How can binary search algorithm improve time complexity?

For me binary search is like looking for a word in a dictionary, so instead of reading every word, you start in the middle. If your word is earlier, you look in the first half; if later, in the second half. Then, you repeat this step in the chosen half. This way, you quickly narrow down to the word, skipping most of the dictionary. It's fast because each step cuts the search area in half.

The binary search algorithm's improvement in time complexity lies in its logarithmic nature, allowing it to efficiently narrow down search spaces with each iteration. This results in a faster search process compared to linear search algorithms, particularly advantageous for large datasets where the logarithmic growth rate significantly reduces the number of required comparisons.

One time at work, I optimized a database query using binary search principles. It dramatically reduced the time from linear to logarithmic complexity, showcasing how binary search transforms time-consuming tasks into more efficient processes, especially with large datasets

In implementing binary search, the key lies in accurately updating the 'low', 'high', and 'mid' pointers. It's critical to calculate the mid-point to avoid overflows — typically using mid = low + (high - low) / 2. Ensuring correct updates of 'low' and 'high' in each iteration avoids infinite loops and ensures convergence, especially in arrays with unique edge cases. Such precision in implementation guarantees the efficiency and reliability of binary search.

Binary search optimal use is for sorted arrays with little or no duplicated elements, and a somehow linear distribution. The bigger the duplication, the worst it will perform. Other thing to consider is when you have additional information on your data. With additional information you can change initial parameters, or the number or the number of idexes to improve performance even more, considering the medium case

Para encontrar la primera o última aparición de un elemento se pueden hacer ligeras modificaciones en el invariante, lo que es algo standar y se implementa en las funciones lower_bound/upper_bound por ejemplo en C++. Una posible implementación de lower_bound(v, x), que nos da la primer posición mayor o igual a x, es: def lower_bound(v,x): l, r = -1, len(v) while r-l>1: m = (l+r)//2 if v[m] < x: l = m else: r = m return r

I’ve found helpful testing binary search with a combination of automated unit tests and manual boundary checks. This ensures robustness against common pitfalls like off-by-one errors and handling of special cases like empty arrays or single-element arrays.

Hay que tener en cuenta que la busqueda binaria no solo permite buscar un valor en un vector ordenado, en terminos generales buscar la preimagen de un elemento para cualquier función monotona, sea su dominio discreto (los vectores son funciones cuyo dominio es un intervalo de números enteros) o continuo. Se puede usar, por ejemplo, para buscar una raiz n-ésima de un número no negativo cualquiera.