This repository will be my documentation while I learn Data structures and algorithms. I will add my solved problems here 😄
https://www.topcoder.com/community/competitive-programming/tutorials/binary-search/
This is where we start to abstract binary search. A sequence (array) is really just a function which associates integers (indices) with the corresponding values. However, there is no reason to restrict our usage of binary search to tangible sequences. In fact, we can use the same algorithm described above on any monotonic function f whose domain is the set of integers. The only difference is that we replace an array lookup with a function evaluation: we are now looking for some x such that f(x) is equal to the target value. The search space is now more formally a subinterval of the domain of the function, while the target value is an element of the codomain. The power of binary search begins to show now: not only do we need at most O(log N) comparisons to find the target value, but we also do not need to evaluate the function more than that many times. Additionally, in this case we aren’t restricted by practical quantities such as available memory, as was the case with arrays.
What we can call the main theorem states that binary search can be used if and only if for all x in S, p(x) implies p(y) for all y > x. This property is what we use when we discard the second half of the search space. It is equivalent to saying that ¬p(x) implies ¬p(y) for all y < x (the symbol ¬ denotes the logical not operator), which is what we use when we discard the first half of the search space. The theorem can easily be proven, although I’ll omit the proof here to reduce clutter.
These two parts are most often interleaved: when we think a problem can be solved by binary search, we aim to design the predicate so that it satisfies the condition in the main theorem.