Abstract:

We prove new cell-probe lower bounds for dynamic data structures that maintain a subset of \{1,2,...,n\}, and compute various statistics of the set. The data structure is said to handle insertions \emph{non-adaptively} if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. For any such data structure that can compute the median of the set, we prove that:

where \tins is the number of memory locations accessed during insertions, \tmed is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. When the data structure is able to perform deletions non-adaptively and compute the minimum non-adaptively, we prove

where \tmin is the number of locations accessed to compute the minimum, and \tdel is the number of locations accessed to perform deletions. For the predecessor search problem, where the data structure 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

were \tp is the number of locations accessed to compute predecessors.

These bounds are nearly matched by Binary Search Trees in some range of parameters. Our results follow from using the Sunflower Lemma of Erd\H{o}s and Rado \cite{ErdosR60} together with several kinds of encoding arguments.

Changes to previous version:

Fixed an error in the statement of past work.

Abstract:

We prove new cell-probe lower bounds for dynamic data structures that maintain a subset of \{1,2,...,n\}, and compute various statistics of the set. The data structure is said to handle insertions \emph{non-adaptively} if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. For any such data structure that can compute the median of the set, we prove that:

where \tins is the number of memory locations accessed during insertions, \tmed is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. When the data structure is able to perform deletions non-adaptively and compute the minimum non-adaptively, we prove

where \tmin is the number of locations accessed to compute the minimum, and \tdel is the number of locations accessed to perform deletions. For the predecessor search problem, where the data structure 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

were \tp is the number of locations accessed to compute predecessors.

These bounds are nearly matched by Binary Search Trees in some range of parameters. Our results follow from using the Sunflower Lemma of Erd\H{o}s and Rado \cite{ErdosR60} together with several kinds of encoding arguments.

Changes to previous version:

Added several references that came to our attention after the initial version was posted, as well as a new proof that proves lower bounds for computing the minimum of a set.

Abstract:

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:
t_m \geq \Omega\left(\frac{n^{\frac{1}{2(t_i+1)}}}{w \cdot t_i}\right),
where t_i is the number of memory locations accessed during insertions, t_m 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
t_p \geq \Omega\left(\frac{\log n}{\log \log n + \log w}\right) or t_i \geq \Omega\left(\frac{t_p\cdot n^{\frac{1}{2(t_p+1)}}}{\log n}\right),
were t_p 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 Erdos and Rado to these questions.