Abstract: This paper describes two fused floating-point operations and applies them to the implementation of fast Fourier transform (FFT) processors. The fused operations are a two-term dot product and an add-subtract unit. The FFT processors use "butterfly” operations that consist of multiplications, additions, and subtractions of complex valued data. Both radix-2 and radix-4 butterflies are implemented efficiently with the two fused floating-point operations. When placed and routed using a high performance standard cell technology, the fused FFT butterflies are about 15 percent faster and 30 percent smaller than a conventional implementation. Also the numerical results of the fused implementations are slightly more accurate, since they use fewer rounding operations.

Abstract: In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NP-hard, even when centers have no capacity restrictions and optimal factor 2 approximation algorithms are known. With capacities, when all centers have identical capacities, a 6 approximation is known with no better lower bounds than for the infinite capacity version. While many generalizations and variations of this problem have been studied extensively, no progress was made on the capacitated version for a general capacity function. We develop the first constant factor approximation algorithm for this problem. Our algorithm uses an LP rounding approach to solve this problem, and works for the case of non-uniform hard capacities, when multiple copies of a node may not be chosen and can be extended to the case when there is a hard bound on the number of copies of a node that may be selected. Finally, for non-uniform soft capacities we present a much simpler 11-approximation algorithm, which we find as one more evidence that hard capacities are much harder to deal with.

Abstract: We investigate analysts' motives for rounding annual EPS forecasts (placing a zero or five in the penny location of the forecast). We first show that an intuitive reason for analysts to engage in rounding is in circumstances where the penny digit of the forecast is of less economic significance. By rounding, analysts reveal that their forecasts are not intended to be precise to the penny. We also show that analyst incentives impact the likelihood of rounding. Specifically, we predict that analysts will exert less effort forecasting earnings for firms that generate less brokerage or investment banking business since such firms create less value for the analysts' employers. As a consequence of this reduced effort and attention, the analyst will be more uncertain about the penny digit of the forecast and so will round. Our results are consistent with this prediction. One implication of our findings is that a rounded forecast is a simple and easily observable proxy for a more noisy measure of the ma...

Abstract: We study network optimization that considers power minimization as an objective. Studies have shown that mechanisms such as speed scaling can significantly reduce the power consumption of telecommunication networks by matching the consumption of each network element to the amount of processing required for its carried traffic. Most existing research on speed scaling focuses on a single network element in isolation. We aim for a network-wide optimization. Specifically, we study a routing problem with the objective of provisioning guaranteed speed/bandwidth for a given demand matrix while minimizing power consumption. Optimizing the routes critically relies on the characteristic of the speed-power curve f(s), which is how power is consumed as a function of the processing speed s. If f is superadditive, we show that there is no bounded approximation in general for integral routing, i.e., each traffic demand follows a single path. This contrasts with the well-known logarithmic approximation for subadditive functions. However, for common speed-power curves such as polynomials f(s) = μsα, we are able to show a constant approximation via a simple scheme of randomized rounding. We also generalize this rounding approach to handle the case in which a nonzero startup cost σ appears in the speed-power curve, i.e., f(s) = {σ + μsα, if s >; 0; 0, if s = 0. We present an O((σ/μ)1/α)-approximation, and we discuss why coming up with an approximation ratio independent of the startup cost may be hard. Finally, we provide simulation results to validate our algorithmic approaches.

Abstract: Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB 2010 benchmark instances.

Abstract: We improve the well-known Wilkinson-type estimates for the error of standard floating-point recursive summation and dot product by up to a factor 2. The bounds are valid when computed in rounding to nearest, no higher order terms are necessary, and they are best possible. For summation there is no restriction on the number of summands. The proofs are short by using a new tool for the estimation of errors in floating-point computations which cures drawbacks of the “unit in the last place (ulp)”. The presented estimates are nice and simple, and closer to what one may expect.

Abstract: Programs using floating-point arithmetic are prone to accuracy problems caused by rounding and catastrophic cancellation. These phenomena provoke bugs that are notoriously hard to track down: the p...

Abstract: Embodiments of the present invention may provide methods and circuits for energy efficient floating point multiply and/or add operations. A variable precision floating point circuit may determine the certainty of the result of a multiply-add floating point calculation in parallel with the floating-point calculation. The variable precision floating point circuit may use the certainty of the inputs in combination with information from the computation, such as, binary digits that cancel, normalization shifts, and rounding, to perform a calculation of the certainty of the result. A floating point multiplication circuit may determine whether a lowest portion of a multiplication result could affect the final result and may induce a replay of the multiplication operation when it is determined that the result could affect the final result.

Abstract: Introduction of an evidence-based practice change, such as hourly rounding, can be difficult in the hospital setting. This study used ethnographic methods to examine problems with the implementation of hourly rounding on 2 similar inpatient units at our hospital. Results indicate that careful planning, communication, implementation, and evaluation are required for successful implementation of a nursing practice change.

Abstract: We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixed-integer nonlinear programs.

Abstract: This paper provides two methods to improve the reliability of carrier phase integer ambiguity resolution: The first one is a group of multi-frequency linear combinations that include both code and carrier phase measurements, and allow an arbitrary scaling of the geometry, an arbitrary scaling of the ionospheric delay, and any preferred wavelength. The maximization of the ambiguity discrimination results in combinations with a wavelength of several meters and a noise level of a few centimeters. These combinations could be beneficial for both Real-Time Kinematics (RTK) and Precise Point Positioning (PPP). The second method incorporates some statistical a priori knowledge of attitude into the actual fixing. The a priori knowledge includes the length and direction of the baseline between two receivers and is given either as a uniform or Gaussian distribution. It enables a substantial reduction of the search space volume but also ensures a large robustness over errors in the a priori information. Both methods improve the accuracy of the float solution, which motivates a simple rounding for ambiguity fixing. A method is described, which enables an efficient computation of its success rate with a few integral transformations. Copyright © 2012 Institute of Navigation.

Abstract: Systems and methods for implementing a floating point fused multiply and accumulate with scaling (FMASc) operation. A floating point unit receives input multiplier, multiplicand, addend, and scaling factor operands. A multiplier block is configured to multiply mantissas of the multiplier and multiplicand to generate an intermediate product. Alignment logic is configured to pre-align the addend with the intermediate product based on the scaling factor and exponents of the addend, multiplier, and multiplicand, and accumulation logic is configured to add or subtract a mantissa of the pre-aligned addend with the intermediate product to obtain a result of the floating point unit. Normalization and rounding are performed on the result, avoiding rounding during intermediate stages.

Abstract: We present the best known algorithms for approximating the minimum-size undirected $k$-edge connected spanning subgraph. For simple graphs our approximation ratio is $1+ {1}/(2k) + O({1}/{k^2})$. The more precise version of this bound requires $k\ge 7$, and for all such $k$ it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], $1+2/(k+1)$. The improvement comes in two steps. First we show that for simple $k$-edge connected graphs, any laminar family of degree $k$ sets is smaller than the general bound ($n(1+ {3}/{k} + O(1/k\sqrt k))$ versus $2n$). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is $1+(21/11)k 1$. This improves the previous ratio $1+2/k$ [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant $c>0$, an approximation ratio $\le 1+c/k$ implies $P=NP$. Our approximation ratio extends to the minimum-size Steiner network problem, where $k$ denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding.

Abstract: We study numerically thermal effects at the depinning transition of an elastic string driven in a two-dimensional uncorrelated disorder potential. The velocity of the string exactly at the sample critical force is shown to behave as V∼Tψ, with ψ the thermal rounding exponent. We show that the computed value of the thermal rounding exponent, ψ=0.15, is robust and accounts for the different scaling properties of several observables both in the steady state and in the transient relaxation to the steady state. In particular, we show the compatibility of the thermal rounding exponent with the scaling properties of the steady-state structure factor, the universal short-time dynamics of the transient velocity at the sample critical force, and the velocity scaling function describing the joint dependence of the steady-state velocity on the external drive and temperature.

Abstract: We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a randomized algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm, surprisingly, closely approximates the idealized process (where the tokens are divisible) on important network topologies. On $d$-dimensional torus graphs with $n$ nodes it deviates from the idealized process only by an additive constant. In contrast, the randomized rounding approach of Friedrich and Sauerwald [Proceedings of the \textup41st Annual ACM Symposium on Theory of Computing, 2009, pp. 121--130] can deviate up to $\Omega(\operatorname{polylog}(n))$, and the deterministic algorithm of Rabani, Sinclair, and Wanka [Proceedings of the \textup39th Annual IEEE Symposium on Foundations of Computer Science, 1998, pp. 694--705] has a deviation of $\Omega(n^{1/d})$. This makes our quasirandom algorithm the first known algorithm for t...

Abstract: Several methods for the multiplication of point and/or interval matrices with interval result are discussed. Some are based on new priori estimates of the error of floating-point matrix products. The amount of overestimation including all rounding errors is analyzed. In particular, algorithms for conversion of infimum-supremum to midpoint-radius representation are discussed and analyzed, one of which is proved to be optimal. All methods are much faster than the classical method because almost no switch of the rounding mode is necessary, and because our methods are based on highly optimized BLAS3 routines. We discuss several possibilities to trade overestimation against computational effort. Numerical examples focussing in particular on applications using interval matrix multiplications are presented.

Abstract: We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.

Abstract: If a system for lossless compression of images applies a decorrelation step, this step must map integer input values to integer output values. This can be achieved, for example, using the integer wavelet transform (IWT). The non-linearity, introduced by the obligatory rounding steps, is the main drawback of the IWT, since it deteriorates the desired filter characteristic. This paper discusses different methods for reducing the influence of rounding in 5/3 and 9/7 filter banks. A novel combination of two-dimensional implementations of the JPEG2000 9/7 filter bank with new filter coefficients is proposed and the effects of the methods on lossless image compression are investigated. In addition, these filter banks are compared to the 9/7 Deslauriers-Dubuc filter bank (97DD). The analysed two-dimensional implementations generally perform better than their one-dimensional counterparts in terms of compression ratio for natural images. On average, the 2D 97DD filter bank performs best. In addition, it has been found that the compression results cannot be improved by simply reducing the number of lifting steps via 2D implementations of the JPEG2000 9/7 filter bank. Only the 2D implementation with a minimum number of lifting steps, in combination with modified lifting coefficients, leads to fewer bits per pixel than the separable implementation on average for a selected set of images.

Abstract: Implementing integer division in hardware is expensive when compared to multiplication. In the case where the divisor is a constant, expensive integer division algorithms can be replaced by cheaper integer multiplications and additions. This paper presents the conditions for multiply-add schemes to perform correctly rounded unsigned invariant integer division under one of three rounding modes. We propose a heuristic to explore the space of implementations meeting the conditions we derive. Experiments show that an average speed up of 20% and area reduction of 50% can be achieved compared to existing correctly rounded approaches. Extension to two's complement numbers is also presented.

Abstract: Staff members on a medical-surgical unit in a large community teaching hospital adapted the hourly rounding concept to their specific patient population. Lessons learned and strategies to assure continuous success with the rounding process are addressed.

Abstract: Multiplications by simple rational constants often appear in fixed- or floating-point application code, for instance, in the form of division by an integer constant. The hardware implementation of such operations is of practical interest to reconfigurable computing. It is well known that the binary representation of rational constants is eventually periodic. This brief shows how this feature can be exploited to implement multiplication by a rational constant in a number of additions that is logarithmic in the precision. An open-source implementation of these techniques is provided and is shown to be practically relevant for constants with small numerators and denominators, where it provides improvements of 20% to 40% in area with respect to the state of the art. It is also shown that, for such constants, the additional cost for a correctly rounded result is very small and that correct rounding very often comes for free in practice.

Abstract: We study the k-level uncapacitated facility location problem, where clients need to be connected with paths crossing open facilities of k types (levels). In this paper we give an approximation algorithm that for any constant k, in polynomial time, delivers solutions of cost at most αk times OPT, where αk is an increasing function of k, with limk→∞αk=3. Our algorithm rounds a fractional solution to an extended LP formulation of the problem. The rounding builds upon the technique of iteratively rounding fractional solutions on trees (Garg, Konjevod, and Ravi SODA'98) originally used for the group Steiner tree problem. We improve the approximation ratio for k-UFL for all k≥3, in particular we obtain the ratio equal 2.02, 2.14, and 2.24 for k=3,4, and 5.

Abstract: This paper presents the algorithm and architecture of the decimal floating-point (DFP) logarithmic converter, based on the digit-recurrence algorithm with selection by rounding. The proposed approach can compute faithful DFP logarithm results for any one of the three DFP formats specified in the IEEE 754-2008 standard. In order to optimize the latency for the proposed design, we mainly integrate the following novel features: 1) using the redundant carry-save representation of the data path; 2) reducing the number of iterations by determining the number of initial iteration; and 3) retiming and balancing the delay of the proposed architecture. The proposed architecture is synthesized with STM 90-nm standard cell library and the results show that the critical path delay and the number of clock cycles of the proposed Decimal64 logarithmic converter are 1.55 ns (34.4 FO4) and 19, respectively, and the total hardware complexity is 43,572 NAND2 gates. The delay estimation results of the proposed architecture show that its latency is close to that of the binary radix-16 logarithmic converter, and that it has a significant decrease on latency compared with a recently published high performance CORDIC implementation.

Abstract: In performance modeling of parallel synchronous iterative applications, the longest individual execution time among parallel processors determines the iteration time and often must be estimated for performance analysis. This involves the mean maximum calculation which has been a challenge in computer modeling for a long time. For large systems, numerical methods are not suitable because of heavy computation requirements and inaccuracy caused by rounding. On the other hand, previous approximation methods face challenges of accuracy and generality, especially for heterogeneous computing environments. This paper presents an interesting property of extreme values to enable Effective Mean Maximum Approximation (EMMA). Compared to previous mean maximum execution time approximation methods, this method is more accurate and general to different computational environments.

Abstract: A 76-year-old white male presented to his primary care physician with a 40-pound weight loss and gradual decline in function over the prior 6 months. In addition, over the previous 2 months, he had begun to suffer a constant, non-bloody, and non-productive cough accompanied by night sweats. Associated complaints included a decline in physical activity, increased sleep needs, decreased appetite, irritability, and generalized body aches.

Abstract: We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$ where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity requirement $k$, our algorithm computes a solution that is a $4$-approximation on the cost, and the degree of each node $v$ in the solution is at most $4b(v)+O(k)$. These ratios improve the previous $O(\log k)$-approximation on the cost and $O(2^k b(v))$ approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k$-out-connectivity, directed and undirected $k$-connectivity, and undirected rooted $k$-connectivity and subset $k$-connectivity.

Abstract: Convex relaxations based on different hierarchies of linear/semi-definite programs have been used recently to devise approximation algorithms for various optimization problems. The approximation guarantee of these algorithms improves with the number of {\em rounds} $r$ in the hierarchy, though the complexity of solving (or even writing down the solution for) the $r$'th level program grows as $n^{\Omega(r)}$ where $n$ is the input size. In this work, we observe that many of these algorithms are based on {\em local} rounding procedures that only use a small part of the SDP solution (of size $n^{O(1)} 2^{O(r)}$ instead of $n^{\Omega(r)}$). We give an algorithm to find the requisite portion in time polynomial in its size. The challenge in achieving this is that the required portion of the solution is not fixed a priori but depends on other parts of the solution, sometimes in a complicated iterative manner. Our solver leads to $n^{O(1)} 2^{O(r)}$ time algorithms to obtain the same guarantees in many cases as the earlier $n^{O(r)}$ time algorithms based on $r$ rounds of the Lasserre hierarchy. In particular, guarantees based on $O(\log n)$ rounds can be realized in polynomial time. We develop and describe our algorithm in a fairly general abstract framework. The main technical tool in our work, which might be of independent interest in convex optimization, is an efficient ellipsoid algorithm based separation oracle for convex programs that can output a {\em certificate of infeasibility with restricted support}. This is used in a recursive manner to find a sequence of consistent points in nested convex bodies that "fools" local rounding algorithms.