Question

Here’s a question for interview crackers-

Given that you are receiving samples from an instrument at a constant rate, and you have constant storage space, how would you design a storage algorithm that would allow me to get a representative readout of data, no matter when I looked at it? In other words, representative of the behavior of the system to date.

I couldn’t get any idea of it. So, I am looking for ideas.

Answer 1

Assume that you have the memory to store k elements. Store the first k elements in the memory in an array. Now when you receive the nth element (where n > k), generate a random number r between 1 and n. If r > k discard the n^th element. Otherwise replace the r^th element in the array with the n^th element.

This approach will ensure that at any stage your array would contain k elements that are uniformly randomly selected from the input elements received so far.

Proof We can show by induction that the k representative elements at any stage are distributed in a uniformly random way. Assume that after receiving n-1 elements, any element is present in the array with probability k/(n-1).

After receiving the nth element, the probability that the element will be inserted into the array = k/n.

For any other element, the probability that it is represented in the current iteration = probability that it is represented in the previous iteration * probability that it is not replaced in the current iteration

= (k/(n-1)) * (n-1)/n = k/n.