wanyu/IteraTeR_full_sent huggingface dataset:
Original:  i.e., p_{u,v = p_v for all (u,v) \in E. }%DIFDELCMD < \\ %%% (3) When the patience constraint \ell_v satisfies \ell_v \in \{ 1,|U|\} for every online vertex; i.e., every online vertex either has unit or full patience. Setting the probability p_e = 1 for all e \in E, the stochastic problem becomes the classical online bipartite matching problem .
Revised:  i.e., p_{u,v = p_v for all (u,v) \in E. }%DIFDELCMD < \\ %%% (3) When the patience constraint \ell_v satisfies \ell_v \in \{ This model generalizes the various settings of the classical bipartite matching problem .
Edit intent: meaning-changed
Original:  Setting the probability p_e = 1 for all e \in E, the stochastic problem becomes the classical online bipartite matching problem . Our competitive ratios thus generalize corresponding results for the classical ROM bipartite matching setting .
Revised:  Setting the probability p_e = 1 for all e \in E, the stochastic problem becomes the classical online bipartite matching problem , and so our main contribution is in making progress towards understanding which classical results extend to the stochastic probing model .
Edit intent: clarity
Original:  Given a parameter \epsilon \in (0, 1 , the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance 2+\epsilon , or answering that there exists no curve in the input within distance 1.
Revised:  For an attainable approximation factor c> 1 , the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance 2+\epsilon , or answering that there exists no curve in the input within distance 1.
Edit intent: meaning-changed
Original:  Given a parameter \epsilon \in (0, 1 , the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance 2+\epsilon , or answering that there exists no curve in the input within distance 1.
Revised:  Given a parameter \epsilon \in (0, 1 , the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance c , or answering that there exists no curve in the input within distance 1.
Edit intent: meaning-changed
Original:  Our data structure uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and  query time in O\left(k\right).
Revised:  Our first data structure achieves a (5+\epsilon) approximation factor, uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and query time in O\left(k\right).
Edit intent: meaning-changed
Original:  Our data structure uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and  query time in O\left(k\right).
Revised:  Our data structure uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and has query time in O\left(k\right).
Edit intent: coherence
Original:  We show that under some conditions the approximation factor achieved by our data structure is optimal in the cell-probe model of computation. Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Revised:  Our second data structure achieves a (2+\epsilon) approximation factor, uses space in n\cdot \mathcal{O}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Edit intent: meaning-changed
Original:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Revised:  Concretely}\left(\frac{m{k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Edit intent: meaning-changed
Original:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Revised:  Concretely}\left({k\epsilon}}\right)^{k + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Edit intent: fluency
Original:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Revised:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(k\cdot 2^k\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Edit intent: meaning-changed
Original:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Revised:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right). As an indicator of hardness , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
Edit intent: meaning-changed
Original:  Our data structure uses only a constant number of probes per query and does not have any dependency on L. In particular, this shows that our solution is optimal if only a constant number of probes is allowed. Our second positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in \mathcal{O}(nm) and query time in \mathcal{O}(k), and which answers queries with an approximation factor in \mathcal{O}(k).
Revised:  Our third positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in \mathcal{O}(nm) and query time in \mathcal{O}(k), and which answers queries with an approximation factor in \mathcal{O}(k).
Edit intent: clarity
Original: Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments . ORB-SLAM2  is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe.
Revised: Indirect methods for visual SLAM are gaining popularity due to their robustness to environmental variations . ORB-SLAM2 is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe.
Edit intent: clarity
Original: Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments . ORB-SLAM2  is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe.
Revised: Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments . ORB-SLAM2 \mbox{%DIFAUXCMD orbslam2 is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe.
Edit intent: meaning-changed
Original: Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments . ORB-SLAM2  is a benchmark method in this domain, however, the computation of descriptors in ORB-SLAM2 is time-consuming and the descriptors cannot be reused unless a frame is selected as a keyframe.
Revised: Indirect methods for visual SLAM are gaining popularity due to their robustness to varying environments . ORB-SLAM2 is a benchmark method in this domain, however, it consumes significant time for computing descriptors that never get reused unless a frame is selected as a keyframe.
Edit intent: clarity
Original:  We test FastORB-SLAM with an RGBD camera on \textit{TUM} and \textit{ICL-NUIM}  datasets and compare its accuracy and efficiency to nine existing RGBD SLAM methods.
Revised:  We test FastORB-SLAM on \textit{TUM} and \textit{ICL-NUIM} datasets and compare its accuracy and efficiency to nine existing RGBD SLAM methods.
Edit intent: clarity
Original:  We test FastORB-SLAM with an RGBD camera on \textit{TUM} and \textit{ICL-NUIM}  datasets and compare its accuracy and efficiency to nine existing RGBD SLAM methods.
Revised:  We test FastORB-SLAM with an RGBD camera on \textit{TUM} and \textit{ICL-NUIM} RGB-D datasets and compare its accuracy and efficiency to nine existing RGBD SLAM methods.
Edit intent: meaning-changed
Original:  We test FastORB-SLAM with an RGBD camera on \textit{TUM} and \textit{ICL-NUIM}  datasets and compare its accuracy and efficiency to nine existing RGBD SLAM methods.
Revised:  We test FastORB-SLAM with an RGBD camera on \textit{TUM} and \textit{ICL-NUIM} datasets and compare its accuracy and efficiency to nine existing RGB-D SLAM methods.
Edit intent: fluency
Original:  However, when restricted to lattice triangulations- i. e. triangulations of a finite subset of the integer lattice, or an affine transformation of this lattice, bounded by a simple, closed polygon with lattice points as vertices - the problem has a polynomial time algorithm. Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right)  .
Revised:  However, for minimum flip paths between lattice triangulations, Eppstein and Caputo et al. gave algorithms running in O\left(\right) .
Edit intent: meaning-changed
Original:  Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right)  .
Revised:  Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(n^2\right) .
Edit intent: meaning-changed
Original:  Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right)  .
Revised:  Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right) time, where n is the number of points in the point-set. Eppstein proved this result for lattice point-sets bounded by convex polygons. Caputo et al. extended this result to the cases of non-convex bounding polygons and constrained flip paths that preserve a set of edges. In fact, Eppstein's approach readily extends to both cases .
Edit intent: meaning-changed
Original:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Revised:  Our first result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Edit intent: clarity
Original:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Revised:  Our first main result shows that there is a unique, partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Edit intent: clarity
Original:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Revised:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of flips whose valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Edit intent: clarity
Original:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations .
Revised:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings are exactly the constrained, minimum flip paths between two lattice triangulations .
Edit intent: meaning-changed
Original:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations . This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
Revised:  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations , leading to an algorithm to compute such a minimum flip path in O\left({2}}}\right) .
Edit intent: clarity
Original:  This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
Revised:  This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right) time. As a further improvement over previous results, in many natural cases, our algorithm runs in time linear in the length of the minimum flip path .
Edit intent: meaning-changed
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Revised:  Our second result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Edit intent: clarity
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Revised:  Our second main result characterizes pairs of triangulations T and T' that contain given sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Edit intent: meaning-changed
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Revised:  Our second main result characterizes pairs of triangulations , containing sets of edges G and G ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Edit intent: meaning-changed
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Revised:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, and attain the minimum flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Edit intent: coherence
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively.
Revised:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between each other, where the minimum is taken over such pairs of triangulations , containing sets of edges E and E' respectively.
Edit intent: clarity
Original:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations , containing sets of edges E and E' respectively. Remarkably, all of our results are derived from a simple relationship between edges in triangulations and Farey sequences .
Revised:  Our second main result characterizes pairs of triangulations , containing sets of edges E and E ' respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations . Finally, we demonstrate how our results can model crack propagation in crystalline materials caused by Stone-Wales defects. Notably, the above results follow from simple number-theoretic and geometric concepts .
Edit intent: meaning-changed
Original:  In addition to learning approaches, we also compare to an inverse-kinematics (IK) based footstep planner , and show that our fully-learned framework is competitive in performance with IK under normal conditions, and outperforms it in adverse settings .
Revised:  In addition to learning approaches, we also compare to inverse-kinematics (IK) based footstep planner , and show that our fully-learned framework is competitive in performance with IK under normal conditions, and outperforms it in adverse settings .
Edit intent: fluency
Original:  In addition to learning approaches, we also compare to an inverse-kinematics (IK) based footstep planner , and show that our fully-learned framework is competitive in performance with IK under normal conditions, and outperforms it in adverse settings .
Revised:  In addition to learning approaches, we also compare to an inverse-kinematics (IK) acting on planned footsteps , and show that our fully-learned framework is competitive in performance with IK under normal conditions, and outperforms it in adverse settings .
Edit intent: clarity
Original:  In addition to learning approaches, we also compare to an inverse-kinematics (IK) based footstep planner , and show that our fully-learned framework is competitive in performance with IK under normal conditions, and outperforms it in adverse settings .
Revised:  In addition to learning approaches, we also compare to an inverse-kinematics (IK) based footstep planner , and show that our fully-learned framework outperforms IK in adverse settings .
Edit intent: coherence