Tight Bounds for the Advice Complexity of the Online Minimum Steiner Tree Problem.

In this work, we study the advice complexity of the online minimum Steiner tree problem (ST). Given a (known) graph G = (V, E) endowed with a weight function on the edges, a set of N terminals are revealed in a step-wise manner. The algorithm maintains a sub-graph of chosen edges, and at each stage, chooses more edges from G to its solution such that the terminals revealed so far are connected in it. In the standard online setting this problem was studied and a tight bound of O(log(N)) on its competitive ratio is known. Here, we study the power of non-uniform advice and fully characterize it. As a first result we show that using q . log(vertical bar V vertical bar) advice bits, where 0 <= q <= N - 1, it is possible to obtain an algorithm with a competitive ratio of O(log(N/q). We then show a matching lower bound for all values of q, and thus settle the question.