Search results for "upper"
showing 10 items of 987 documents
A Polynomial Quantum Query Lower Bound for the Set Equality Problem
2004
The set equality problem is to tell whether two sets A and B are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any ω(1) query lower bound when sets A and B are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles \(\Omega(\sqrt[5]{\frac{n}{\ln n}})\) times, where n=|A|=|B|.
Symmetry-assisted adversaries for quantum state generation
2011
We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the $backslash$sc Graph Isomorphism problem. We show that for the related problem of $backslash$sc Index Erasure our method leads to a lower bound of $backslash Omega(backslash sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is …
Nonmalleable encryption of quantum information
2008
We introduce the notion of "non-malleability" of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a "unitary 2-design" [Dankert et al.], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of (d^2-1)^2+1 on the number of unitaries in a 2-design [Gross et al.], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with =…
On n–Fold Blocking Sets
1986
An n-fold blocking set is a set of n-disjoint blocking sets. We shall prove upper and lower bounds for the number of components in an n-fold blocking set in projective and affine spaces.
Conditional Random Quantities and Compounds of Conditionals
2013
In this paper we consider finite conditional random quantities and conditional previsions assessments in the setting of coherence. We use a suitable representation for conditional random quantities; in particular the indicator of a conditional event $E|H$ is looked at as a three-valued quantity with values 1, or 0, or $p$, where $p$ is the probability of $E|H$. We introduce a notion of iterated conditional random quantity of the form $(X|H)|K$ defined as a suitable conditional random quantity, which coincides with $X|HK$ when $H \subseteq K$. Based on a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of cohere…
Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test
2017
We explore multi-round quantum memoryless communication protocols. These are restricted version of multi-round quantum communication protocols. The “memoryless” term means that players forget history from previous rounds, and their behavior is obtained only by input and message from the opposite player. The model is interesting because this allows us to get lower bounds for models like automata, Ordered Binary Decision Diagrams and streaming algorithms. At the same time, we can prove stronger results with this restriction. We present a lower bound for quantum memoryless protocols. Additionally, we show a lower bound for Disjointness function for this model. As an application of communicatio…
Real Line Arrangements and Surfaces with Many Real Nodes
2008
A long standing question is if the maximum number μ(d) of nodes on a surface of degree d in P( ) can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, μ(d) 5 12 d, is provided by Chmutov’s construction from 1992 which gives surfaces whose nodes have non-real coordinates. Using explicit constructions of certain real line arrangements we show that Chmutov’s construction can be adapted to give only real singularities. All currently best known constructions which exceed Chmutov’s lower bound (i.e., for d = 3, 4, . . . , 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up t…
Frequency Assignment and Multicoloring Powers of Square and Triangular Meshes
2005
The static frequency assignment problem on cellular networks can be abstracted as a multicoloring problem on a weighted graph, where each vertex of the graph is a base station in the network, and the weight associated with each vertex represents the number of calls to be served at the vertex. The edges of the graph model interference constraints for frequencies assigned to neighboring stations. In this paper, we first propose an algorithm to multicolor any weighted planar graph with at most $\frac{11}{4}W$ colors, where W denotes the weighted clique number. Next, we present a polynomial time approximation algorithm which garantees at most 2W colors for multicoloring a power square mesh. Fur…
Online Scheduling of Task Graphs on Heterogeneous Platforms
2020
Modern computing platforms commonly include accelerators. We target the problem of scheduling applications modeled as task graphs on hybrid platforms made of two types of resources, such as CPUs and GPUs. We consider that task graphs are uncovered dynamically, and that the scheduler has information only on the available tasks, i.e., tasks whose predecessors have all been completed. Each task can be processed by either a CPU or a GPU, and the corresponding processing times are known. Our study extends a previous $4\sqrt{m/k}$ 4 m / k -competitive online algorithm by Amaris et al. [1] , where $m$ m is the number of CPUs and $k$ k the number of GPUs ( $m\geq k$ m ≥ k ). We prove that no online…
A genetic system based on simulated crossover of sequences of two-bit genes
2006
AbstractWe introduce a genetic model based on simulated crossover of fixed sequences of two-bit genes. Results are(1)a lower bound on population size is exhibited such that a transition takes the stochastic finite population genetic system near the next state of the deterministic infinite population genetic system (provided both begin in the same state);(2)states and dynamics of the deterministic infinite population genetic system are derived for arbitrary (finite) fitness functions (expressed in terms of multivariate polynomials);(3)in the case of quadratic fitness defined by weight matrices with m nonnull entries it is shown that each state transition can be implemented in time O(m+l), wh…