Online algorithms with advice

Abstract

In online problems, the input forms a finite sequence of requests. Each request must be processed, i. e., a partial output has to be computed only depending on the requests having arrived so far, and it is not allowed to change this partial output subsequently. The aim of an online algorithm is to produce a sequence of partial outputs that optimizes some global measure. The most frequently used tool for analyzing the quality of online algorithms is the competitive analysis which compares the solution quality of an online algorithm to the optimal solution for the whole input sequence, and in fact measures the degradation in the solution quality caused by the lack of any information about the input. In this paper, we investigate to what extent the solution quality can be improved by allowing the algorithm to extract a given amount of information about the input. We consider the recently introduced notion of advice complexity where the algorithm, in addition to being fed the requests one by one, has access to a tape of advice bits that were computed by some oracle function from the complete input. The advice complexity is the number of advice bits read. We introduce an improved model of advice complexity and investigate the connections of advice complexity to the competitive ratio of both deterministic and randomized online algorithms using the paging problem, job shop scheduling, and the routing problem on a line as sample problems. Our results for all of these problems show that very small advice (only three bits in the case of paging) already suffices to significantly improve over the best deterministic algorithm. Moreover, to achieve the same competitive ratio as any randomized online algorithm, a logarithmic number of advice bits is sufficient. On the other hand, to obtain optimality, much larger advice is necessary.