wanyu/IteraTeR_full_sent huggingface dataset:
Original:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Revised:  Then if the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Edit intent: coherence
Original:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Revised:  Then the crossing edges have a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Edit intent: clarity
Original:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Revised:  Then the crossing edges are incident to a common vertex , that is they form a fan. Both are prominent examples for beyond-planar graphs. Further well-known beyond-planar classes are the k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Edit intent: meaning-changed
Original:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Revised:  Then the crossing edges are incident to a common vertex . Graphs are k-planar , k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Edit intent: clarity
Original:  Then  the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge .
Revised:  Then the crossing edges are incident to a common vertex . Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar , quasi-planar, and right angle crossing graphs .
Edit intent: meaning-changed
Original:  We use the s-subdivision, path-addition, and node-to-circle expansion operations to show that there are fan-crossing free graphs that are not fan-crossing, k-planar, and k-gap-planar for k >= 1, respectively.
Revised:  We use the subdivision, node-to-circle expansion operations to show that there are fan-crossing free graphs that are not fan-crossing, k-planar, and k-gap-planar for k >= 1, respectively.
Edit intent: clarity
Original:  We use the s-subdivision, path-addition, and node-to-circle expansion operations to show that there are fan-crossing free graphs that are not fan-crossing, k-planar, and k-gap-planar for k >= 1, respectively. A path-addition adds a long path between any two vertices to a graph . An s-subdivision replaces an edge by a path of length s, and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex.
Revised:  We use the s-subdivision, path-addition, and node-to-circle expansion and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex.
Edit intent: clarity
Original:  A path-addition adds a long path between any two vertices to a graph . An s-subdivision replaces an edge by a path of length s, and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex. We introduce universality for an operation and a graph class, so the every graph has an image in the graph class. In particular, we show the fan22 crossing free graphs are universal for 2-subdivision and for node-to-circle 3 expansion  .
Revised:  A path-addition adds a long path between any two vertices to a graph . An s-subdivision replaces an edge by a path of length s, and path-addition operations to distinguish all these graph classes. In particular, we show that the 2-subdivision and for node-to-circle 3 expansion .
Edit intent: coherence
Original:  In particular, we show the fan22 crossing free graphs are universal for 2-subdivision and for node-to-circle 3 expansion  .
Revised:  In particular, we show the fan22 crossing free graphs are universal for 2-subdivision and for node-to-circle 3 expansion of any graph is fan-crossing free, which does not hold for fan-crossing and k-(gap)-planar graphs, respectively. Thereby, we obtain graphs that are fan-crossing free and neither fan-crossing nor k-(gap)-planar .
Edit intent: meaning-changed
Original:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are  maximal fan-crossing free graphs with less edges than the density , and that the recognition problem for fan-crossing free graphs is NP-complete.
Revised:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are thinned maximal fan-crossing free graphs with less edges than the density , and that the recognition problem for fan-crossing free graphs is NP-complete.
Edit intent: meaning-changed
Original:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are  maximal fan-crossing free graphs with less edges than the density , and that the recognition problem for fan-crossing free graphs is NP-complete.
Revised:  Finally, we show that some graphs have a unique fan-crossing free embedding, that there are maximal fan-crossing free graphs , and that the recognition problem for fan-crossing free graphs is NP-complete.
Edit intent: clarity
Original:  However, GCN faces challenges in flexibility due to the fact of requiring the full graph Laplacian available in the training phase. Moreover, with the depth of layers increases, the computational and memory cost of GCN grows explosively on account of the recursive neighborhood expansion , which leads to a limitation in processing large graphs .
Revised:  However, GCN-based models (GCNs) is an inherently coupled training framework repetitively conducting the complex neighboring aggregation, which leads to the limitation of flexibility in processing large-scale graph. With the depth of layers increases, the computational and memory cost of GCN grows explosively on account of the recursive neighborhood expansion , which leads to a limitation in processing large graphs .
Edit intent: meaning-changed
Original:  To tackle these issues, we take advantage of image processing in agility and present Node2Img , a flexible architecture for large-scale graph learning. Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Revised:  To tackle these issues, we take advantage of image processing in agility and present Node2Img , a flexible uncoupled training framework that leverages the independent mapped data for obtaining the embedding . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Edit intent: clarity
Original:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Revised:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of directly processing the coupled nodes as GCNs, Node2Grids supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Edit intent: meaning-changed
Original:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Revised:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious method in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Edit intent: clarity
Original:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, where the batch size can be set elastically and the data in the same batch can be calculated parallelly.
Revised:  Node2Img maps the nodes to "images" (i.e. grid-like data in Euclidean space) which can be the inputs of Convolutional Neural Network (CNN) . Instead of leveraging the fixed whole network as a batch to train the model, Node2Img supports a more efficacious framework in practice, mapping the coupled graph data into the independent grid-like data which can be fed into the efficient Convolutional Neural Network (CNN). This simple but valid strategy significantly saves memory and computational resource while achieving comparable results with the leading GCN-based models.
Edit intent: meaning-changed
Original:  Specifically, by ranking each node's influence through degree, Node2Img selects the most influential first-order as well as second-order neighbors with central node fusion information to construct the grid-like data.
Revised:  Specifically, by ranking each node's influence through degree, Node2Grids selects the most influential first-order as well as second-order neighbors with central node fusion information to construct the grid-like data.
Edit intent: meaning-changed
Original:  For further improving the efficiency of downstream tasks, a simple CNN-based neural network is employed to capture the significant information from the Euclidean grids. Additionally, the attention mechanism is implemented, which enables implicitly specifying the different weights for neighboring nodes with different influences.
Revised:  For further improving the efficiency of downstream tasks, a simple CNN-based neural network is employed to capture the significant information from the mapped grid-like data. Moreover, the grid-level attention mechanism is implemented, which enables implicitly specifying the different weights for neighboring nodes with different influences.
Edit intent: clarity
Original:  Extensive experiments on real graphs' transductive and inductive learning tasks  demonstrate the superiority of the proposed Node2Img model against the state-of-the-art GCN-based approaches.
Revised:  Extensive experiments on real graphs' transductive and inductive learning tasks , we also verify our framework on million-scale graphs to demonstrate the superiority of the proposed Node2Img model against the state-of-the-art GCN-based approaches.
Edit intent: meaning-changed
Original:  Extensive experiments on real graphs' transductive and inductive learning tasks  demonstrate the superiority of the proposed Node2Img model against the state-of-the-art GCN-based approaches.
Revised:  Extensive experiments on real graphs' transductive and inductive learning tasks demonstrate the superiority of the proposed Node2Grids model against the state-of-the-art GCN-based approaches.
Edit intent: meaning-changed
Original:  We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Revised:  For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that \Omega( local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Edit intent: meaning-changed
Original:  We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Revised:  We demonstrate the application ofLIC local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Edit intent: meaning-changed
Original:  We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Revised:  We demonstrate the application of_\gamma(P)/ \log\tau \log n) bits are required for solving a graph problem P with a \tau-round algorithm that errs with probability at most \gamma. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Edit intent: meaning-changed
Original:  We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon})
Revised:  We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ 1/t + \epsilon})
Edit intent: meaning-changed
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% ( 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: fluency
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / %DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: fluency
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% t^2) . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: meaning-changed
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: meaning-changed
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% ((1/t^2) n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: meaning-changed
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% ) bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: fluency
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ;
Edit intent: meaning-changed
Original:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ; in fact, our result is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem under the KT1 assumption .
Revised:  edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% 1 / t^2%DIFDELCMD < } \right)%%% . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem .
Edit intent: meaning-changed
Original: Based on the Kac random walk on the orthogonal group, we present a fast Johnson-Lindenstrauss transform: given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised: Based on the Kac ^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: coherence