Combinatorial algorithms for distributed graph coloring

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.     

An improved FPTAS for maximizing the weighted number of just-in-time jobs in a two-machine flow shop problem

Recently, Shabtay and Bensoussan (2012) developed an original exact pseudo-polynomial algorithm and an efficient $\upvarepsilon $ -approximation algorithm (FPTAS) for maximizing the weighted number of just-in-time jobs in a two-machine flow shop problem. The complexity of the FPTAS is $O$ (( $n^{4}/\upvarepsilon $ )log( $n$ / $\upvarepsilon $ )), where $n$ is the number of jobs. In this note we suggest another pseudo-polynomial algorithm that can be converted to a new FPTAS which improves Shabtay–Bensoussan’s complexity result and runs in $O(n^{3}/\upvarepsilon )$ time.     

Efficient broadcasting in radio networks with long-range interference

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.     

Order optimal information spreading using algebraic gossip

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.     

Null controllability of Kolmogorov-type equations

We study the null controllability of Kolmogorov-type equations $\partial _t f + v^\gamma \partial _x f - \partial _v^2 f = u(t,x,v) 1_{\omega }(x,v)$ in a rectangle $\Omega $ , under an additive control supported in an open subset $\omega $ of $\Omega $ . For $\gamma =1$ , with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support $\omega $ . This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009), in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if $\gamma =1$ or if $\gamma =2$ and $\omega $ contains the segment $\{v=0\}$ , and only in large time if $\gamma =2$ and $\omega $ does not contain the segment $\{v=0\}$ . Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995), is based on two key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.     

An efficient simulation algorithm on Kripke structures

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.     

Toward more localized local algorithms: removing assumptions concerning global knowledge

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and $(\Delta +1)$ -coloring algorithms by Barenboim and Elkin (Distrib Comput 22(5–6):363–379, 2010), by Kuhn (2009), and by Panconesi and Srinivasan (J Algorithms 20(2):356–374, 1996), as well as the $O\mathopen {}(\Delta ^2)$ -coloring algorithm by Linial (J Comput 21:193, 1992). Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree $\Delta $ or the number of nodes $n$ . This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem (In particular, it applies to all aforementioned algorithms). To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.     

A new PAC bound for intersection-closed concept classes

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.     

A study of single-machine scheduling problem to maximize throughput

We study inherent structural properties of a strongly NP-hard problem of scheduling $n$ jobs with release times and due dates on a single machine to minimize the number of late jobs. Our study leads to two polynomial-time algorithms. The first algorithm with the time complexity $O(n^3\log n)$ solves the problem if during its execution no job with some special property occurs. The second algorithm solves the version of the problem when all jobs have the same length. The time complexity of the latter algorithm is $O(n^2\log n)$ , which is an improvement over the earlier known algorithm with the time complexity $O(n^5)$ .     

Optimized Parameter Search for Large Datasets of the Regularization Parameter and Feature Selection for Ridge Regression

In this paper we propose mathematical optimizations to select the optimal regularization parameter for ridge regression using cross-validation. The resulting algorithm is suited for large datasets and the computational cost does not depend on the size of the training set. We extend this algorithm to forward or backward feature selection in which the optimal regularization parameter is selected for each possible feature set. These feature selection algorithms yield solutions with a sparse weight matrix using a quadratic cost on the norm of the weights. A naive approach to optimizing the ridge regression parameter has a computational complexity of the order $O(R K N^{2} M)$ with $R$ the number of applied regularization parameters, $K$ the number of folds in the validation set, $N$ the number of input features and $M$ the number of data samples in the training set. Our implementation has a computational complexity of the order $O(KN^3)$ . This computational cost is smaller than that of regression without regularization $O(N^2M)$ for large datasets and is independent of the number of applied regularization parameters and the size of the training set. Combined with a feature selection algorithm the algorithm is of complexity $O(RKNN_s^3)$ and $O(RKN^3N_r)$ for forward and backward feature selection respectively, with $N_s$ the number of selected features and $N_r$ the number of removed features. This is an order $M$ faster than $O(RKNN_s^3M)$ and $O(RKN^3N_rM)$ for the naive implementation, with $N \ll M$ for large datasets. To show the performance and reduction in computational cost, we apply this technique to train recurrent neural networks using the reservoir computing approach, windowed ridge regression, least-squares support vector machines (LS-SVMs) in primal space using the fixed-size LS-SVM approximation and extreme learning machines.     

Chain conditions on \textit{BL}-algebras

The aim of this paper is to solve the open problem appeared in Motamed and Moghaderi (Soft Comput 2012), about the relation between Noetherian (Artinian) $\textit{BL}$ -algebras in short exact sequences. Also, a better theorem to improve its results is suggested. The relation between Noetherian and Artinian $\textit{BL}$ -algebras is found, the concept of length for a filter in $\textit{BL}$ -algebras is introduced and properties of finite length $\textit{BL}$ -algebras are developed. Finally, it is proved that any $\textit{BL}$ -algebra has finite length if and only if be Noetherian and Artinian.     

A new set of generators and a physical interpretation for the SU (3) finite subgroup D(9,1,1;2,1,1)

After 100 years of effort, the classification of all the finite subgroups of $SU(3)$ is yet incomplete. The most recently updated list can be found in Ludl (J Phys A Math Theory 44:255204, 2011), where the structure of the series $(C)$ and $(D)$ of $SU(3)$ -subgroups is studied. We provide a minimal set of generators for one of these groups which has order $162$ . These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones–Kauffman version of $SU(2)$ Chern–Simons theory at level $4$ . In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product $\mathbb Z _9\times \mathbb Z _3\rtimes S_3$ with respect to conjugation.     

A quantum multiply-accumulator

This paper proposes a quantum multiply-accumulator circuit (QMAC), which can perform the calculation on conventional integers faster than its classical counterpart. Whereas classically applying a multiply–adder (MAC) $n$ times to $k$ bit integers would require $O(n \log k)$ parallel steps, the hybrid QMAC needs only $O(n + k)$ steps for the exact result and $O(n + \log k)$ steps for an approximate result. The proposed circuit could potentially be embedded in a conventional computer architecture as a quantum device or accelerator, enabling a wide range of applications to execute faster.     

A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem $1||L_{\max }$ are studied in [2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: $\ell _{1},\ell _{2},\ell _{\infty },\ell _{H}^{\Sigma } $ , and $\ell _{H}^{\max }$ . It appears that the $O(n^{2})$ -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms $\ell _{1},\ell _{\infty },\ell _{H}^{\Sigma }$ , and $ \ell _{H}^{\max }$ , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm $\ell _{2}$ is treated separately. As a by-product, we resolve an open question on the complexity of problem $1||\sum \alpha _{j}T_{j}^{2}$ .     

Arithmetizing Classes Around \textsf{NC} 1 and \textsf{L}

The parallel complexity class $\textsf{NC}$ ¹ has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, $\textsf{\#NC}$ ¹ and $\textsf{\#BWBP}$ . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in $\textsf{FLogDCFL}$ , while counting proof-trees in logarithmic width formulae has the same power as $\textsf{\#NC}$ ¹. We also consider polynomial-degree restrictions of $\textsf{SC}$ ⁱ , denoted $\textsf{sSC}$ ⁱ , and show that the Boolean class $\textsf{sSC}$ ¹ is sandwiched between $\textsf{NC}$ ¹ and $\textsf{L}$ , whereas $\textsf{sSC}$ ⁰ equals $\textsf{NC}$ ¹. On the other hand, the arithmetic class $\textsf{\#sSC}$ ⁰ contains $\textsf{\#BWBP}$ and is contained in $\textsf{FL}$ , and $\textsf{\#sSC}$ ¹ contains $\textsf{\#NC}$ ¹ and is in $\textsf{SC}$ ². We also investigate some closure properties of the newly defined arithmetic classes.     

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. The PDE, which is a nonlinear, fourth-order parabolic equation, is the $L^2$ gradient flow of the energy $ \int _\Omega \left( - \frac{1}{2} \ln \left( 1 + | \nabla \phi |^2 \right) + \frac{\epsilon ^2}{2}|\Delta \phi (\mathbf{x})|^2 \right) \mathrm{d}\mathbf{x}$ . The energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work (Shen et al. in SIAM J Numer Anal 50:105–125, 2012). The resulting scheme is highly nonlinear, and its implementation is non-trivial. In this paper, we propose a linear iteration algorithm to solve the resulting nonlinear system. To accomplish this we introduce an $O(s^2)$ (with $s$ the time step size) artificial diffusion term, a Douglas-Dupont-type regularization, that leads to a contraction mapping property. As a result, the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be very efficiently implemented with the help of FFT in a collocation Fourier spectral setting. We present a careful analysis showing convergence for the numerical scheme in a discrete $L^\infty (0, T; H^1) \cap L^2 (0,T; H^3)$ norm. Some numerical simulation results are presented to demonstrate the efficiency of the linear iteration solver and the convergence of the scheme as a whole.     

Validating the Knuth-Morris-Pratt Failure Function,Fast and Online

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 $\mathcal{O}(n)$ .     

Improved Approximation of Linear Threshold Functions

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.     

CLOVER: a faster prior-free approach to rare-category detection

Rare-category detection helps discover new rare classes in an unlabeled data set by selecting their candidate data examples for labeling. Most of the existing approaches for rare-category detection require prior information about the data set without which they are otherwise not applicable. The prior-free algorithms try to address this problem without prior information about the data set; though, the compensation is high time complexity, which is not lower than $O(dN^2)$ where $N$ is the number of data examples in a data set and $d$ is the data set dimension. In this paper, we propose CLOVER a prior-free algorithm by introducing a novel rare-category criterion known as local variation degree (LVD), which utilizes the characteristics of rare classes for identifying rare-class data examples from other types of data examples and passes those data examples with maximum LVD values to CLOVER for labeling. A remarkable improvement is that CLOVER’s time complexity is $O(dN^{2-1/d})$ for $d > 1$ or $O(N\log N)$ for $d = 1$ . Extensive experimental results on real data sets demonstrate the effectiveness and efficiency of our method in terms of new rare classes discovery and lower time complexity.