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In its simplest form, binary search is used to quickly find a value in a sorted sequence (consider a sequence an ordinary array for now).We’ll call the sought value the target value for clarity. At each step, the algorithm compares the median value in the search space to the target value.From this we conclude not only that the element at index 6 is not the target value, but also that no element at indices between 1 and 5 can be the target value, because all elements at these indices are smaller than 41, which is smaller than the target value.This brings the search space down to indices 7 through 11: Depending on how we choose the median of an even number of elements we will either find 55 in the next step or chop off 68 to get a search space of only one element.
As described above, we now choose the median value, which is the value at index 6 (the midpoint between 1 and 11): this value is 41 and it is smaller than the target value.
We use the predicate to verify if a candidate solution is legal (does not violate some constraint) according to the definition of the problem.
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 This is a series of no answers followed by a series of yes answers, as we were expecting.
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.