11,238 research outputs found
The Partition Bound for Classical Communication Complexity and Query Complexity
We describe new lower bounds for randomized communication complexity and
query complexity which we call the partition bounds. They are expressed as the
optimum value of linear programs. For communication complexity we show that the
partition bound is stronger than both the rectangle/corruption bound and the
\gamma_2/generalized discrepancy bounds. In the model of query complexity we
show that the partition bound is stronger than the approximate polynomial
degree and classical adversary bounds. We also exhibit an example where the
partition bound is quadratically larger than polynomial degree and classical
adversary bounds.Comment: 28 pages, ver. 2, added conten
Separating decision tree complexity from subcube partition complexity
The subcube partition model of computation is at least as powerful as
decision trees but no separation between these models was known. We show that
there exists a function whose deterministic subcube partition complexity is
asymptotically smaller than its randomized decision tree complexity, resolving
an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is
based on the information-theoretic techniques first introduced to lower bound
the randomized decision tree complexity of the recursive majority function.
We also show that the public-coin partition bound, the best known lower bound
method for randomized decision tree complexity subsuming other general
techniques such as block sensitivity, approximate degree, randomized
certificate complexity, and the classical adversary bound, also lower bounds
randomized subcube partition complexity. This shows that all these lower bound
techniques cannot prove optimal lower bounds for randomized decision tree
complexity, which answers an open question of Jain and Klauck (2010) and Jain,
Lee, and Vishnoi (2014).Comment: 16 pages, 1 figur
Query-to-Communication Lifting for BPP
For any -bit boolean function , we show that the randomized
communication complexity of the composed function , where is an
index gadget, is characterized by the randomized decision tree complexity of
. In particular, this means that many query complexity separations involving
randomized models (e.g., classical vs. quantum) automatically imply analogous
separations in communication complexity.Comment: 21 page
The quantum adversary method and classical formula size lower bounds
We introduce two new complexity measures for Boolean functions, or more
generally for functions of the form f:S->T. We call these measures sumPI and
maxPI. The quantity sumPI has been emerging through a line of research on
quantum query complexity lower bounds via the so-called quantum adversary
method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the
realization that these many different formulations are in fact equivalent.
Given that sumPI turns out to be such a robust invariant of a function, we
begin to investigate this quantity in its own right and see that it also has
applications to classical complexity theory.
As a surprising application we show that sumPI^2(f) is a lower bound on the
formula size, and even, up to a constant multiplicative factor, the
probabilistic formula size of f. We show that several formula size lower bounds
in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93],
including a key lemma of [Has98], are in fact special cases of our method.
The second quantity we introduce, maxPI(f), is always at least as large as
sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a
lower bound on formula size. While sumPI(f) is always a lower bound on the
quantum query complexity of f, this is not the case in general for maxPI(f). A
strong advantage of sumPI(f) is that it has both primal and dual
characterizations, and thus it is relatively easy to give both upper and lower
bounds on the sumPI complexity of functions. To demonstrate this, we look at a
few concrete examples, for three functions: recursive majority of three, a
function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200
Generalizations of the distributed Deutsch-Jozsa promise problem
In the {\em distributed Deutsch-Jozsa promise problem}, two parties are to
determine whether their respective strings are at the {\em
Hamming distance} or . Buhrman et al. (STOC' 98)
proved that the exact {\em quantum communication complexity} of this problem is
while the {\em deterministic communication complexity} is
. This was the first impressive (exponential) gap between
quantum and classical communication complexity.
In this paper, we generalize the above distributed Deutsch-Jozsa promise
problem to determine, for any fixed , whether
or , and show that an exponential gap between exact
quantum and deterministic communication complexity still holds if is an
even such that , where is given. We also deal with a promise version of the
well-known {\em disjointness} problem and show also that for this promise
problem there exists an exponential gap between quantum (and also
probabilistic) communication complexity and deterministic communication
complexity of the promise version of such a disjointness problem. Finally, some
applications to quantum, probabilistic and deterministic finite automata of the
results obtained are demonstrated.Comment: we correct some errors of and improve the presentation the previous
version. arXiv admin note: substantial text overlap with arXiv:1309.773
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
A strong direct product theorem says that if we want to compute k independent
instances of a function, using less than k times the resources needed for one
instance, then our overall success probability will be exponentially small in
k. We establish such theorems for the classical as well as quantum query
complexity of the OR function. This implies slightly weaker direct product
results for all total functions. We prove a similar result for quantum
communication protocols computing k instances of the Disjointness function.
Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for
sorting N items on a quantum computer, which is optimal up to polylog factors.
They also give several tight time-space and communication-space tradeoffs for
the problems of Boolean matrix-vector multiplication and matrix multiplication.Comment: 22 pages LaTeX. 2nd version: some parts rewritten, results are
essentially the same. A shorter version will appear in IEEE FOCS 0
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