Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model [article]

Huacheng Yu
2015 arXiv   pre-print
In this paper, we develop a new communication model to prove a data structure lower bound for the dynamic interval union problem. The problem is to maintain a multiset of intervals I over [0, n] with integer coordinates, supporting the following operations: - insert(a, b): add an interval [a, b] to I, provided that a and b are integers in [0, n]; - delete(a, b): delete a (previously inserted) interval [a, b] from I; - query(): return the total length of the union of all intervals in I. It is
more » ... ated to the two-dimensional case of Klee's measure problem. We prove that there is a distribution over sequences of operations with O(n) insertions and deletions, and O(n^0.01) queries, for which any data structure with any constant error probability requires Ω(n n) time in expectation. Interestingly, we use the sparse set disjointness protocol of Håstad and Wigderson [ToC'07] to speed up a reduction from a new kind of nondeterministic communication games, for which we prove lower bounds. For applications, we prove lower bounds for several dynamic graph problems by reducing them from dynamic interval union.
arXiv:1512.01293v1 fatcat:yl32fnk2pzda3oi7dxdj2zml6a