wanyu/IteraTeR_full_sent huggingface dataset:
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised:  1%DIFDELCMD < ]%%% -hard problem of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without edge crossings.
Edit intent: coherence
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised: Fast constant factor approximation algorithms are devised for an NP- and W %DIFDELCMD < ]%%% -hard problem of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without edge crossings.
Edit intent: clarity
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% An NP-hard problem is considered of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without edge crossings.
Edit intent: clarity
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a given set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without edge crossings.
Edit intent: clarity
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a set of n straight line segments on the plane with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without edge crossings.
Edit intent: meaning-changed
Original: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a  set of n straight line segments  with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without  edge crossings.
Revised: Fast constant factor approximation algorithms are devised for an NP- and W 1%DIFDELCMD < ]%%% -hard problem of intersecting a set of n straight line segments with the smallest cardinality set of disks of fixed radii r>0, where the set of segments forms a straight line drawing G=(V,E) of a planar graph without proper edge crossings.
Edit intent: clarity
Original:   Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm  .
Revised:  To the best of our knowledge, related work only tackles a setting where E consists of (generally, properly overlapping) axis-parallel segments, resulting in an O(n\log n)-time and O(n\log n)-space 8-approximation algorithm. Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm .
Edit intent: meaning-changed
Original:   Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm  .
Revised:  Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \nu \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm .
Edit intent: meaning-changed
Original:   Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm  .
Revised:  Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is devised based on the modified Agarwal-Pan algorithm .
Edit intent: clarity
Original:   Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm  .
Revised:  Exploiting tough connection of the problem with the geometric Hitting Set problem, an \left(50+52\frac{12{13}}+ \varepsilon \right)-approximate O\left(n^4\log n\right)-time and O\left(n^2\log n\right)-space algorithm is given based on the modified Agarwal-Pan algorithm , which uses epsilon nets .
Edit intent: meaning-changed
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \nu )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Edit intent: fluency
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \varepsilon )- and \left( \frac{144 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Edit intent: meaning-changed
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \varepsilon )- and \left( 34 + 32\sqrt{\frac{3 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Edit intent: meaning-changed
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \nu \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Edit intent: fluency
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either a generalized outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Edit intent: clarity
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant.
Revised:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \nu >0 is an arbitrary small constant.
Edit intent: fluency
Original:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrary small constant. Moreover, an O(n^2\log n)-time and O(n^2)-space 18-approximation is designed for the case where G is any subgraph of a Gabriel graph. To the best of our knowledge, related work only tackles the case where E consists of axis-parallel segments, resulting in an O(n\log n)-time and O(n\log n)-space 8-approximation .
Revised:  More accurate (34+242+ \varepsilon )- and \left( 34 + 44\sqrt{\frac{6 + \varepsilon \right)-approxi\-mate algorithms are also proposed for cases where G is any subgraph of either an outerplane graph or a Delaunay triangulation respectively, which work within the same time and space complexity bounds, where \varepsilon >0 is an arbitrarily small constant .
Edit intent: meaning-changed
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised:  Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Edit intent: clarity
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised: A sequence of pathological changes takes place in Alzheimer's disease (AD) is characterized by a sequence of pathological changes, which are commonly assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Edit intent: meaning-changed
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using MRI and PET . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Edit intent: meaning-changed
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, the most approaches to analyze statistical associations between brain regions rely on Pearson correlation. However, these are prone to spurious correlations arising from uninformative shared variance. Notably, there are no appropriate multivariate statistical models available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Edit intent: meaning-changed
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate dozens of variables derived from such data , being able to utilize the additional information provided from the combined data .
Edit intent: meaning-changed
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data .
Revised: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to use the additional information provided from the combined data .
Edit intent: fluency
Original: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combined data . We applied Gaussian graphical models (GGMs) for analyzing the conditional dependency networks of multimodal neuroimaging dataand assessed alterations of the network structure in mild cognitive impairment (MCI) and Alzheimer's dementia (AD) compared to cognitively healthy controls.
Revised: A sequence of pathological changes takes place in Alzheimer's disease , which can be assessed in vivo using various brain imaging methods . Currently, there is no appropriate statistical model available that can easily integrate multiple imaging modalities , being able to utilize the additional information provided from the combination of data sources. Gaussian graphical models (GGMs) for analyzing the conditional dependency networks of multimodal neuroimaging dataand assessed alterations of the network structure in mild cognitive impairment (MCI) and Alzheimer's dementia (AD) compared to cognitively healthy controls.
Edit intent: clarity
Original:  We applied Gaussian graphical models (GGMs) for analyzing the conditional dependency networks of multimodal neuroimaging dataand assessed alterations of the network structure in mild cognitive impairment (MCI) and Alzheimer's dementia (AD) compared to cognitively healthy controls.
Revised:  We applied Gaussian graphical models (GGMs) can estimate the conditional dependency networks of multimodal neuroimaging dataand assessed alterations of the network structure in mild cognitive impairment (MCI) and Alzheimer's dementia (AD) compared to cognitively healthy controls.
Edit intent: clarity
Original:  We applied Gaussian graphical models (GGMs) for analyzing the conditional dependency networks of multimodal neuroimaging dataand assessed alterations of the network structure in mild cognitive impairment (MCI) and Alzheimer's dementia (AD) compared to cognitively healthy controls. Data from N= 667 subjects were obtained from the Alzheimer's Disease Neuroimaging Initiative.
Revised:  We applied Gaussian graphical models (GGMs) for analyzing the conditional dependency from given data, which is expected to reflect the underlying causal relationships. We applied GGMs to assess multimodal regional brain alterations in AD. We obtained data from N= 667 subjects were obtained from the Alzheimer's Disease Neuroimaging Initiative.
Edit intent: meaning-changed
Original:  Mean amyloid load (AV45-PET), glucose metabolism (FDG-PET), and gray matter volume (MRI) was calculatedfor each brain region. Separate GGMs were estimated using a Bayesian framework for the combined multimodal data for each diagnostic category.
Revised:  Mean amyloid load (AV45-PET), glucose metabolism (FDG-PET), and gray matter volume (MRI) were calculated. GGMs were estimated using a Bayesian framework for the combined multimodal data for each diagnostic category.
Edit intent: clarity
Original:  Separate GGMs were estimated using a Bayesian framework for the combined multimodal data for each diagnostic category.
Revised:  Separate GGMs were estimated using a Bayesian framework for the combined multimodal data to obtain conditional dependency networks. Conditional dependency matrices were much sparser (10\% density) than Pearson correlation matrices (50\% density). Within modalities, conditional dependency networks yielded clusters connecting anatomically adjacent regions. For associations between different modalities, only few region-specific connections remained.
Edit intent: meaning-changed
Original:  Graph-theoretical statistics were calculated to determine network alterations associated with disease severity. Network measures clustering coefficient, path length and small-world coefficient were significantly altered across diagnostic groups, with a biphasic u-shape trajectory , i. e. increased small-world coefficient in early MCI , intermediate values in late MCI, and decreased values in ADpatients compared to controls.
Revised:  Graph-theoretical network statistics were significantly altered across diagnostic groups, with a biphasic u-shape trajectory , i. e. increased small-world coefficient in early MCI , intermediate values in late MCI, and decreased values in ADpatients compared to controls.
Edit intent: clarity
Original:  Network measures clustering coefficient, path length and small-world coefficient were significantly altered across diagnostic groups, with a biphasic u-shape trajectory , i. e. increased small-world coefficient in early MCI , intermediate values in late MCI, and decreased values in ADpatients compared to controls.
Revised:  Network measures clustering coefficient, path length and small-world coefficient were significantly altered between groups, with a biphasic u-shape trajectory , i. e. increased small-world coefficient in early MCI , intermediate values in late MCI, and decreased values in ADpatients compared to controls.
Edit intent: clarity
Original:  Network measures clustering coefficient, path length and small-world coefficient were significantly altered across diagnostic groups, with a biphasic u-shape trajectory , i. e. increased small-world coefficient in early MCI , intermediate values in late MCI, and decreased values in ADpatients compared to controls. In contrast, no group differences were found for clustering coefficient and small-world coefficient when estimating conditional dependency networkson single imaging modalities. GGMs provide a useful methodology to analyze the conditional dependency networks of multimodal neuroimaging data .
Revised:  Network measures clustering coefficient, path length and small-world coefficient were significantly altered across diagnostic groups, with a biphasic u-shape trajectory . GGMs removed shared variance among multimodal measures of regional brain alterations in MCI and AD, and yielded sparser matrices compared to Pearson correlation networks. Therefore, GGMs may be used as alternative to thresholding-approaches typically applied to correlation networks to obtain the most informative relations between variables .
Edit intent: clarity
Original:  It is achieved by a systematic multi-step approach , that allows 1. to quantify the nonlinearity of coupling , 2. to correct its effects wherever it is caused by simple univariate non-Gaussianity , 3. to potentially localize in space and time any remaining strong sources of this nonlinearity , andfinally, 4. to study the effect the nonlinearity has on global network properties.
Revised:  This is achieved by a systematic multi-step approach , that allows 1. to quantify the nonlinearity of coupling , 2. to correct its effects wherever it is caused by simple univariate non-Gaussianity , 3. to potentially localize in space and time any remaining strong sources of this nonlinearity , andfinally, 4. to study the effect the nonlinearity has on global network properties.
Edit intent: fluency