Publication Information

M.-Y. Kao and S. R. Tate. On-Line Difference Maximization, in SIAM Journal on Discrete Mathematics, Vol. 12, No. 1, 1999, pp. 78--90. Online AlgorithmsJournal

Abstract

In this paper we examine problems motivated by on-line financial 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 $n-O(1)$, and so differs from the best possible off-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 $n^2/8-\Theta(n \log n)$. An analysis shows that the optimal expected off-line gain is $n^2/6+\Theta(1)$, so the performance of our on-line algorithm is within a factor of 3/4 of the best off-line strategy.

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