees and red-black trees both maintain sorted order, you could perform a traversal of both trees to generate a sorted list of all keys. In-order traversal of a 2-3 tree and an LL red-black tree would give you the keys in sorted order.
• Once you have the two sorted lists, merge them to create a single sorted list of all keys. This merge operation is similar to the one used in the merge sort algorithm.
• Finally, using the merged sorted list, construct a new 2-3 tree or a new red-black tree.

In terms of time complexity, doing an in-order traversal for both trees would take O(m+n), merging the sorted lists also takes O(m+n), and constructing the new tree from the sorted list would take O(m+n). Thus, the overall time bound for this task is O(m+n).