We prove that in comparison to the corresponding model of Rabani, Sinclair, and Wanka (1998) with arbitrary round- ings, the randomization yields an improvement of roughly a square root of the achieved discrepancy in the same num- ber of time-steps on all graphs. For the important case of expanders we can even achieve a constant discrepancy in O(log n (log log n) 3 ) rounds. This is optimal up to log log n- factors while the best previous algorithms in this setting either require Ω(log 2 n) time or can only achieve a logarith- mic discrepancy. This result also demonstrates that with **randomized** **rounding** the difference between discrete and continuous load balancing vanishes almost completely. Categories and Subject Descriptors: F.2 [Theory of Computation]: Analysis of Algorithms and Problem Com- plexity

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In this paper we present an LP-based approximation algorithm for Steiner tree with an im- proved approximation factor. Our algorithm is based on a, seemingly novel, iterative **randomized** **rounding** technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + ε < 1.39 times the cost of an optimal Steiner tree. The algorithm can be derandomized using the method of limited independence.

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Many algorithms for combinatorial problems that are modeled as PIPs rely heavily on the structure of the problem; though these problem-specific techniques often lead to good results, they may not be more broadly applicable. Thus, there is also a need for general techniques that can be applied to a large class of problems. One natural candidate is **randomized** **rounding**, which involves first solving a linear programming relaxation to obtain a fractional solution x ∗ , and then converting/**rounding** x ∗ to a true integral solution by setting variable j to one with probability x ∗ j divided by a suitable scaling factor. However, the difficulty in applying this technique to all packing problems is that for some instances, unless the scaling factor is extremely large (resulting in a solution with low expected weight), the probability of obtaining a feasible solution is extremely small.

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metric labeling consists of arbitrary unary potentials and pairwise potentials that are proportional to the distance between the labels assigned to them. The problem is known to be NP -hard (Veksler, 1999). Two popular approaches for semi-metric labeling are: (i) move-making algorithms (Boykov et al., 1999; Gupta and Tardos, 2000; Kumar and Koller, 2009; Kumar and Torr, 2008; Veksler, 2007), which iteratively improve the labeling by solving a minimum st-cut problem; and (ii) linear programming ( LP ) relaxation (Chekuri et al., 2001; Koster et al., 1998; Schlesinger, 1976; Wainwright et al., 2005), which is obtained by dropping the integral constraints in the corresponding integer programming formulation. Move-making algorithms are very efficient due to the availability of fast minimum st-cut solvers (Boykov and Kolmogorov, 2004) and are very popular in the computer vision community. In contrast, the LP relaxation is significantly slower, despite the development of several specialized solvers (Globerson and Jaakkola, 2007; Hazan and Shashua, 2008; Kolmogorov, 2006; Komodakis et al., 2007; Ravikumar et al., 2008; Tarlow et al., 2011; Wainwright et al., 2005; Weiss et al., 2007; Werner, 2007, 2010). However, when used in conjunction with **randomized** **rounding** algorithms, the LP relaxation provides the best known polynomial-time theoretical guarantees for semi-metric labeling (Archer et al., 2004; Chekuri et al., 2001; Kleinberg and Tardos, 1999).

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Semi-metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are propor- tional to a given semi-metric distance function over the label set. Popular methods for solving semi-metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear programming ( LP ) relaxation based approach. In order to convert the frac- tional solution of the LP relaxation to an integer solution, several **randomized** **rounding** procedures have been developed in the literature. We consider a large class of parallel **rounding** procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding **rounding** procedure for any arbitrary distance function. Our analysis includes all known results for move-making algorithms as special cases.

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The fact that LP/SDP solutions can be viewed as expectations of distributions is well known, and several **rounding** algorithms can be considered as trying to “reverse engineer” a relaxation solution to get a good distribution over actual solutions.
Techniques such as **randomized** **rounding**, the hyperplane **rounding** of [ GW95 ], and the **rounding** for TSP [ GSS11 , AKS12 ] can all be viewed in this way. One way to summarize the conceptual difference be- tween our techniques and those approaches is that these previous algorithms often considered the relaxation solution as giving moments of an actual distribution on “fake” solutions. For example, in [ GW95 ]’s Max Cut algorithm, where actual solutions are modeled as vectors in {±1} n , the SDP solution is treated as the moment matrix of a Gaussian distribution over real vectors that are not necessarily ±1-valued. Similarly in the TSP setting one often considers the LP solution to yield moments of a distribution over spanning trees that are not necessarily TSP tours. In contrast, in our setting we view the solution as providing moments of a “fake” distribution on actual solutions.

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Survey (PSS). The PSS was developed by the research team to measure the effects of patient comfort **rounding**. The PSS included similar questions related to timeliness of patients’ needs being met, as the currently used HCAHPS Survey. Both survey tools include a scale range from 1 to 5, with 5 being the most positive response. This study, which is a replication of Gardner et al.’s (2009) study, utilized the HCAHPS tool for the added benefit of comparison data. HCAHPS scores are available on Hospital Compare (2012) along with a data base which allows comparisons to other hospitals. HCAHPS scores influence the CMS reimbursement rate, therefore this survey tool was selected to test for patient satisfaction, and also the knowledge of how that satisfaction will affect the facility’s reimbursement dollars. Further study is needed on patient **rounding** as a

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The first specification in Table 3 shows the results for probit estimation that a loan will subsequently experience delinquency for the first time. We report the average marginal effects calculated around mean points using all loans in our dataset. As expected, the results indicate that the occurrence of **rounding** is associated with adverse outcomes. Borrowers who report a rounded income are almost 0.4 percent more likely to experience first-time delinquency than those who reported a more accurate figure, controlling for the credit score groups and origination year. Given that the average delinquency rate for the whole sample is 7 percent, detecting borrowers with misreported income may decrease the average delinquency rate by 0.4 percentage points. For borrowers’ characteristics, we find that a 1 percent increase in the debt-to-income ratio is associated with 0.2 percent increase in the probability that a current loan will experience delinquency. Further, having a long history of credit decreases the probability of delinquency and an increase in the number of delinquencies in the last 2 years increases the possibility of delinquency.

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This section conducts an RD analysis of the retirement-consumption puzzle in China, taking advantage of the Chinese mandatory retirement rule. Since age is reported in years in the dataset used here, I apply the proposed approach to correct the associated **rounding** bias. The Chinese case is interesting due to its unique social and cultural environment, which differs in many ways from developed Western countries, and hence may shed additional light on the underlying mechanism of consumption changes around retirement. One way China differs from these other countries is that China has very high savings rates, so most households may have saved enough to avoid signi cant drops in consumption at retirement. In addition, cash transfers from adult children to retirees are a common practice in urban China. This may also help to prevent consumption declines that would otherwise result from inadequate accumulated wealth.

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In June 2010 - 2011 a particular ward (not a study site included in this thesis) that had been involved in the ‘falls **rounding**’ project recorded 71 low level falls, a higher number of falls compared to other wards within the organisation. On analysis these falls typically occurred at peak activity times on the ward, peak activity being defined by ‘Productive Ward Activity Clocks’ (NHS Institute 2008), between 05:00-07:00 hours and 10:00-12:00 hours. The increase in patient falls was despite the ‘falls **rounding**’ document being introduced in August 2010. On further investigation the compliance with the completion of the ‘falls **rounding**’ document was found to be inconsistent, ward staff reported that it was too prescriptive, the seven questions were time consuming and the process too long, with individual document sheets being kept in every patient’s folder. Feedback from patients highlighted the repetitive nature of questions. On reviewing the national agendas at the time (Mid Staffordshire NHS Foundation Trust Inquiry 2010; Patients Association 2010; Care Quality Commission Report 2011; Health Services Management Centre 2011; Parliamentary and Health Service Ombudsman’s 2011), the process of **rounding** was considered important by the senior nursing team and should be reviewed. In June 2011 the senior nursing team for the ward met to discuss a strategy for addressing many of the key issues raised by the policy documents/reports and how **rounding** practice could address these issues.

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Model 2.1 is clearly not always realistic. For example, in some cases δ is neces- sarily zero, such as when the operands are (not too large) integers in an addition, subtraction, or multiplication; when the operands in a subtraction differ by at most a factor two and so are subtracted exactly (by Sterbenz’s result [14, Thm. 2.5]); or when one of the operands is a power of two in a multiplication or division. Or pairs of operands might be repeated, so that different occurrences of δ are in the fact the same. Indeed non-pathological examples can be found where **rounding** errors are strongly correlated—notably a rational function example of Kahan [14, sec. 1.17]. More subtly, if an operand comes from an earlier computation it will depend on an earlier δ and so the new δ will depend on the previous one, violating the independence assumption.

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Section 2: Background and Context
Introduction
The clinical practice problems to be addressed by this DNP project were the low
HCAHPS scores and the high number of inpatient falls on a hospital unit. The change made on a critical care step-down unit was implementing hourly **rounding** with a purpose, which means focusing hourly nursing rounds on the four basic needs of the patient: pain, pottying, positioning, and proximity of personal care items (Brosey & March, 2015). This purposeful **rounding** has been shown in research to greatly reduce the stress of the

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From Table 5.5 we can see that iterative methods were much slower than threshold meth- ods. The number of timeouts also increased considerably going from threshold methods to iterative. However, all **rounding** approaches managed to find a solution in under one second for most of the of the instances. For most instances a solution was obtained fast but a few of the instances took a long time to solve which can be seen from the fact that mean is high but median is low. This can be seen in Table 5.6. Half of the instances could be solved by any method within one second and around 500 instances could be solved within five seconds by any method. Increasing time limit to one minute does not increase the number of solved instances dramatically for any **rounding** method. However, there are still few instances that none of the **rounding** methods could solve within the given 300-second time limit.

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©IJRASET: All Rights are Reserved 308 Thus, one can perform the multiplication operation using three shift and two addition/subtraction operations. In this approach, the nearest values for A and B in the form of 2 n should be determined. When the value of A (or B) is equal to the 3 × 2p−2 (where p is an arbitrary positive integer larger than one), it has two nearest values in the form of 2n with equal absolute differences that are 2p and 2p−1. While both values lead to the same effect on the accuracy of the proposed multiplier, selecting the larger one (except for the case of p = 2) leads to a smaller hardware implementation for determining the nearest rounded value, and hence, it is considered in this paper. It originates from the fact that the numbers in the form of 3 × 2p−2 are considered as do not care in both **rounding** up and down simplifying the process, and smaller logic expressions may be achieved if they are used

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size is expected to result in significant cost‐savings, minimization of waste disposal 8 , and increase in the institution’s inventory. In this prospective, interventional study conducted at a not‐for profit community hospital, we hypothesized that **rounding** the dose of single‐dose IV chemotherapy is a feasible process resulting in significant cost‐savings for our pharmacy department.

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This section deals with the derivation of the statistical word length (SWL) required to quantize the Kalman gain vector elements in order to meet certain performance measures.. The metho[r]

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gains nothing. An often-overlooked fact about LWR is that it would likely be much more resistant to side channel attacks than LWE, since noise sampling can be difficult and take a lot of effort to inure against these sorts of attack [RRVV14]. This fact implies that LWR would be a very good post-quantum candidate for certain hardware implementations. There have been many recent schemes that utilized the power of LWR. A natural use case was lattice-based PRFs, including [BLMR13] and [BP14]. Papers with a practical focus on things like key exchange have also been build using **rounding**, including [CKLS16] and [DFH + 16]. More powerful PRFs that provide increased functionality and also might provide

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education to staff working all shifts regarding the importance of appropriate use of documentation codes on the **rounding** sheet. The first and second months demonstrated inconsistent use of the appropriate codes on the legend. In addition, staff initials were occasionally used on the hourly **rounding** form instead of the codes on the legend. Additional codes were also written on the forms, which made it difficult to identify what interventions were being addressed. This was a clear indication that additional feedback from the staff was necessary during the re-evaluation phase of the PDCA cycle to gather feedback as to whether there was a deficiency in education provided or if the codes on the form were inadequate to meet the **rounding** actions for resident’s needs. The unit culture and customs also played a role in the documentation changes. Incomplete forms made it

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IBR was piloted in small Plan–Do–Study–Act cycles dur- ing June 2013 with direct feedback solicited from participating nurses and physicians. Early responses to the pilot indicated that IBR interactions were frequently unstructured, and this sometimes led to ineffective communication. To address this barrier, a **rounding** “checklist” (Figure 1) was created and utilized by the nurse during IBR to ensure that key issues were addressed. The content of the checklist was derived from feed- back of physician and nurse participants and input from the QI leadership team and included key patient care-related issues. Daily audits were performed by the charge nurse to assess the rates of participation. IBR was fully implemented on all the medicine teams at the start of the academic year in July 2013.

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communication is often associated with poor patient outcomes (Corley & Spooner, 2016). The practice-focused question for this project was, Will the implementation of SBAR for **rounding** in SAR facilities decrease the incidence of 30-day rehospitalizations? The purpose of the QI project was to evaluate whether a 2017 QI initiative involving the implementation of SBAR during ITRA in the SAR setting was effective in reducing 30- day return to hospital admissions. The problem statement addressed a decrease in quality scores and patient safety concerns with transfers back to the acute care setting within 30 days of discharge. Evidence for this project came from de-identified data collected from the institution’s electronic health record database. Analytical strategies evolved from the presentation of 3-month report of 30-day readmissions data before and after the 2017 QI initiative was implemented, the time frame between discharge and readmission, the provider type (MD or ARNP), shift when discharge occurred, and the reason(s) for readmission.

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