Final answer:
To achieve a lookup time complexity of log n for an arbitrary integer 'm', a balanced binary search tree (BST) such as an AVL or Red-Black tree should be used, or alternatively, a binary search in a pre-sorted array. Both data structures offer the required logarithmic search time complexity.
Step-by-step explanation:
Designing a Lookup Table for Integer 'm' with Logarithmic Search Time Complexity
When designing a lookup table to achieve a search time complexity of log n, it is essential to utilize a data structure that supports logarithmic time complexity operations. The most suitable data structure for such a requirement is a balanced binary search tree (BST), such as an AVL or Red-Black tree. Alternatively, a binary search on a sorted array can also achieve logarithmic search complexity.
For a balanced BST, each insertion, deletion, and lookup operation has an average time complexity of O(log n). To create such a table for an arbitrary integer number 'm', one would first need to insert 'm' into the BST if it is not already present. Once the BST is constructed, searching for any integer will result in a search time of O(log n). If the integers are not known in advance or if dynamic insertion and deletion are required, a self-balancing BST is the best choice.
A binary search algorithm can be used in a sorted array to find an integer 'm'. The algorithm compares 'm' with the middle element of the array; if they are not equal, it divides the search interval in half and recursively proceeds to search the half in which 'm' could reside. This approach will also provide a O(log n) search time complexity, assuming that the array is sorted ahead of time.