wanyu/IteraTeR_full_sent huggingface dataset:
Original:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons  , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Revised:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and for convex polygons , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Edit intent: fluency
Original:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons  , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Revised:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons of constant size , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Edit intent: meaning-changed
Original:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons  , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Revised:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( {\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Edit intent: fluency
Original:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons  , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
Revised:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} \frac{\log (L/\delta){\epsilon^{5/2}}} n^3\log n), respectively.
Edit intent: fluency
Original:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Revised:  There also exist two algorithms with approximation factor more than 2, for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Edit intent: meaning-changed
Original:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Revised:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned hypercubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Edit intent: meaning-changed
Original:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Revised:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, where they consider the shapes centered at the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks.
Edit intent: meaning-changed
Original:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness and therefore contradict the original definition of the c-packedness that has no assumptions on the placement of the disks. Call this measure the relative c-packedness.
Revised:  Recently, two other algorithms were claimed to be constant-factor approximations for the c-packedness using axis-aligned cubes instead of disks, which only consider the vertices of the curve when determining the c-packedness . We call this measure the relative c-packedness.
Edit intent: coherence
Original:  Using an example, we show that c-packedness and relative c-packedness are not within a constant factor of each other .
Revised:  For axis-aligned squares instead of disks, a related concept called relative-length was already discussed and it has an exact O(n^3) algorithm. We show that vertex-relative c-packedness and relative c-packedness are not within a constant factor of each other .
Edit intent: meaning-changed
Original:  Also, we show that there are no constant-factor approximation algorithms that solve the problem in sub-quadratic timein the worst-case and only uses intersection information to solve the problem .
Revised:  We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem. It has O(n^6\log n) construction time, uses O(n^6) space, and has query time O(\log n+k), where k is the number of intersected segments with the query shape .
Edit intent: meaning-changed
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the diameter of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: clarity
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between points on disjoint edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: clarity
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, we show the approximation factor of the existing O(\frac{\log (L/\delta){\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: meaning-changed
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: fluency
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation {\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: meaning-changed
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} algorithm. The massively parallel versions of these algorithms run in O(\log (L/\delta)) rounds. We improve the existing O((\frac{n{\epsilon^3})^{\frac 4 3}}\polylog n ), respectively.
Edit intent: meaning-changed
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog \frac n \epsilon) time (6+\epsilon)-approximation algorithm by providing a (4+\epsilon)-approximation O( n ), respectively.
Edit intent: meaning-changed
Original:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog  n ), respectively. There also exist two algorithms with approximation factor more than 2, for the c-packedness using axis-aligned hypercubes instead of disks, where they consider the shapes centered at the vertices of the curve when determining the c-packedness. We call this measure the vertex-relative c-packedness. For axis-aligned squares instead of disks, a related concept called relative-length was already discussed and it has an exact{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^3) algorithm.
Revised:  Assuming L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, ({\epsilon}n^3) time algorithm is } 1+\epsilon ) -approximation algorithms exist for disks, and for convex polygons of constant size, with time complexities O(\frac{\log (L/\delta){\epsilon}n ^3) and O(\log (L/\delta){\epsilon^{5/2}}n^3\log} {\epsilon^3})^{\frac 4 3}}\polylog n (\log^2 n)(\log^2 \frac{1{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^3) algorithm.
Edit intent: meaning-changed
Original:  For axis-aligned squares instead of disks, a related concept called relative-length was already discussed and it has an exact{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^3) algorithm. We show that vertex-relative c-packedness has approximation factor at least 2 for c-packedness. We design the first exact algorithmfor computing the minimum c for which a given curve P is c-packed. Our algorithm runs in O(n^5) time .
Revised:  For axis-aligned squares instead of disks, a related concept called relative-length was already discussed and it has an exact{\epsilon})+n{\epsilon}) time algorithm, and the existing } O(n ^2) time 2-approximation algorithm improving the existing O(n^2\log n) time 2-approximation algorithm. Our exact c-packedness algorithm takes O(n^5) time .
Edit intent: meaning-changed
Original:  Our algorithm runs in O(n^5) time . We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem .
Revised:  Our algorithm runs in O(n^5) time , which is the first exact algorithm for disks. We show using \alpha-fat shapes instead of disks adds a factor \alpha^2 to the approximation. We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem .
Edit intent: meaning-changed
Original:  It has O(n^6\log n) construction time, uses O(n^6) space, and has query time O(\log n+k), where k is the number of intersected segments with the query shape  .
Revised:  It has O(n^6\log n) construction time, uses O(n^6) space, and has query time O(\log n+k), where k is the number of intersected segments with the query shape . We also give a massively parallel algorithm for relative c-packedness with O(1) rounds .
Edit intent: meaning-changed
Original:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables . Using the monthly number  FDA Center for Drug Evaluation and Research (CDER)
Revised:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in the long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables . Using the monthly number FDA Center for Drug Evaluation and Research (CDER)
Edit intent: clarity
Original:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables . Using the monthly number  FDA Center for Drug Evaluation and Research (CDER)
Revised:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with changes in such influences . Using the monthly number FDA Center for Drug Evaluation and Research (CDER)
Edit intent: clarity
Original:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables . Using the monthly number  FDA Center for Drug Evaluation and Research (CDER)
Revised:  Here, a novel approach, termed the Chronological Hurst Exponent (CHE), is proposed, which hypothesizes that changes in intrinsic long-range memory latent within the dynamics of time series data may be temporally associated with extrinsic variables . Using the monthly number the FDA Center for Drug Evaluation and Research (CDER)
Edit intent: fluency
Original:  approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped (trichotomized) long-range CHE structure : an 8-year (1939-1947) Stagnated (random;
Revised:  approvals from 1939 to 2019 as the data source, it is demonstrated that the CHE has a distinct S-shaped (trichotomized) long-range CHE structure : an 8-year (1939-1947) Stagnated (random;
Edit intent: clarity
Original:  approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped (trichotomized) long-range CHE structure : an 8-year (1939-1947) Stagnated (random;
Revised:  approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped structure demarcated by an 8-year (1939-1947) Stagnated (random;
Edit intent: clarity
Original:  approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped (trichotomized) long-range CHE structure : an 8-year (1939-1947) Stagnated (random; H of 0.5) , a 27-year (1947-1974) Emergent (time-varying persistent;
Revised:  approvals from 1939 to 2019 as the data source, it is demonstrated that US approvals are found to have a distinct S-shaped (trichotomized) long-range CHE structure : an 8-year (1939-1947) Stagnation Period , a 27-year (1947-1974) Emergent (time-varying persistent;
Edit intent: clarity
Original:  H of 0.5) , a 27-year (1947-1974) Emergent (time-varying persistent; H between 0.5 and 0.9), and the 45-year (1974-2019) Saturated (persistent;
Revised:  H of 0.5) , a 27-year (1947-1974) Emergent (time-varying Period, and a 45-year (1974-2019) Saturated (persistent;
Edit intent: meaning-changed
Original:  This work suggests that (1) macro-factors in the Emergent Period  have led to persistent growth in US  approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Revised:  As discussed, this work suggests that (1) macro-factors in the Emergent Period have led to persistent growth in US approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Edit intent: coherence
Original:  This work suggests that (1) macro-factors in the Emergent Period  have led to persistent growth in US  approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Revised:  This work suggests that (1) changes in extrinsic factors (e.g., of economic and/or policy origin ) during the Emergent Period have led to persistent growth in US approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Edit intent: meaning-changed
Original:  This work suggests that (1) macro-factors in the Emergent Period  have led to persistent growth in US  approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Revised:  This work suggests that (1) macro-factors in the Emergent Period may have led to persistent growth in US approvals enjoyed since 1974, (2) the CHE may be a valued method to explore the constancy of extrinsic pressures on time series data, and (3) adds further evidence that innovation-related economic cycles exist (as supported via the proxy metric of US drug approvals).
Edit intent: fluency