838 research outputs found
Ambiguity and Communication
The ambiguity of a nondeterministic finite automaton (NFA) N for input size n
is the maximal number of accepting computations of N for an input of size n.
For all k, r 2 N we construct languages Lr,k which can be recognized by NFA's
with size k poly(r) and ambiguity O(nk), but Lr,k has only NFA's with
exponential size, if ambiguity o(nk) is required. In particular, a hierarchy
for polynomial ambiguity is obtained, solving a long standing open problem
(Ravikumar and Ibarra, 1989, Leung, 1998)
Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method
A formulation of "Ne\v{c}iporuk's lower bound method" slightly more inclusive
than the usual complexity-measure-specific formulation is presented. Using this
general formulation, limitations to lower bounds achievable by the method are
obtained for several computation models, such as branching programs and Boolean
formulas having access to a sublinear number of nondeterministic bits. In
particular, it is shown that any lower bound achievable by the method of
Ne\v{c}iporuk for the size of nondeterministic and parity branching programs is
at most
Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication
This paper develops a new technique for proving amortized, randomized
cell-probe lower bounds on dynamic data structure problems. We introduce a new
randomized nondeterministic four-party communication model that enables
"accelerated", error-preserving simulations of dynamic data structures.
We use this technique to prove an cell-probe
lower bound for the dynamic 2D weighted orthogonal range counting problem
(2D-ORC) with updates and queries, that holds even
for data structures with success probability. This
result not only proves the highest amortized lower bound to date, but is also
tight in the strongest possible sense, as a matching upper bound can be
obtained by a deterministic data structure with worst-case operational time.
This is the first demonstration of a "sharp threshold" phenomenon for dynamic
data structures.
Our broader motivation is that cell-probe lower bounds for exponentially
small success facilitate reductions from dynamic to static data structures. As
a proof-of-concept, we show that a slightly strengthened version of our lower
bound would imply an lower bound for the
static 3D-ORC problem with space. Such result would give a
near quadratic improvement over the highest known static cell-probe lower
bound, and break the long standing barrier for static data
structures
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