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Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model
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 over with integer coordinates,
supporting the following operations:
- insert(a, b): add an interval to , provided that
and are integers in ;
- delete(a, b): delete a (previously inserted) interval from
;
- query(): return the total length of the union of all intervals in
.
It is related to the two-dimensional case of Klee's measure problem. We prove
that there is a distribution over sequences of operations with
insertions and deletions, and queries, for which any data
structure with any constant error probability requires time
in expectation. Interestingly, we use the sparse set disjointness protocol of
H\aa{}stad 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
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