Binary Search

Binary search is an exceptionally efficient search algorithm with a time complexity of O(log n). It operates on the principle of 'divide and conquer' but requires the data to be sorted. The algorithm repeatedly divides the search interval in half. It compares the middle element of the interval with the target value. If the target value matches the middle element, its position is returned. If the target value is less than the middle element, the search continues in the lower half of the interval. If it's greater, the search continues in the upper half. This process is repeated until the value is found or the interval is empty. While the standard use case is finding an element in a sorted array, the true power of binary search is revealed when it's applied to the 'answer space'. For many optimization problems that ask for the minimum or maximum value that satisfies a certain condition (e.g., 'what is the minimum capacity needed to ship all packages in D days?'), if we can determine for any given value `x` whether it's a possible answer (i.e., if a check function `is_possible(x)` exists and is monotonic), we can binary search for the optimal `x` in the range of all possible answers. This transforms the problem from a potentially complex calculation into a simple search, showcasing the versatility of the binary search concept.