问题
I understand that binary search is O(logn) for a sorted array. However, I want to know if it is possible to calculate the average runtime with the guarantee that the search value is in the sorted array. This guarantee gives the binary search a much greater probability of finding the search value in less than logn time.
i.e. there is a 1/n chance of finding it in 1 step, 2/n chance of finding it in the 2nd step and so on.
I'm not sure how to go from this intuition into calculating the average runtime. Any ideas?
回答1:
So the probability of finding it in step i is 2^(i-1). To confirm this is correct, the sum of the probabilities for every step should be equal to 1. So, the summation from i = 1 to logn of 2^(i-1)/n is the equation we get, and that is equal to 1.
The average runtime then is the probability at each step * step #:
Summation from i=1 to logn of i * 2^(i-1)/n
Say n = 16, the average runtime is then 1/16( 1*1 + 2*2 + 3*4 + 4*8) = 3.125 which is less than log16 = 4.
来源：https://stackoverflow.com/questions/49269114/calculate-average-runtime-of-binary-search-with-guarantee-that-search-value-is-i