On-line difference maximization

In this paper we examine problems motivated by on-line nancial problems and stochastic games. In particular, we consider a sequence of entirely arbitrary distinct values arriving in random order, and must devise strategies for selecting low values followed by high values in such a way as to maximize the expected gain in rank from low values to high values. First, we consider a scenario in which only one low value and one high value may be selected. We give an optimal on-line algorithm for this scenario, and analyze it to show that, surprisingly, the expected gain is niO(1), and so diers from the best possible o-line gain by only a constant additive term (which is, in fact, fairly small|at most 15). In a second scenario, we allow multiple nonoverlapping low/high selections, where the total gain for our algorithm is the sum of the individual pair gains. We also give an optimal on-line algorithm for this problem, where the expected gain is n 2 =8i (n logn). An analysis shows that the optimal expected o-line gain is n 2 =6 + (1), so the performance of our on-line algorithm is within a factor of 3=4 of the best o-line strategy.