Competitive Analysis

The ski rental problem arises from the following scenario. You want to start skiing, but you don’t own any skis. On each day you go skiing, you can either choose to rent skis for cost 1, or you can buy skis for cost s. Since you don’t know when you might get sick of skiing, (or when you might break your leg and be unable to ski again), the number of days d that you will end up skiing is unknown to you. For this problem, assume s is a positive integer bigger than 1. In examples, we will use s = 10. We model the ski rental problem as online problem and analyze it using competitive analysis. For the ski rental problem, an input sequence σ represents a sequence of some number of requests to ski. An algorithm A describes some strategy for deciding when to rent skis and when to buy skis. For now, we consider only deterministic algorithms A; in this case, the space of all “reasonable” deterministic algorithms can be represented by a single integer marking when to buy skis. To describe σ and A more precisely, we use the following notation: