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1.
This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send $\tilde{\varOmega}(m)$ messages and incur $\tilde{\varOmega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\tilde{\varOmega}(m)$ messages and $\tilde{\varOmega}(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using ${\tilde{O}}(\sqrt {n} + D)$ time) requires O(m+n 3/2) messages, and the best known message-optimal algorithm (using ${\tilde{O}}(m)$ messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.  相似文献   

2.
In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate $k$ distinct messages to all $n$ nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of $O((k+\log n + D)\varDelta )$ rounds with high probability, where $D$ and $\varDelta $ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of $k$ this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is order optimal and the stopping time is $\varTheta (k + D)$ . To eliminate the factor of $\varDelta $ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol $\mathcal{S } $ . The stopping time of TAG is $O(k+\log n +d(\mathcal{S })+t(\mathcal{S }))$ , where $t(\mathcal{S })$ is the stopping time of the spanning tree protocol, and $d(\mathcal{S })$ is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for $k=\varOmega (n)$ , where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after $\varTheta (n)$ rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for $k=\varOmega (\text{ polylog }(n))$ . The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis.  相似文献   

3.
We present a technique for numerically solving convection-diffusion problems in domains $\varOmega $ with curved boundary. The technique consists in approximating the domain $\varOmega $ by polyhedral subdomains $\mathsf{{D}}_h$ where a finite element method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain $\varOmega $ . This approach allows for the use of only polyhedral elements; there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. To achieve this, the boundary condition on the border of $\varOmega $ is transferred to the border of $\mathsf{D }_h$ by using simple line integrals. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that, whenever the distance of $\mathsf{{D}}_h$ to $\partial \varOmega $ is of order of the meshsize $h$ , the convergence properties of the resulting method are the same as those for the case in which $\varOmega =\mathsf{{D}}_h$ . We also show numerical evidence indicating that the ratio of the $L^2(\varOmega )$ norm of the error in the scalar variable computed with $d>0$ to that of that computed with $d=0$ remains constant (and fairly close to one), whenever the distance $d$ is proportional to $\min \{h,Pe^{-1}\}/(k+1)^2$ , where $Pe$ is the so-called Péclet number.  相似文献   

4.
We give matching upper and lower bounds of \(\varTheta(\min(\frac{\log m}{\log \log m},\, n))\) for the individual step complexity of a wait-free m-valued adopt-commit object implemented using multi-writer registers for n anonymous processes. While the upper bound is deterministic, the lower bound holds for randomized adopt-commit objects as well. Our results are based on showing that adopt-commit objects are equivalent, up to small additive constants, to a simpler class of objects that we call conflict detectors. Our anonymous lower bound also applies to the individual step complexity of m-valued wait-free anonymous consensus, even for randomized algorithms with global coins against an oblivious adversary. The upper bound can be used to slightly improve the cost of randomized consensus against an oblivious adversary. For deterministic non-anonymous implementations of adopt-commit objects, we show a lower bound of \(\varOmega(\min(\frac{\log m}{\log \log m}, \frac{\sqrt{\log n}}{\log \log n}))\) and an upper bound of \(O(\min(\frac{\log m}{\log \log m},\, \log n))\) on the worst-case individual step complexity. For randomized non-anonymous implementations, we show that any execution contains at least one process whose steps exceed the deterministic lower bound.  相似文献   

5.
We study broadcasting, also known as one-to-all communication, in synchronous radio networks with known topology modeled by undirected (symmetric) graphs, where the interference range of a node is likely exceeding its transmission range. In this model, if two nodes are connected by a transmission edge they can communicate directly. On the other hand, if two nodes are connected by an interference edge they cannot communicate directly and transmission of one node disables recipience of any message at the other node. For a network $G,$ we term the smallest integer $d$ , s.t., for any interference edge $e$ there exists a simple path formed of at most $d$ transmission edges connecting the endpoints of $e$ as its interference distance $d_I$ . In this model the schedule of transmissions is precomputed in advance. It is based on the full knowledge of the size and the topology (including location of transmission and interference edges) of the network. We are interested in the design of fast broadcasting schedules that are energy efficient, i.e., based on a bounded number of transmissions executed at each node. We adopt $n$ as the number of nodes, $D_T$ is the diameter of the subnetwork induced by the transmission edges, and $\varDelta $ refers to the maximum combined degree (formed of transmission and interference edges) of the network. We contribute the following new results: (1) We prove that for networks with the interference distance $d_I\ge 2$ any broadcasting schedule requires at least $D_T+\varOmega (\varDelta \cdot \frac{\log {n}}{\log {\varDelta }})$ rounds. (2) We provide for networks modeled by bipartite graphs an algorithm that computes $1$ -shot (each node transmits at most once) broadcasting schedules of length $O(\varDelta \cdot \log {n})$ . (3) The main result of the paper is an algorithm that computes a $1$ -shot broadcasting schedule of length at most $4 \cdot D_T + O(\varDelta \cdot d_I \cdot \log ^4{n})$ for networks with arbitrary topology. Note that in view of the lower bound from (1) if $d_I$ is poly-logarithmic in $n$ this broadcast schedule is a poly-logarithmic factor away from the optimal solution.  相似文献   

6.
For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.  相似文献   

7.
Given a DNF formula f on n variables, the two natural size measures are the number of terms or size s(f) and the maximum width of a term w(f). It is folklore that small DNF formulas can be made narrow: if a formula has m terms, it can be ${\epsilon}$ -approximated by a formula with width ${{\rm log}(m/{\epsilon})}$ . We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be ${\epsilon}$ -approximated by a width w DNF with at most ${(w\, {\rm log}(1/{\epsilon}))^{O(w)}}$ terms. We combine our sparsification result with the work of Luby & Velickovic (1991, Algorithmica 16(4/5):415–433, 1996) to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic ${n^{\tilde{O}({\rm log}\, {\rm log} (n))}}$ time algorithm that computes an additive ${\epsilon}$ approximation to the fraction of satisfying assignments of f for ${\epsilon = 1/{\rm poly}({\rm log}\, n)}$ . The previous best result due to Luby and Velickovic from nearly two decades ago had a run time of ${n^{{\rm exp}(O(\sqrt{{\rm log}\, {\rm log} n}))}}$ (Luby & Velickovic 1991, in Algorithmica 16(4/5):415–433, 1996).  相似文献   

8.
A pair of unit clauses is called conflicting if it is of the form (x), $(\bar{x})$ . A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28:411?C421, 1981) showed that for each UCF CNF formula with m clauses we can simultaneously satisfy at least $\hat{ \varphi } m$ clauses, where $\hat{ \varphi }=(\sqrt{5}-1)/2$ . We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula F with m clauses we can find, in polynomial time, a?subformula F?? with m?? clauses such that we can simultaneously satisfy at least $\hat{ \varphi } m+(1-\hat{ \varphi })m'+(2-3\hat {\varphi })n''/2$ clauses (in F), where n?? is the number of variables in F which are not in F??. We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds m/2 and $m(\sqrt{5}-1)/2$ . The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31:335?C354, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most 6k+3 variables and 10k clauses. We improve this to 4k variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above.  相似文献   

9.
Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just few examples of this phenomenon. The quest for combinatorial algorithms that do not use heavy algebraic machinery, but are roughly as efficient, has become a central field of study in this area. Combinatorial algorithms are often simpler than their algebraic counterparts. Moreover, in many cases, combinatorial algorithms and proofs provide additional understanding of studied problems. In this paper we initiate the study of combinatorial algorithms for Distributed Graph Coloring problems. In a distributed setting a communication network is modeled by a graph $G=(V,E)$ of maximum degree $\varDelta $ . The vertices of $G$ host the processors, and communication is performed over the edges of $G$ . The goal of distributed vertex coloring is to color $V$ with $(\varDelta + 1)$ colors such that any two neighbors are colored with distinct colors. Currently, efficient algorithms for vertex coloring that require $O(\varDelta + \log ^* n)$ time are based on the algebraic algorithm of Linial (SIAM J Comput 21(1):193–201, 1992) that employs set-systems. The best currently-known combinatorial set-system free algorithm, due to Goldberg et al. (SIAM J Discret Math 1(4):434–446, 1988), requires $O(\varDelta ^2+\log ^*n)$ time. We significantly improve over this by devising a combinatorial $(\varDelta + 1)$ -coloring algorithm that runs in $O(\varDelta + \log ^* n)$ time. This exactly matches the running time of the best-known algebraic algorithm. In addition, we devise a tradeoff for computing $O(\varDelta \cdot t)$ -coloring in $O(\varDelta /t + \log ^* n)$ time, for almost the entire range $1 < t < \varDelta $ . We also compute a Maximal Independent Set in $O(\varDelta + \log ^* n)$ time on general graphs, and in $O(\log n/ \log \log n)$ time on graphs of bounded arboricity. Prior to our work, these results could be only achieved using algebraic techniques. We believe that our algorithms are more suitable for real-life networks with limited resources, such as sensor networks.  相似文献   

10.
We consider transactional memory contention management in the context of balanced workloads, where if a transaction is writing, the number of write operations it performs is a constant fraction of its total reads and writes. We explore the theoretical performance boundaries of contention management in balanced workloads from the worst-case perspective by presenting and analyzing two new polynomial time contention management algorithms. We analyze the performance of a contention management algorithm by comparison with an optimal offline contention management algorithm to provide a competitive ratio. The first algorithm Clairvoyant is $O(\sqrt{s})$ -competitive, where s is the number of shared resources. This algorithm depends on explicitly knowing the conflict graph at each time step of execution. The second algorithm Non-Clairvoyant is $O(\sqrt{s} \cdot \log n)$ -competitive, with high probability, which is only a O(log?n) factor worse, but does not require knowledge of the conflict graph, where n is the number of transactions. Both of these algorithms are greedy. We also prove that the performance of Clairvoyant is close to optimal, since there is no polynomial time contention management algorithm for the balanced transaction scheduling problem that is better than $O((\sqrt{s})^{1-\varepsilon})$ -competitive for any constant ε>0, unless NP?ZPP. To our knowledge, these results are significant improvements over the best previously known O(s) competitive ratio bound.  相似文献   

11.
We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $\mathbb R ^{d}$ , $d\in \{1,2,3\}$ . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $hp$ -version of the finite element method explicitly in terms of the mesh width $h$ , polynomial degree $p$ and wavenumber $k$ . It is shown that the optimal convergence order estimate is obtained under the conditions that $kh/\sqrt{p}$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$ . On regular meshes, the first condition is improved to the requirement that $kh/p$ be sufficiently small.  相似文献   

12.
A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by ${\mathbb{F}}^{*}$ -graphs— $\mathbb {F}$ -coloured graphs where each edge has exactly one colour from $\mathbb{F}\setminus \{0\},\ \mathbb{F}$ a field—and named respectively $\mathbb{F}$ -rank-width and $\mathbb {F}$ -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for $\mathbb{F}^{*}$ -graphs and prove that $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width are characterised by a list of $\mathbb{F}^{*}$ -graphs to exclude as vertex-minors (this list is finite if $\mathbb{F}$ is finite). An algorithm that decides in time O(n 3) whether an $\mathbb{F}^{*}$ -graph with n vertices has $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) at most k, for fixed k and fixed finite field $\mathbb{F}$ , is also given. Graph operations to check MSOL-definable properties on $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.  相似文献   

13.
In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC0[p], the class of constant-depth circuits with unbounded fan-in MOD p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ${\epsilon > 0}$ . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route of constructing our PRG, we prove two structural results for low-degree polynomials over finite fields that can be of independent interest.
  1. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . Then, for every ${\epsilon > 0}$ , there exists a subset ${S \subset [n]}$ , whose size depends only on d and ${\epsilon}$ , such that ${\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}$ . Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.
  2. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . If the distribution of f when applied to uniform zero–one bits is ${\epsilon}$ -far (in statistical distance) from its distribution when applied to biased bits, then for every ${\delta > 0}$ , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on ${\epsilon,\delta}$ and d) of lower degree polynomials.
  相似文献   

14.
The discrete logarithm problem modulo a composite??abbreviate it as DLPC??is the following: given a (possibly) composite integer n??? 1 and elements ${a, b \in \mathbb{Z}_n^*}$ , determine an ${x \in \mathbb{N}}$ satisfying a x ?=?b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem ${{\rm DLPC}_\varepsilon}$ obtained by adding in the DLPC the constraint ${x\le (1-\varepsilon)n}$ , where ${\varepsilon}$ is an arbitrary fixed number, ${0 < \varepsilon\le\frac{1}{2}}$ . We prove that factoring n reduces in deterministic subexponential time to the ${{\rm DLPC}_\varepsilon}$ with ${O_\varepsilon((\ln n)^2)}$ queries for moduli less or equal to n.  相似文献   

15.
In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model (Clementi et al. in J Parallel Distrib Comput 64:89–96, 2004; Kuhn et al. in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process $u$ is provided with a local link detector set consisting of every neighbor connected to $u$ by a reliable link. The algorithm solves the CCDS problem in $O\left( \frac{\varDelta \log ^2{n}}{b} + \log ^3{n}\right) $ rounds, with high probability, where $\varDelta $ is the maximum degree in the reliable link graph, $n$ is the network size, and $b$ is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in $\log ^3{n}$ time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in $O(\varDelta $ polylog $(n))$ time under the assumption of $O(1)$ unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require $\varOmega (\varDelta )$ rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.  相似文献   

16.
We consider the problem of leader election (LE) in single-hop radio networks with synchronized time slots for transmitting and receiving messages. We assume that the actual number n of processes is unknown, while the size u of the ID space is known, but is possibly much larger. We consider two types of collision detection: strong (SCD), whereby all processes detect collisions, and weak (WCD), whereby only non-transmitting processes detect collisions. We introduce loneliness detection (LD) as a key subproblem for solving LE in WCD systems. LD informs all processes whether the system contains exactly one process or more than one. We show that LD captures the difference in power between SCD and WCD, by providing an implementation of SCD over WCD and LD. We present two algorithms that solve deterministic and probabilistic LD in WCD systems with time costs of ${\mathcal{O}(\log \frac{u}{n})}$ and ${\mathcal{O}(\min( \log \frac{u}{n}, \frac{\log (1/\epsilon)}{n}))}$ , respectively, where ${\epsilon}$ is the error probability. We also provide matching lower bounds. Assuming LD is solved, we show that SCD systems can be emulated in WCD systems with factor-2 overhead in time. We present two algorithms that solve deterministic and probabilistic LE in SCD systems with time costs of ${\mathcal{O}(\log u)}$ and ${\mathcal{O}(\min ( \log u, \log \log n + \log (\frac{1}{\epsilon})))}$ , respectively, where ${\epsilon}$ is the error probability. We provide matching lower bounds.  相似文献   

17.
A number of algorithms for computing the simulation preorder (and equivalence) on Kripke structures are available. Let $\varSigma $ denote the state space, ${\rightarrow }$ the transition relation and $P_{\mathrm {sim}}$ the partition of $\varSigma $ induced by simulation equivalence. While some algorithms are designed to reach the best space bounds, whose dominating additive term is $|P_{\mathrm {sim}}|^2$ , other algorithms are devised to attain the best time complexity $O(|P_{\mathrm {sim}}||{\rightarrow }|)$ . We present a novel simulation algorithm which is both space and time efficient: it runs in $O(|P_ {\mathrm {sim}}|^2 \log |P_{\mathrm {sim}}| + |\varSigma |\log |\varSigma |)$ space and $O(|P_{\mathrm {sim}}||{\rightarrow }|\log |\varSigma |)$ time. Our simulation algorithm thus reaches the best space bounds while closely approaching the best time complexity.  相似文献   

18.
Let $\pi'_{w}$ denote the failure function of the Knuth-Morris-Pratt algorithm for a word w. In this paper we study the following problem: given an integer array $A'[1 \mathinner {\ldotp \ldotp }n]$ , is there a word w over an arbitrary alphabet Σ such that $A'[i]=\pi'_{w}[i]$ for all i? Moreover, what is the minimum cardinality of Σ required? We give an elementary and self-contained $\mathcal{O}(n\log n)$ time algorithm for this problem, thus improving the previously known solution (Duval et al. in Conference in honor of Donald E. Knuth, 2007), which had no polynomial time bound. Using both deeper combinatorial insight into the structure of π′ and advanced algorithmic tools, we further improve the running time to $\mathcal{O}(n)$ .  相似文献   

19.
We prove two main results on how arbitrary linear threshold functions ${f(x) = {\rm sign}(w \cdot x - \theta)}$ over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is ${\epsilon}$ -close to a threshold function depending only on ${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$ many variables, where ${{\rm Inf}(f)}$ denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s well-known theorem (Friedgut in Combinatorica 18(1):474–483, 1998), which states that every Boolean function f is ${\epsilon}$ -close to a function depending only on ${2^{O({\rm Inf}(f)/\epsilon)}}$ many variables, for the case of threshold functions. We complement this upper bound by showing that ${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$ many variables are required for ${\epsilon}$ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is ${\epsilon}$ -close to a threshold function with integer weights at most ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$ This is an improvement, in the dependence on the error parameter ${\epsilon}$ , on an earlier result of Servedio (Comput Complex 16(2):180–209, 2007) which gave a ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180–209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.  相似文献   

20.
We employ geometric discord and measurement induced nonlocality to quantify quantumness of some well-known bipartite bound entangled states, namely the two families of Horodecki’s ( $2\otimes 4, 3\otimes 3$ and $4\otimes 4$ dimensional) bound entangled states and that of Bennett et al.’s in $3\otimes 3$ dimension. In most of the cases our results are analytic and both the measures attain relatively small value. The amount of quantumness in the $4\otimes 4$ bound entangled state of Benatti et al. and the $2\otimes 8$ state having the same matrix representation (in computational basis) is same. Coincidently, the $2m\otimes 2m$ Werner and isotropic states also exhibit the same property, when seen as $2\otimes 2m^2$ dimensional states.  相似文献   

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