wanyu/IteraTeR_v2 huggingface dataset:
Original:  The n^(log n) barrier for group isomorphism has withstood all attacks - even for the special cases of p-groups and nilpotent groups - ever since Tarjan's n^(log n + O(1))  algorithm.
Revised:  The n^(log n) barrier for group isomorphism has withstood all attacks - even for the special cases of p-groups and nilpotent groups --- ever since n^(log n + O(1)) algorithm.
Edit intent: clarity
Original:  In this work, we present the first significant improvement over n^( \log n) by showing that group isomorphism is n^((1 / 2) log_p n + O(1)) Turing reducible to composition-series isomorphism where p is the smallest prime dividing the order of the group.
Revised:  In this work, we present the first significant improvement over n^( log n) by showing that group isomorphism is n^((1 / 2) log_p n + O(1)) Turing reducible to composition-series isomorphism where p is the smallest prime dividing the order of the group.
Edit intent: clarity
Original:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^O(p / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Edit intent: fluency
Original:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Edit intent: fluency
Original:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Edit intent: clarity
Original:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( log n / log log n)) algorithm for p-group isomorphism and an n^(( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^(O(log n / log log n) ) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log _p n + O( 1 / 2)log n + O(log n / log log n) ) algorithm for nilpotent-group isomorphism.
Edit intent: coherence
Original:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and nilpotent groups --- ever since n^(log n + O(1)) generator enumeration algorithm.
Revised:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since n^(log n + O(1)) generator enumeration algorithm.
Edit intent: clarity
Original:  Combining our reduction with an n ^O (p / log p)  algorithm for p-group composition-series isomorphism, we obtain  n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^(O (p / log p) algorithm for p-group composition-series isomorphism, we obtain n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Edit intent: fluency
Original:  Combining our reduction with an n ^O (p / log p)  algorithm for p-group composition-series isomorphism, we obtain  n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^O (p / log p) ) algorithm for p-group composition-series isomorphism, we obtain n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Edit intent: fluency
Original:  Combining our reduction with an n ^O (p / log p)  algorithm for p-group composition-series isomorphism, we obtain  n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Revised:  Combining our reduction with an n ^O (p / log p) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log n + O(1)) algorithms for p- and nilpotent-group isomorphism.
Edit intent: fluency
Original: We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G \cong H .
Revised: We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G and H are isomorphic .
Edit intent: clarity
Original:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since  n^(log n + O(1)) generator enumeration algorithm.
Revised:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since the n^(log n + O(1)) generator enumeration algorithm.
Edit intent: fluency
Original:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since  n^(log n + O(1)) generator enumeration algorithm.
Revised:  The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since n^(log n + O(1)) generator-enumeration algorithm.
Edit intent: fluency
Original:  Finally, we show a new relationship between group isomorphism and the birthday problem which allows us replace the 1 / 2 in the exponents with 1 / 4 using randomized algorithms and 1 / 6 using quantum algorithms.
Revised:  Finally, we relate group isomorphism to the collision problem which allows us replace the 1 / 2 in the exponents with 1 / 4 using randomized algorithms and 1 / 6 using quantum algorithms.
Edit intent: clarity
Original:  These various indicators are tested against the hypothesis that the classification of journals into 11 broad fields by PatentBoard/National Science Foundation provides (statistically ) significant between-field effects.
Revised:  These various indicators are tested against the hypothesis that the classification of journals into 11 broad fields by PatentBoard/National Science Foundation provides statistically significant between-field effects.
Edit intent: fluency
Original:  The winner of the existential k-pebble game on two given finite structures can easily be determined in time O(n^{2k})  .
Revised:  The winner of the existential k-pebble game on two given finite structures can be determined in time O(n^{2k}) .
Edit intent: clarity
Original:  We show that there is no O(n^{(k-3)/12}) time algorithm that decides which player can win the existential k-pebble game on two given structures.
Revised:  We show that there is no O(n^{(k-3)/12}) -time algorithm that decides which player can win the existential k-pebble game on two given structures.
Edit intent: fluency
Original:  This lower bound is unconditional and does not rely on any unproven complexity theoretic assumptions.
Revised:  This lower bound is unconditional and does not rely on any complexity-theoretic assumptions.
Edit intent: clarity
Original:  By the game characterization of Kolaitis and Vardi our result implies that there is no O(n^{(k-3)/12}) time algorithm that decides if strong k-consistency can be established for a given CSP-instance.
Revised:  By the game characterization of Kolaitis and Vardi our result implies that there is no O(n^{(k-3)/12}) -time algorithm that decides if strong k-consistency can be established for a given CSP-instance.
Edit intent: fluency
Original:  The winner of the existential k-pebble game on two given finite structures can be determined in time O( n^{2k ) by dynamic programming on the graph of game configurations.
Revised:  The winner of the existential k-pebble game on two given finite structures can be determined in time O( n2k ) by dynamic programming on the graph of game configurations.
Edit intent: clarity
Original:  By the game characterization of Kolaitis and Vardi our result implies that there is no O(n ^{(k-3)/12 )-time algorithm that decides if strong k-consistency can be established for a given CSP-instance.
Revised:  By the game characterization of Kolaitis and Vardi our result implies that there is no O(n (k-3)/12 )-time algorithm that decides if strong k-consistency can be established for a given CSP-instance.
Edit intent: clarity
Original: In 1998, Reed conjectured that for any graph G, \chi(G) \leq \lceil \omega(G) + \Delta(G)+1{2}\rceil, where \chi(G), \omega(G), and \Delta(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we study this conjecture for some %DIFDELCMD < {\em %DIFDELCMD < %%% expansions of graphs, that is graphs obtained with the well known operation %DIFDELCMD < {\em %%% composition of graphs.
Revised: In 1998, Reed conjectured that for any graph G, \chi(G) \leq \lceil \omega(G) + \Delta(G)+1{2}\rceil, where \chi(G), \omega(G), and \Delta(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we study this conjecture for some %DIFDELCMD < {\em %DIFDELCMD < %%% expansions of graphs, that is graphs obtained with the well known operation %DIFDELCMD < {\em %%% composition of graphs.
Edit intent: fluency
Original: In this paper we prove that given any lattice \Lambda_1 \in R^n and any (n-k)-dimensional lattice \Lambda_2\nobreak  , there is a set of k vectors v_i \in \Lambda_1 such that \Lambda_2 is, up to similarity, arbitrarily close to the projection of \Lambda_1 onto the orthogonal complement of the subspace spanned by v_1, ... , v_k.
Revised: In this paper we prove that given any lattice \Lambda_1 \in R^n and \Lambda_2 \subset\nobreak , there is a set of k vectors v_i \in \Lambda_1 such that \Lambda_2 is, up to similarity, arbitrarily close to the projection of \Lambda_1 onto the orthogonal complement of the subspace spanned by v_1, ... , v_k.
Edit intent: clarity
Original: Given a graph G and an integer k, the Feedback Vertex Set (FVS ) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph.
Revised: Given a graph G and an integer k, the (FVS ) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph.
Edit intent: coherence
Original: Given a graph G and an integer k, the Feedback Vertex Set (FVS ) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph.
Revised: Given a graph G and an integer k, the Feedback Vertex Set FVS ) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph.
Edit intent: clarity
Original:  Bodlaender (WG '91) gave the first fixed-parameter algorithm forFVS in undirected graphs.
Revised:  FVS in undirected graphs.
Edit intent: clarity
Original:  Bodlaender (WG '91) gave the first fixed-parameter algorithm forFVS in undirected graphs.
Revised:  Bodlaender (WG '91) gave the first fixed-parameter algorithm for in undirected graphs.
Edit intent: clarity
Original:  Bodlaender (WG '91) gave the first fixed-parameter algorithm forFVS in undirected graphs.
Revised:  Bodlaender (WG '91) gave the first fixed-parameter algorithm forFVS 
Edit intent: clarity
Original:  The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is fixed-parameter tractable by giving an 4^{k}k!n^{O(1)}  algorithm.
Revised:  The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is fixed-parameter tractable by giving an 4^{k}k!n^{O(1)} algorithm.
Edit intent: fluency
Original:  The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is fixed-parameter tractable by giving an 4^{k}k!n^{O(1)}  algorithm.
Revised:  The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a 4^{k}k!n^{O(1)} algorithm.
Edit intent: clarity
Original:  In the subset versions of this problems, we are given an additional subset S of vertices (resp.  edges) and we want to hit all cycles passing through a vertex of S (resp. an edge of S).
Revised:  In the subset versions of this problems, we are given an additional subset S of vertices (resp. , edges) and we want to hit all cycles passing through a vertex of S (resp. an edge of S).
Edit intent: fluency
Original:  Recentlythe\textsc{Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and  Kakimura et al. (SODA '12).
Revised:  Recentlythe\textsc{Recently, the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and Kakimura et al. (SODA '12).
Edit intent: clarity
Original:  We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time 2^{ 2^{O(k) }n^{O(1)} , i. e., FPT parameterized by size k of the solution.
Revised:  We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time 2^{ 2^{O(k) }n^{O(1)} , i. e., FPT parameterized by size k of the solution.
Edit intent: fluency
Original:  We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time 2^{ 2^{O(k) }n^{O(1)} , i. e., FPT parameterized by size k of the solution.
Revised:  We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time 2^{ 2^{O(k) }n^{O(1)} .
Edit intent: coherence
Original:  In this paper we give a general family of problems (which includes\textsc{Directed Multiway Cut and\textsc{Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Revised:  \textsc{Directed Multiway Cut and\textsc{Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Edit intent: clarity
Original:  In this paper we give a general family of problems (which includes\textsc{Directed Multiway Cut and\textsc{Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Revised:  In this paper we give a general family of problems (which includes\textsc{Directed Multiway Cut \textsc{Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Edit intent: fluency
Original:  In this paper we give a general family of problems (which includes\textsc{Directed Multiway Cut and\textsc{Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Revised:  In this paper we give a general family of problems (which includes\textsc{Directed Multiway Cut and\textsc{ among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow".
Edit intent: coherence
Original:  The design algorithm of 1-byte in 1-clock is complex since it takes ]  two consecutive bytes together  .
Revised:  The design algorithm in ] two consecutive bytes together .
Edit intent: clarity
Original:  The design algorithm of 1-byte in 1-clock is complex since it takes ]  two consecutive bytes together  .
Revised:  The design algorithm of 1-byte in 1-clock is complex since it takes ] considers two consecutive bytes together .
Edit intent: fluency
Original:  In this paper a simpler embedded system design of RC4 in 1-byte  in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Revised:  In this paper considering the RC4 in 1-byte in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Edit intent: clarity
Original:  In this paper a simpler embedded system design of RC4 in 1-byte  in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Revised:  In this paper a simpler embedded system design of RC4 in 1-byte is processed in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Edit intent: clarity
Original:  In this paper a simpler embedded system design of RC4 in 1-byte  in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Revised:  In this paper a simpler embedded system design of RC4 in 1-byte in 1-clock . In the design two sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Edit intent: clarity
Original:  In this paper a simpler embedded system design of RC4 in 1-byte  in 1-clock is proposed where two successive sequential tasks are executed as two independent events by using rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Revised:  In this paper a simpler embedded system design of RC4 in 1-byte in 1-clock is proposed where two successive sequential tasks are executed as two independent events during rising and falling edges of the same clock and swapping is done using a MUX-DEMUX combination , thereby increasing its throughput .
Edit intent: clarity
Original:  The encryption and decryption designs are respectively implemented on two FPGA boards to achieve Ethernet communication.
Revised:  The encryption and decryption designs are respectively embedded on two FPGA boards to achieve Ethernet communication.
Edit intent: clarity
Original:  Thus far, SELEX experiments have been very good at suggesting the highest affinity oligonucleotide, but modeling but modelling lower affinity recognition site variants has been difficult.
Revised:  Thus far, SELEX experiments have been very good at suggesting the highest affinity oligonucleotide, but modeling lower affinity recognition site variants has been difficult.
Edit intent: clarity
Original: We study a new geometric graph parameter egd \egd (G), defined as the smallest integer r\ge 1 for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of G, can be completed to a matrix in the convex hull of correlation matrices of \rank at most r.
Revised: We study a new geometric graph parameter \egd (G), defined as the smallest integer r\ge 1 for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of G, can be completed to a matrix in the convex hull of correlation matrices of \rank at most r.
Edit intent: clarity
Original:  This graph parameter is motivated by its relevance to the  bounded rank Grothendieck constant : egd\egd (G)\le r if and only if the rank-r Grothendieck constant of G is equal to 1.
Revised:  This graph parameter is motivated by its relevance to the bounded rank Grothendieck constant . Indeed,\egd (G)\le r if and only if the rank-r Grothendieck constant of G is equal to 1.
Edit intent: coherence
Original:  The parameter egd\egd (G) is minor monotone .
Revised:  We show that the parameter\egd (G) is minor monotone .
Edit intent: clarity
Original:  We identify several classes of forbidden minors for egd \egd (G)\le r and  give the full characterization for the case r=2.
Revised:  We identify several classes of forbidden minors for \egd (G)\le r and give the full characterization for the case r=2.
Edit intent: fluency
Original:  We identify several classes of forbidden minors for egd \egd (G)\le r and  give the full characterization for the case r=2.
Revised:  We identify several classes of forbidden minors for egd \egd (G)\le r and we give the full characterization for the case r=2.
Edit intent: fluency
Original:  We  show an upper bound for egd \egd (G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest r for which G is a minor of the strong product of a tree and K_r.
Revised:  We also show an upper bound for egd \egd (G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest r for which G is a minor of the strong product of a tree and K_r.
Edit intent: clarity
Original:  We  show an upper bound for egd \egd (G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest r for which G is a minor of the strong product of a tree and K_r.
Revised:  We show an upper bound for \egd (G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest r for which G is a minor of the strong product of a tree and K_r.
Edit intent: clarity
Original:  We show that, for  G\ne K_{3,3} 2-connected on at least 6 nodes, egd \egd (G)\le 2 if and only if \sla(G)\le 2.
Revised:  We show that, for any 2-connected graph G\ne K_{3,3} 2-connected on at least 6 nodes, egd \egd (G)\le 2 if and only if \sla(G)\le 2.
Edit intent: clarity
Original:  We show that, for  G\ne K_{3,3} 2-connected on at least 6 nodes, egd \egd (G)\le 2 if and only if \sla(G)\le 2.
Revised:  We show that, for G\ne K_{3,3} 2-connected on at least 6 nodes, \egd (G)\le 2 if and only if \sla(G)\le 2.
Edit intent: clarity
Original:  Such clones emerge , when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population , despite being broken down by recombination.
Revised:  Such clones emerge when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population , despite being broken down by recombination.
Edit intent: fluency
Original:  Such clones emerge , when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population , despite being broken down by recombination.
Revised:  Such clones emerge , when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population despite being broken down by recombination.
Edit intent: fluency