2,951 research outputs found
Tight lower bounds on the ambiguity of strong, total, associative, one-way functions
AbstractWe study the ambiguity, or âmany-to-oneâ-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 Pâ UP, then, for every dâN+, there is an O(log1dn)-to-one, strong, total, associative, one-way function Ïd. We argue that this bound is tight in the sense that any total, associative function having similar properties to Ïd but not necessarily strong or one-way must have at least the same order of magnitude of ambiguity as Ïd 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 dâR+ it holds that M has on each string x at most O(logdn) accepting paths, where n=|x|. We show that Pâ PolylogP if and only for some dâR+ there exists an O(logdn)-to-one, strong, total, associative, one-way function
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Cell-Probe Lower Bounds from Online Communication Complexity
In this work, we introduce an online model for communication complexity.
Analogous to how online algorithms receive their input piece-by-piece, our
model presents one of the players, Bob, his input piece-by-piece, and has the
players Alice and Bob cooperate to compute a result each time before the next
piece is revealed to Bob. This model has a closer and more natural
correspondence to dynamic data structures than classic communication models do,
and hence presents a new perspective on data structures.
We first present a tight lower bound for the online set intersection problem
in the online communication model, demonstrating a general approach for proving
online communication lower bounds. The online communication model prevents a
batching trick that classic communication complexity allows, and yields a
stronger lower bound. We then apply the online communication model to prove
data structure lower bounds for two dynamic data structure problems: the Group
Range problem and the Dynamic Connectivity problem for forests. Both of the
problems admit a worst case -time data structure. Using online
communication complexity, we prove a tight cell-probe lower bound for each:
spending (even amortized) time per operation results in at best an
probability of correctly answering a
-fraction of the queries
Random Neural Networks and Optimisation
In this thesis we introduce new models and learning algorithms for the Random
Neural Network (RNN), and we develop RNN-based and other approaches for the
solution of emergency management optimisation problems.
With respect to RNN developments, two novel supervised learning algorithms are
proposed. The first, is a gradient descent algorithm for an RNN extension model
that we have introduced, the RNN with synchronised interactions (RNNSI), which
was inspired from the synchronised firing activity observed in brain neural circuits.
The second algorithm is based on modelling the signal-flow equations in RNN as a
nonnegative least squares (NNLS) problem. NNLS is solved using a limited-memory
quasi-Newton algorithm specifically designed for the RNN case.
Regarding the investigation of emergency management optimisation problems,
we examine combinatorial assignment problems that require fast, distributed and
close to optimal solution, under information uncertainty. We consider three different
problems with the above characteristics associated with the assignment of
emergency units to incidents with injured civilians (AEUI), the assignment of assets
to tasks under execution uncertainty (ATAU), and the deployment of a robotic
network to establish communication with trapped civilians (DRNCTC).
AEUI is solved by training an RNN tool with instances of the optimisation problem
and then using the trained RNN for decision making; training is achieved using
the developed learning algorithms. For the solution of ATAU problem, we introduce
two different approaches. The first is based on mapping parameters of the
optimisation problem to RNN parameters, and the second on solving a sequence of
minimum cost flow problems on appropriately constructed networks with estimated
arc costs. For the exact solution of DRNCTC problem, we develop a mixed-integer
linear programming formulation, which is based on network flows. Finally, we design
and implement distributed heuristic algorithms for the deployment of robots
when the civilian locations are known or uncertain
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