136 research outputs found
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
A parent-centered radial layout algorithm for interactive graph visualization and animation
We have developed (1) a graph visualization system that allows users to
explore graphs by viewing them as a succession of spanning trees selected
interactively, (2) a radial graph layout algorithm, and (3) an animation
algorithm that generates meaningful visualizations and smooth transitions
between graphs while minimizing edge crossings during transitions and in static
layouts.
Our system is similar to the radial layout system of Yee et al. (2001), but
differs primarily in that each node is positioned on a coordinate system
centered on its own parent rather than on a single coordinate system for all
nodes. Our system is thus easy to define recursively and lends itself to
parallelization. It also guarantees that layouts have many nice properties,
such as: it guarantees certain edges never cross during an animation.
We compared the layouts and transitions produced by our algorithms to those
produced by Yee et al. Results from several experiments indicate that our
system produces fewer edge crossings during transitions between graph drawings,
and that the transitions more often involve changes in local scaling rather
than structure.
These findings suggest the system has promise as an interactive graph
exploration tool in a variety of settings
Cluster Computing and the Power of Edge Recognition
We study the robustness--the invariance under definition changes--of the
cluster class CL#P [HHKW05]. This class contains each #P function that is
computed by a balanced Turing machine whose accepting paths always form a
cluster with respect to some length-respecting total order with efficient
adjacency checks. The definition of CL#P is heavily influenced by the defining
paper's focus on (global) orders. In contrast, we define a cluster class,
CLU#P, to capture what seems to us a more natural model of cluster computing.
We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the
more natural, flexible features of CLU#P to prove new robustness results for
CL#P and to expand what is known about the closure properties of CL#P.
The complexity of recognizing edges--of an ordered collection of computation
paths or of a cluster of accepting computation paths--is central to this study.
Most particularly, our proofs exploit the power of unique discovery of
edges--the ability of nondeterministic functions to, in certain settings,
discover on exactly one (in some cases, on at most one) computation path a
critical piece of information regarding edges of orderings or clusters
Tight lower bounds on the ambiguity of strong, total, associative, one-way functions
AbstractWe study the ambiguity, or “many-to-one”-ness, of two-argument, one-way functions that are strong (that is, hard to invert even if one of their arguments is given), total, and associative. Such powerful one-way functions are the basis of a cryptographic paradigm described by Rabi and Sherman (Inform. Process. Lett. 64(2) (1997) 239) and were shown by Hemaspaandra and Rothe (J. Comput. System Sci. 58(3) (1999) 648) to exist exactly if standard one-way functions exist.Rabi and Sherman (1997) show that no total, associative function defined over a universe having at least two elements is one-to-one. We show that if P≠UP, then, for every d∈N+, there is an O(log1dn)-to-one, strong, total, associative, one-way function σd. We argue that this bound is tight in the sense that any total, associative function having similar properties to σd but not necessarily strong or one-way must have at least the same order of magnitude of ambiguity as σd has. We demonstrate that the techniques used in proving the above-stated results easily apply to other classes of total, associative functions.We provide a complete characterization for the existence of strong, total, associative, one-way functions whose ambiguity approaches the lower bounds we provide. We say a language is in PolylogP if there exists a polynomial-time Turing machine M accepting the language such that for some d∈R+ it holds that M has on each string x at most O(logdn) accepting paths, where n=|x|. We show that P≠PolylogP if and only for some d∈R+ there exists an O(logdn)-to-one, strong, total, associative, one-way function
Tuning the Diversity of Open-Ended Responses from the Crowd
Crowdsourcing can solve problems that current fully automated systems cannot.
Its effectiveness depends on the reliability, accuracy, and speed of the crowd
workers that drive it. These objectives are frequently at odds with one
another. For instance, how much time should workers be given to discover and
propose new solutions versus deliberate over those currently proposed? How do
we determine if discovering a new answer is appropriate at all? And how do we
manage workers who lack the expertise or attention needed to provide useful
input to a given task? We present a mechanism that uses distinct payoffs for
three possible worker actions---propose,vote, or abstain---to provide workers
with the necessary incentives to guarantee an effective (or even optimal)
balance between searching for new answers, assessing those currently available,
and, when they have insufficient expertise or insight for the task at hand,
abstaining. We provide a novel game theoretic analysis for this mechanism and
test it experimentally on an image---labeling problem and show that it allows a
system to reliably control the balance betweendiscovering new answers and
converging to existing ones
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}. For a class F of boolean functions and a class G of graphs, an (F, G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #Pcomplete to compute the number of fixed points in an (F, G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas
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