Factorial Lower Bounds for (Almost) Random Order Streams
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In this paper we introduce and study the StreamingCycles problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph G, comprising n/ℓ disjoint length-ℓ cycles on n vertices, are partitioned randomly among n players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output the parity of some cycle in G after one round of sequential communication. Our main result is an ℓ^Ω(ℓ) lower bound on the communication complexity of StreamingCycles, which is tight up to constant factors in ℓ. Applications of our lower bound for StreamingCycles include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA'18] and the first exponential space lower bounds for random walk generation.
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