4,672 research outputs found
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
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
Fixed-Parameter Algorithms for Computing Kemeny Scores - Theory and Practice
The central problem in this work is to compute a ranking of a set of elements
which is "closest to" a given set of input rankings of the elements. We define
"closest to" in an established way as having the minimum sum of Kendall-Tau
distances to each input ranking. Unfortunately, the resulting problem Kemeny
consensus is NP-hard for instances with n input rankings, n being an even
integer greater than three. Nevertheless this problem plays a central role in
many rank aggregation problems. It was shown that one can compute the
corresponding Kemeny consensus list in f(k) + poly(n) time, being f(k) a
computable function in one of the parameters "score of the consensus", "maximum
distance between two input rankings", "number of candidates" and "average
pairwise Kendall-Tau distance" and poly(n) a polynomial in the input size. This
work will demonstrate the practical usefulness of the corresponding algorithms
by applying them to randomly generated and several real-world data. Thus, we
show that these fixed-parameter algorithms are not only of theoretical
interest. In a more theoretical part of this work we will develop an improved
fixed-parameter algorithm for the parameter "score of the consensus" having a
better upper bound for the running time than previous algorithms.Comment: Studienarbei
On the Computational Power of Radio Channels
Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity.
Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature
An Improved Distributed Algorithm for Maximal Independent Set
The Maximal Independent Set (MIS) problem is one of the basics in the study
of locality in distributed graph algorithms. This paper presents an extremely
simple randomized algorithm providing a near-optimal local complexity for this
problem, which incidentally, when combined with some recent techniques, also
leads to a near-optimal global complexity.
Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86]
provide the global complexity guarantee that, with high probability, all nodes
terminate after rounds. In contrast, our initial focus is on the
local complexity, and our main contribution is to provide a very simple
algorithm guaranteeing that each particular node terminates after rounds, with probability at least
. The guarantee holds even if the randomness outside -hops
neighborhood of is determined adversarially. This degree-dependency is
optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04].
Interestingly, this local complexity smoothly transitions to a global
complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider
[FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high
probability global complexity of ,
where denotes the maximum degree. This improves over the result of Barenboim et al., and gets close
to the lower bound of Kuhn et al.
Corollaries include improved algorithms for MIS in graphs of upper-bounded
arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local
Computation Algorithms (LCA) model, and a faster distributed algorithm for the
Lov\'asz Local Lemma
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