Ski Rental Conundrum: A Randomized Algorithm for Optimal Cost

This article tackles the classic ski rental problem, a fascinating example in online algorithms. The problem: a skier doesn't know how many days they'll ski; renting costs 1 unit per day, buying costs B units. The article details an optimal offline solution, then analyzes a simple online algorithm with a competitive ratio of 2. Crucially, it dives into a randomized algorithm using a continuous probability distribution to approximate the discrete problem, achieving an expected competitive ratio of approximately e/(e-1), significantly better than the simple approach. While not directly applicable in reality for single decisions, this algorithm offers a theoretically optimal strategy for scenarios involving many similar choices.