Measuring the problem-relevant information in input

¹ Institute of Mathematics, Slovak Academy of Sciences, Slovakia; Stefan.Dobrev@savba.sk
² Department of Computer Science, Comenius University, Bratislava, Slovakia; kralovic@dcs.fmph.uniba.sk;pardubska@dcs.fmph.uniba.sk

Received: 1 August 2007
Accepted: 19 January 2009

Abstract

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of the output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated by the algorithm to the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver upper and lower bounds in both communication modes; in the case of DiffServ problem in helper mode the bounds are tight.