I came to know that Interpolation Search is a modification of Binary Search where in binary search the input is divided into two equal halves in each iteration by computing
mid = (low + high) / 2
and in Interpolation search the mid is computed as
mid = low + (key - arr[low]) * ((high - low) / (arr[high] - arr[low]))
Now I need to understand this formula of calculating mid in interpolation search.
Ref: https://en.wikipedia.org/wiki/Interpolation_search#Sample_implementation
You can think of array arr as a function f that acts on index and return a value, which is monotone (because array is sorted). So you have your initial data f(low) = m and f(high) = M. Now you can interpolate your function f with a straight line, which is quite reasonable to do because your f is monotone and you have only 2 points.
So your interpolation should be line (linear function) that pass throw points (low, m) and (high, M). This is it's equation
(y - f(low))/(f(high) - f(low)) = (x - low)/(high - low)
So y here is the element of search space and x is from domain (x is index of array). So if your function f would be the same as it's interpolation, then index of your key would be:
x = low + (high - low)*(key - f(low))/(f(high) - f(low))
So, assuming that your function f is close to it's interpolation, you should just check the value f at x to see if it is the goal. Otherwise you just shrink your interpolation interval.
An extention to @JustAnotherCurious answer
the equation perposed by him was based on Interpolation Formula (equ of a line)
Now, thinkf() as function that takes an index and return its y axis value,
y1 = f(low)
y2 = f(high)
x1 = low
x2 = high
(y - f(low)) = [(f(high) - f(low)) / (high - low)] * (x - low);
OR
x = low + [(high - low) * (y - f(low))] / (f(high) - f(low))
here: y = key
we are looking for position(value) of x.
