Time-space tradeoffs, multiparty communication complexity, and nearest-neighbor problems

The first non-trivial time-space tradeoff lower bounds have been shown for decision problems in P using notions derived from the study of two-party communication complexity. These results are proven directly for branching programs, natural generalizations of decision trees to directed graphs that provide elegant models of both non-uniform time T and space S simultaneously. We develop a new lower bound criterion, based on extending two-party communication complexity ideas to multiparty communication complexity. Applying this criterion to an explicit Boolean function based on a multilinear form over F ₂. for suitable s, we show lower bounds that yield T = Ω(n log² n) when S ⩽ n^1-ε log |D| for large input domain D. Finally, we develop lower bounds for nearest-neighbor problems involving n data points in a variety of d-dimensional metric spaces