wanyu/IteraTeR_v2 huggingface dataset:
Original:  More precisely, in these applications , we are given a graph with non-negative edge weights, and a monotone function such as sum, which aggregates edge weights into a single value that defines a path weight .
Revised:  More precisely, in these applications , we are given a graph with non-negative edge weights, along with a monotone function such as sum, which aggregates edge weights into a single value that defines a path weight .
Edit intent: clarity
Original:  We are required to find the top-k heaviest simple paths, i.e., the k simple ( i.e., cycle-free) paths with the greatest weight, whose length equals a given parameter \ell.
Revised:  We are then interested in finding the top-k heaviest simple paths, i.e., the k simple ( i.e., cycle-free) paths with the greatest weight, whose length equals a given parameter \ell.
Edit intent: coherence
Original:  We are required to find the top-k heaviest simple paths, i.e., the k simple ( i.e., cycle-free) paths with the greatest weight, whose length equals a given parameter \ell.
Revised:  We are required to find the top-k heaviest simple paths, i.e., the k simple ( cycle-free) paths with the greatest weight, whose length equals a given parameter \ell.
Edit intent: fluency
Original:  We call this the Heavy Path Problem (HPP).
Revised:  We call this the Heavy Path Problem (HPP).
Edit intent: fluency
Original:  The problem is known to be NP-Hard , and in general not approximable .
Revised:  It is easy to show that the problem is NP-Hard , and in general not approximable .
Edit intent: clarity
Original:  The problem is known to be NP-Hard , and in general not approximable .
Revised:  The problem is known to be NP-Hard .
Edit intent: clarity
Original:  In this work, we develop a practical approach to solve the Heavy Path problem by establishing a strong connection with the well-known Rank Join paradigm.
Revised:  In this work, we develop a practical approach to solve the Heavy Path problem by leveraging a strong connection with the well-known Rank Join paradigm.
Edit intent: clarity
Original:  We present an adaptation of the Rank Join algorithm and tailor it to self-joins over the list of edges.
Revised:  We first present an algorithm by adapting the Rank Join algorithm and tailor it to self-joins over the list of edges.
Edit intent: clarity
Original:  In addition , our heuristic approach finds paths that are empirically within 50\% of the optimum solution or better under various settings, while taking a fraction of the running time compared to the exact algorithm.
Revised:  Further , our heuristic approach finds paths that are empirically within 50\% of the optimum solution or better under various settings, while taking a fraction of the running time compared to the exact algorithm.
Edit intent: coherence
Original:  In addition , our heuristic approach finds paths that are empirically within 50\% of the optimum solution or better under various settings, while taking a fraction of the running time compared to the exact algorithm.
Revised:  In addition , our heuristic approach finds paths that are empirically within 50\% of the optimum solution or better under various settings, and takes only a fraction of the running time compared to the exact algorithm.
Edit intent: coherence
Original:  A convex Steiner partition is a tiling of CH \conv (S) with empty convex bodies.
Revised:  A Steiner convex partition is a tiling of CH \conv (S) with empty convex bodies.
Edit intent: clarity
Original:  A convex Steiner partition is a tiling of CH \conv (S) with empty convex bodies.
Revised:  A convex Steiner partition is a tiling of \conv (S) with empty convex bodies.
Edit intent: clarity
Original:  For every integer d, we show that S admits a convex Steiner partition with at most \lceil (n-1)/d \rceil tiles.
Revised:  For every integer d, we show that S admits a Steiner convex partition with at most \lceil (n-1)/d \rceil tiles.
Edit intent: fluency
Original:  This bound is the best possible for affine independent points in the plane, and it is best possible apart from constant factors in every dimension d>= 3.
Revised:  This bound is the best possible for affine independent points in the plane, and it is best possible apart from constant factors in every fixed dimension d\geq 3.
Edit intent: clarity
Original:  We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of an affine independent point set in the plane.
Revised:  We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in the plane.
Edit intent: clarity
Original:  Determining the maximum possible volume of a single tile in a Steiner partition is equivalent to a famous problem of Danzer and Rogers.
Revised:  Determining the maximum possible volume of a tile in a Steiner partition is equivalent to a famous problem of Danzer and Rogers.
Edit intent: clarity
Original:  We give a ( 1-epsilon\eps )-approximation for the maximum volume of an empty convex body when S lies in the d-dimensional unit box [0,1]^d.
Revised:  We give a ( 1-epsilon\eps )-approximation algorithm for computing the maximum volume of an empty convex body when S lies in the d-dimensional unit box [0,1]^d.
Edit intent: clarity
Original: Let S be a set of n points in d-space \RR . A Steiner convex partition is a tiling of \conv(S) with empty convex bodies.
Revised: Let S be a set of n points in \RR . A Steiner convex partition is a tiling of \conv(S) with empty convex bodies.
Edit intent: clarity
Original: Let S be a set of n points in %DIFDELCMD < \RR%%%  ^d. A Steiner convex partition is a tiling of %DIFDELCMD < \conv%%% {\rm  (S) with empty convex bodies.
Revised: Let S be a set of n points in %DIFDELCMD < \RR%%% ^d. A Steiner convex partition is a tiling of %DIFDELCMD < \conv%%% {\rm conv (S) with empty convex bodies.
Edit intent: fluency
Original:  We consider numerations and probabilistic properties of these trees when generated under the random coalescent process.
Revised:  We consider enumerations and probabilistic properties of these trees when generated under the random coalescent process.
Edit intent: fluency
Original:  We consider enumerations and probabilistic properties of these trees when generated under the random coalescent process.
Revised:  We consider enumerations and probabilistic properties of ranked trees when generated under the random coalescent process.
Edit intent: clarity
Original: Optimizing link activation is a fundamental aspect in performance engineering of wireless networks  .
Revised: A fundamental aspect in performance engineering of wireless networks .
Edit intent: clarity
Original:  In this paper, we assume multiuser detection receivers, which can decode and cancel strongly interfering signals.
Revised:  In this paper, we assume multiuser detection receivers, which can cancel strongly interfering signals.
Edit intent: clarity
Original:  We provide both theoretical and numerical studies of optimal link activation in this novel setup .
Revised:  Our goal here is to deliver a comprehensive theoretical and numerical studies of optimal link activation in this novel setup .
Edit intent: clarity
Original:  We provide both theoretical and numerical studies of optimal link activation in this novel setup .
Revised:  We provide both theoretical and numerical study on optimal link activation in this novel setup .
Edit intent: fluency
Original:  We provide both theoretical and numerical studies of optimal link activation in this novel setup .
Revised:  We provide both theoretical and numerical studies of optimal link activation under this novel setup .
Edit intent: fluency
Original:  By a simple distributed control rule, the considered multi-agent system  achieves not only a consensus, but also an optimal one by agreeing within the optimal solution set of the group's optimization objective.
Revised:  By a simple distributed control rule, the considered multi-agent system achieves not only a consensus, but also an optimal agreement within the optimal solution set of the group's optimization objective.
Edit intent: clarity
Original:  By a simple distributed control rule, the considered multi-agent system  achieves not only a consensus, but also an optimal one by agreeing within the optimal solution set of the group's optimization objective.
Revised:  By a simple distributed control rule, the considered multi-agent system achieves not only a consensus, but also an optimal one by agreeing within the optimal solution set of the overall optimization objective.
Edit intent: clarity
Original:  Directed and bidirectional communications are studied  respectively, and connectivity conditions are given to ensure a global optimal consensus.
Revised:  Directed and bidirectional communications are studied , respectively, and connectivity conditions are given to ensure a global optimal consensus.
Edit intent: fluency
Original:  We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex analysis and non-smooth analysis , based on which the convergence analysis is presented .
Revised:  We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis , based on which the convergence analysis is presented .
Edit intent: fluency
Original:  We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex analysis and non-smooth analysis , based on which the convergence analysis is presented .
Revised:  We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex analysis and non-smooth analysis .
Edit intent: coherence
Original: This paper investigates the role which the persistent links play for a  network to reach a global  agreement under discrete-time or continuous-time consensus dynamics . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another.
Revised: This paper investigates the role persistent arcs play for a network to reach a global agreement under discrete-time or continuous-time consensus dynamics . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another.
Edit intent: clarity
Original: This paper investigates the role which the persistent links play for a  network to reach a global  agreement under discrete-time or continuous-time consensus dynamics . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another.
Revised: This paper investigates the role which the persistent links play for a network to reach a global agreement under discrete-time or continuous-time evolution . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another.
Edit intent: clarity
Original: This paper investigates the role which the persistent links play for a  network to reach a global  agreement under discrete-time or continuous-time consensus dynamics . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow between the from one node to another.
Revised: This paper investigates the role which the persistent links play for a network to reach a global agreement under discrete-time or continuous-time consensus dynamics . Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function which describes the strength of the information flow from one node to another.
Edit intent: coherence
Original:  Then an arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph . A series of sufficient and necessary conditions on global agreement or \epsilon-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement.
Revised:  An arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph . A series of sufficient and necessary conditions on global agreement or \epsilon-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement.
Edit intent: coherence
Original:  Then an arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph . A series of sufficient and necessary conditions on global agreement or \epsilon-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement.
Revised:  Then an arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph of the underlying network. Three necessary and sufficient conditions on agreement or \epsilon-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement.
Edit intent: clarity
Original:  Then an arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph . A series of sufficient and necessary conditions on global agreement or \epsilon-agreement are established, by which we clearly address that the persistent graph fully determines the convergence to a collective agreement.
Revised:  Then an arc is said to be persistent if its weight function has infinite L_1 or \ell_1 norm , and the subgraph which contains all persistent arcs is called the persistent graph . A series of sufficient and necessary conditions on global agreement or \epsilon-agreement are established, by which we prove that the persistent graph fully determines the convergence to a collective agreement.
Edit intent: coherence
Original:  The results conclude that towards a global agreement, the links which last in a long period contribute fundamentally while the links which are only significant for a short time do not .
Revised:  The results adds to the understanding of the fundamentals behind global agreements, as it is only persistent arcs that contribute to the convergence .
Edit intent: clarity
Original: Baxter permutations are  in bijection with  floorplans that arise in chip design  .
Revised: Baxter permutations are a class of permutations which are in bijection with floorplans that arise in chip design .
Edit intent: clarity
Original:  This naturally leads to studying a sub-family of Baxter permutations.
Revised:  This naturally leads to studying a subclass of Baxter permutations.
Edit intent: clarity
Original:  The sub-family of Baxter permutations are characterized by pattern avoidance.
Revised:  This subclass of Baxter permutations are characterized by pattern avoidance.
Edit intent: clarity
Original:  We establish a bijection between the sub-family of floorplans and a sub-family Baxter permutations based on the analogy between decomposition of a floorplan into smaller blocks and block decomposition of permutations.
Revised:  We establish a bijection between the sub-family of floorplans and a sub-family Baxter permutations based on the analogy between decomposition of a floorplan into smaller blocks and block decomposition of permutations.
Edit intent: fluency
Original:  Apart from the characterization, we also answer combinatorial questions on these families .
Revised:  Apart from the characterization, we also answer combinatorial questions on these classes .
Edit intent: clarity
Original:  We give a rational generating function for number of permutations in each class , an exponential lower bound on growth rate of each class , and a quadratic time algorithm for deciding membership in each class . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Revised:  We give a rational generating function for number of permutations , an exponential lower bound on growth rate of each class , and a quadratic time algorithm for deciding membership in each class . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Edit intent: fluency
Original:  We give a rational generating function for number of permutations in each class , an exponential lower bound on growth rate of each class , and a quadratic time algorithm for deciding membership in each class . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Revised:  We give a rational generating function for number of permutations in each class , an exponential lower bound on growth rate , and a quadratic time algorithm for deciding membership in each class . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Edit intent: clarity
Original:  We give a rational generating function for number of permutations in each class , an exponential lower bound on growth rate of each class , and a quadratic time algorithm for deciding membership in each class . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Revised:  We give a rational generating function for number of permutations in each class , an exponential lower bound on growth rate of each class , and a quadratic time algorithm for deciding membership in each subclass . Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration.
Edit intent: clarity
Original:  Our composite rational strategy involving literature-mined experimental data, graph theoretical analysis and gene ontological studies predicts targets that are more specific and relevant for SBC .
Revised:  Our composite rational strategy involving literature-mined experimental data, graph theoretical analysis and gene ontological studies predicts targets that are of ontological and regulatory relevance to secondary bone cancer .
Edit intent: clarity
Original:  We define a comprehensive pricing framework which allows us to derive earlier results  funding or counterparty risk.
Revised:  We define a comprehensive pricing framework which allows us to derive earlier results on funding or counterparty risk.
Edit intent: fluency
Original: We study the problem of finding a triangulation T of a planar point set S such that the expected distance between two points x and y chosen uniformly at random from S is minimized .
Revised: We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S is minimized .
Edit intent: clarity
Original: We study the problem of finding a triangulation T of a planar point set S such that the expected distance between two points x and y chosen uniformly at random from S is minimized .
Revised: We study the problem of finding a triangulation T of a planar point set S such that the expected distance between two points x and y chosen uniformly at random from S .
Edit intent: clarity
Original:  By distance we mean the shortest path length between x and y along edges of T.
Revised:  By distance we mean the length of the shortest path between x and y along edges of T.
Edit intent: clarity
Original:  The length of a path is the sum of the weights of the edgescontained in it . Edge weights are assumed to be given as part of the problem for every pair of distinct points (x,y) in S^2.
Revised:  The length of a path is the sum of the weights of its edges . Edge weights are assumed to be given as part of the problem for every pair of distinct points (x,y) in S^2.
Edit intent: clarity
Original:  We prove that a general formulation of the problem in which the weights are arbitrary positive numbers is strongly NP-complete , both for point sets and for polygons .
Revised:  We prove that a general formulation of the problem in which the weights are arbitrary positive numbers is strongly NP-complete .
Edit intent: coherence
Original:  The proof develops a new simulation result that may be of an independent interest.
Revised:  The proof develops a new simulation procedure that may be of an independent interest.
Edit intent: clarity
Original: We introduce a class of correlated weighted graphs whose properties are meant to mimic the topological features of idiotypic networks, namely the interaction networks involving the B-core of the immune system.
Revised: We introduce a class of weighted graphs whose properties are meant to mimic the topological features of idiotypic networks, namely the interaction networks involving the B-core of the immune system.
Edit intent: clarity
Original:  Each node is endowed with a bit-string representing the epitopal specificity of the corresponding B cell , a proper distance between any couple of bit-strings provides the coupling strength between the two nodes.
Revised:  Each node is endowed with a bit-string representing the epitopal specificity of the corresponding B cell and a proper distance between any couple of bit-strings provides the coupling strength between the two nodes.
Edit intent: coherence
Original:  They showed that the median problem under the breakpoint distance is solvable in polynomial time in the multichromosomal circular and mixed  models.
Revised:  They showed that the median problem is solvable in polynomial time in the multichromosomal circular and mixed models.
Edit intent: clarity
Original:  This is intriguing, since in all other rearrangement models (DCJ, reversal, unichromosomal or multilinear breakpoint models), the problem is NP-hard.
Revised:  This is intriguing, since in most other rearrangement models (DCJ, reversal, unichromosomal or multilinear breakpoint models), the problem is NP-hard.
Edit intent: clarity
Original:  In other words, while finding an ancestor for 3 species is easy, already finding two ancestors for 4 species is  hard.
Revised:  In other words, while finding an ancestor for 3 species is easy, finding two ancestors for 4 species is hard.
Edit intent: coherence
Original:  In other words, while finding an ancestor for 3 species is easy, already finding two ancestors for 4 species is  hard.
Revised:  In other words, while finding an ancestor for 3 species is easy, already finding two ancestors for 4 species is already hard.
Edit intent: coherence
Original: The Web emerged as the antidoteto rapidly increasing quantity of accumulated knowledge because it successfully enables massive representation and communication with minimum costs.
Revised: The Web initially emerged as an "antidote" to accumulated scientific knowledge since it enables global representation and communication with minimum costs.
Edit intent: clarity
Original:  Despite the fact that its gigantic scale and impact make difficult to anticipate the effects in humans, we claim from it to be fast, secure, reliable, all-inclusive and trustworthy .
Revised:  Its gigantic scale and impact make difficult to anticipate the effects in humans, we claim from it to be fast, secure, reliable, all-inclusive and trustworthy .
Edit intent: coherence