Revisiting the Direct Sum Theorem and Space Lower Bounds in Random Order Streams
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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There is a natural relationship between lower bounds in the multi-pass stream model and lower bounds in multi-round communication. However, this connection is less understood than the connection between single-pass stream computation and one-way communication. In this paper, we consider data-stream problems for which reductions from natural multi-round communication problems do not yield tight bounds or do not apply. While lower bounds are known for some of these data-stream problems, many of these only apply to deterministic or comparison-based algorithms, whereas the lower bounds we present apply to any (possibly randomized) algorithms. Our results are particularly relevant to evaluating functions that are dependent on the ordering of the stream, such as the longest increasing subsequence and a variant of tree pointer jumping in which pointers are revealed according to a post-order traversal.Our approach is based on establishing "pass-elimination" type results that are analogous to the round-elimination results of Miltersen et al. [23] and Sen [29]. We demonstrate our approach by proving tight bounds for a range of data-stream problems including finding the longest increasing sequences (a problem that has recently become very popular [22,16,30,15,12] and we resolve an open question of [30]), constructing convex hulls and fixed-dimensional linear programming (generalizing results of [8] to randomized algorithms), and the "greater-than" problem (improving results of [9]). These results will also clarify one of the main messages of our work: sometimes it is necessary to prove lower bounds directly for stream computation rather than proving a lower bound for a communication problem and then constructing a reduction to a data-stream problem.