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The Karatsuba algorithm is used for multiplying two large numbers. It breaks down the numbers into smaller components, which are then multiplied using a more efficient approach that reduces the total number of multiplication operations. Best used when you need to multiply large integers. Its complexity is $O(n^{\log_2 3})$ which is approximately $O(n^{1.585})$.

Merge Sort is a comparison-based sorting algorithm that divides the unsorted list into n sublists, each containing one element, and then repeatedly merges the sublists to produce new sorted sublists until there is only one sublist remaining. This algorithm is best used when a stable sort is needed and memory space is not a huge concern. The time complexity is always $O(n \log n)$.

Binary Search is an efficient algorithm for finding an item from a sorted list of items. It works by repeatedly dividing in half the portion of the list that could contain the item, until you've narrowed the possible locations to just one. Best used when the list is already sorted and random access is possible. Its time complexity is $O(\log n)$.

The Closest Pair of Points problem is solved using a divide and conquer approach that involves splitting the point set into two halves, solving the problem for each half recursively, and then combining results using a merge step. Best used when you need to efficiently find the closest pair in a set of points in Euclidean space. Its complexity is $O(n \log n)$.

Quicksort is a sorting algorithm that works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively. Best used for large datasets where an in-place sort is desirable. Its average-case complexity is $O(n \log n)$, whereas the worst-case is $O(n^2)$. However, randomized versions of quicksort have expected time complexity $O(n \log n)$.