Non-Adaptive Data Structure Lower Bounds for Median and Predecessor Search from Sunflowers

We prove new cell-probe lower bounds for data structures that maintain a subset of {1, 2, ..., n}, and compute the median of the set. The data structure is said to handle insertions non-adaptively if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. We prove that any such data structure must satisfy: tm ≥ Ω ( n 1 2(ti+1) w · ti ) , where ti is the number of memory locations accessed during insertions, tm is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. Our lower bounds are nearly matched by Binary Search Trees. For the predecessor search problem, where the algorithm is required to compute the predecessor of any element in the set, we prove that if computing the predecessors can be done non-adaptively, then tp ≥ Ω ( log n log log n + logw ) or ti ≥ Ω ( tp · n 1 2(tp+1) log n ) , were tp is the number of locations accessed to compute predecessors. Again, these bounds prove that Binary Search Trees have essentially optimal parameters for the predecessor search problem. Our results follow from a novel application of the Sunflower Lemma of Erdős and Rado [ER60] to these questions.