Greetings, curious minds! Today, we'll explore binary search applications that transcend basic searching. We'll apply binary search to complex data structures, such as bitonic arrays and rotated sorted arrays, to find specific elements efficiently.

Consider a bitonic array as a numerical sequence simulating the trajectory of a roller coaster — first, it rises to a peak, then descends. For example, consider the array [1, 2, 3, 4, 5, 3, 1]: its first part ascends, and the last descends, making it bitonic. Your goal is to find a value in such an array. You might walk the entire path, which is exhaustive and represents our naive approach with a time complexity of $O(n)$. Our aim today is a more efficient method.

To apply binary search, we first locate the peak of the array, then perform binary search on either side of the peak: one for the increasing sub-array and one for the decreasing sub-array.

Now, let's perform a targeted binary search on sub-arrays.

The binarySearchInBitonicArray function locates a target in a bitonic array by first finding the peak. It then performs a binary search on the ascending part up to the peak and, if the target is not found, on the descending part. It returns the index if found, otherwise -1.

Imagine you have a shuffled deck of cards that needs to be reordered. That could be represented with a rotated sorted array. For example, if we rotate the array [1, 2, 3, 4, 5], we could get [3, 4, 5, 1, 2]. You could check each element, one by one, to find the lowest, which is our naive approach with a time complexity of $O(n)$. Or, we could use binary search for a more efficient find.

For our advanced approach by adjusting our binary search logic to this context.

We apply our strategy to solve this problem.

The findMinimumInRotatedSortedArray function uses binary search to locate the minimum element in a rotated sorted array. It adjusts the search range based on whether the middle element is greater than the last element, iterating until start indicates the minimum element, which is returned.

Mastering binary search in TypeScript is akin to mastering quick decision-making in a labyrinth; each choice is a pivot leading to a more efficient exit. Today, you have learned to tackle complex problems using binary search techniques with TypeScript. These tasks, which at first glance seemed intricate, are readily solvable using the power of TypeScript's features, ensuring efficient and reliable solutions.

With this enhanced understanding, we now advance to practice exercises. Apply the binary search principles you've learned to navigate through and solve the forthcoming challenges. Happy coding!