Design an efficient algorithm to find the post-office

Question

Problem

Let x₁ < x₂ < ... < x_n be real numbers representing coordinates of n villages located along a straight road. A post office needs to be built in one of these villages.

a. Design an efficient algorithm to find the post-office location minimizing the average distance between the villages and the post office.

b. Design an efficient algorithm to find the post-office location minimizing the maximum distance from a village to the post office.