When it comes to algorithms and data structures, finding the k-th smallest number in a set of numbers is a classic problem. Traditionally, this challenge is approached using methods that involve sorting or partitioning, which often result in time complexity greater than $O(\log n)$. However, for scenarios where arrays or elements have specific properties, we can leverage more efficient solutions. One prominent approach utilizes specialized data structures and optimization techniques to achieve this.

Understanding the Problem

Given an unsorted collection of numbers, the task is to identify the k-th smallest element. In typical scenarios, with no additional constraints, sorting the entire array would take $O(n \log n)$ and selecting the element would take constant time, resulting in $O(n \log n)$ time complexity, which is not optimal for large data sets. For a more efficient solution, we seek algorithms or data structures with $O(\log n)$ complexity.

Theoretical Possibility of $O(\log n)$ Solution

Achieving a solution in $O(\log n)$ time ordinarily necessitates specific preconditions, as it involves logarithmic time operations relative to the input size. Such performance is usually obtainable in either balanced search tree structures or through binary search on sorted/multi-level pre-arranged data.

1. Perfect Balanced Binary Search Trees

In a scenario where data is already stored in a perfectly balanced Binary Search Tree (BST), finding the k-th smallest element may be possible in logarithmic time due to the structural properties of the tree. Typically, tree traversal operations (like in-order traversal) are performed in $O(\log n)$ for balanced trees.

Example:

1. Construct a balanced BST from the provided dataset.
2. Perform an in-order traversal recursive process, counting nodes until reaching k.

2. Segment Trees or Fenwick Trees

Segment trees or Fenwick trees are useful, especially in the context of range queries. Although their use is more common for sums and other aggregated queries, they can be adapted to maintain elements' rank to support k-th smallest queries.