Continuous Monitoring of Distributed Data Streams over a Time-Based Sliding Window

In this paper we extend the study of algorithms for monitoring distributed data streams from whole data streams to a time-based sliding window. The concern is how to minimize the communication between individual streams and the root, while allowing the root, at any time, to report the global statistics of all streams within a given error bound. This paper presents communication-efficient algorithms for three classical statistics, namely, basic counting, frequent items and quantiles. The worst-case communication cost over a window is $O(\frac{k}{\varepsilon} \log\frac{\varepsilon N}{k})$ bits for basic counting, $O(\frac{k}{\varepsilon} \log\frac{N}{k})$ words for frequent items and $O(\frac{k}{\varepsilon^{2}} \log\frac{N}{k})$ words for quantiles, where k is the number of distributed data streams, N is the total number of items in the streams that arrive or expire in the window, and ε<1 is the given error bound. The performance of our algorithms matches and nearly matches the corresponding lower bounds. We also show how to generalize these results to streams with out-of-order data.     

Tile Complexity of Approximate Squares

The standard Tile Assembly Model (TAM) of Winfree (Algorithmic self-assembly of DNA, Ph.D. thesis, 1998) is a mathematical theory of crystal aggregations via monomer additions with applications to the emerging science of DNA self-assembly. Self-assembly under the rules of this model is programmable and can perform Turing universal computation. Many variations of this model have been proposed and the canonical problem of assembling squares has been studied extensively. We consider the problem of building approximate squares in TAM. Given any $\varepsilon \in (0,\frac{1}{4}]$ we show how to construct squares whose sides are within (1±ε)N of any given positive integer N using $O( \frac{\log \frac{1}{\varepsilon}}{\log \log\frac{1}{\varepsilon}} + \frac{\log \log \varepsilon N}{\log \log \log \varepsilon N} )$ tile types. We prove a matching lower bound by showing that $\varOmega( \frac{\log \frac{1}{\varepsilon}}{\log \log\frac{1}{\varepsilon}} + \frac{\log \log \varepsilon N}{\log \log \log \varepsilon N} )$ tile types are necessary almost always to build squares of required approximate dimensions. In comparison, the optimal construction for a square of side exactly N in TAM uses $O(\frac{\log N}{\log \log N})$ tile types. The question of constructing approximate squares has been recently studied in a modified tile assembly model involving concentration programming. All our results are trivially translated into the concentration programming model by assuming arbitrary (non-zero) concentrations for our tile types. Indeed, the non-zero concentrations could be chosen by an adversary and our results would still hold. Our construction can get highly accurate squares using very few tile types and are feasible starting from values of N that are orders of magnitude smaller than the best comparable constructions previously suggested. At an accuracy of ε=0.01, the number of tile types used to achieve a square of size 10⁷ is just 58 and our constructions are proven to work for all N≥13130. If the concentrations of the tile types are carefully chosen, we prove that our construction assembles an L×L square in optimal assembly time O(L) where (1?ε)N≤L≤(1+ε)N.     

Pseudorandom generators for combinatorial checkerboards

We define a combinatorial checkerboard to be a function f : {1, . . . , m} ^d → {1,?1} of the form ${f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)}$ for some functions f _i : {1, . . . , m} → {1,?1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1,?1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m = 2. We construct a pseudorandom generator that ${\epsilon}$ -fools all combinatorial checkerboards with seed length ${O\bigl(\log m+\log d\cdot\log\log d+\log^{3/2} \frac{1}{\epsilon}\bigr)}$ . Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length ${O\bigl(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\bigr)}$ . Our seed length is better except when ${\frac{1}{\epsilon}\geq d^{\omega(\log d)}}$ .     

Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis

Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen’s (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .     

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.     

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.     

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.     

Property Testing on k-Vertex-Connectivity of?Graphs

We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least $\frac{\epsilon dn}{2}$ edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in $O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})$ time (c>1 is a constant) for (k?1)-vertex-connected graphs, and in $O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})$ time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.     

On reducing factorization to the discrete logarithm problem modulo a composite

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.     

Leader election using loneliness detection

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.     

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.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.     

Rounds in Combinatorial Search

A set system $\mathcal{H} \subseteq2^{[m]}$ is said to be separating if for every pair of distinct elements x,y∈[m] there exists a set $H\in\mathcal{H}$ such that H contains exactly one of them. The search complexity of a separating system $\mathcal{H} \subseteq 2^{[m]}$ is the minimum number of questions of type “x∈H?” (where $H \in\mathcal{H}$ ) needed in the worst case to determine a hidden element x∈[m]. If we receive the answer before asking a new question then we speak of the adaptive complexity, denoted by $\mathrm{c} (\mathcal{H})$ ; if the questions are all fixed beforehand then we speak of the non-adaptive complexity, denoted by $\mathrm{c}_{na} (\mathcal{H})$ . If we are allowed to ask the questions in at most k rounds then we speak of the k-round complexity of $\mathcal{H}$ , denoted by $\mathrm{c}_{k} (\mathcal{H})$ . It is clear that $|\mathcal{H}| \geq\mathrm{c}_{na} (\mathcal{H}) = \mathrm{c}_{1} (\mathcal{H}) \geq\mathrm{c}_{2} (\mathcal{H}) \geq\cdots\geq\mathrm{c}_{m} (\mathcal{H}) = \mathrm{c} (\mathcal{H})$ . A group of problems raised by G.O.H. Katona is to characterize those separating systems for which some of these inequalities are tight. In this paper we are discussing set systems $\mathcal{H}$ with the property $|\mathcal{H}| = \mathrm{c}_{k} (\mathcal{H}) $ for any k≥3. We give a necessary condition for this property by proving a theorem about traces of hypergraphs which also has its own interest.     

Structuring unreliable radio networks

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.     

Tight Bounds for Adopt-Commit Objects

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.     

\lambda -Statistical convergence of fuzzy numbers and fuzzy functions of order \theta

In this paper, we introduce the concept of $\lambda $ -statistical convergence of order $\theta $ and strong $\lambda $ -summability of order $\theta $ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions and introduce the spaces like $S_\lambda ^\theta (\hat{f})$ and $\omega _{\lambda p} ^\theta (\hat{f})$ . Some inclusion relations in those spaces and also the underlying relation between these two spaces are also obtained.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

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.     

External Memory Planar Point Location with Logarithmic Updates

Point location is an extremely well-studied problem both in internal memory models and recently also in the external memory model. In this paper, we present an I/O-efficient dynamic data structure for point location in general planar subdivisions. Our structure uses linear space to store a subdivision with N segments. Insertions and deletions of segments can be performed in amortized O(log? _B N) I/Os and queries can be answered in $O(\log_{B}^{2} N)$ I/Os in the worst-case. The previous best known linear space dynamic structure also answers queries in $O(\log_{B}^{2} N)$ I/Os, but only supports insertions in amortized $O(\log_{B}^{2} N)$ I/Os. Our structure is also considerably simpler than previous structures.     

General DG-Methods for Highly Indefinite Helmholtz Problems

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.