wanyu/IteraTeR_v2 huggingface dataset:
Original:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n}  for some constant C.
Revised:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% {n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n} for some constant C.
Edit intent: clarity
Original:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n}  for some constant C.
Revised:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% {n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n} for some constant C.
Edit intent: clarity
Original:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n}  for some constant C.
Revised:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% {n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n} for some constant C.
Edit intent: clarity
Original:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n}  for some constant C.
Revised:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% {n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n} for some constant C.
Edit intent: clarity
Original:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n}  for some constant C.
Revised:  We show that if m\geq n then \ds %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %DIFDELCMD < %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil%%% \right\rceil \leq \lambda_n(m,k) \leq %DIFDELCMD < \lceil%%% \frac{m{n}}%DIFDELCMD < \lceil %%% \frac{k{n}}%DIFDELCMD < \rceil %%% \left\lceil{n}}\left\lceil {n}}\right\rceil + k{n} %DIFDELCMD < \rceil %%% \right\rceil + C m^2\log k{n} \mbox for some constant C.
Edit intent: fluency
Original: A digraph is %DIFDELCMD < {\it %%% m-labelled if every arcs is labelled by an integer in \{1, ... \dots ,m\}.
Revised: A digraph is %DIFDELCMD < {\it %%% m-labelled if every arc is labelled by an integer in \{1, ... \dots ,m\}.
Edit intent: fluency
Original: A digraph is %DIFDELCMD < {\it %%% m-labelled if every arcs is labelled by an integer in \{1, ... \dots ,m\}.
Revised: A digraph is %DIFDELCMD < {\it %%% m-labelled if every arcs is labelled by an integer in \{1, \dots ,m\}.
Edit intent: fluency
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we %DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings introduce and study n-fibre colourings of labelled digraphs. These are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: coherence
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, and for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: fluency
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \alpha )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \alpha )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \alpha ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \alpha ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Edit intent: clarity
Original:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v,  for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there exists an arc leaving v of label l coloured \lambda.
Revised:  Motivated by wavelength assignment for multicasts in optical star networks, we study%DIFDELCMD < {\it %%% n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour \lambda , in(v, \lambda )+out(v, \lambda )\leq n with in(v, \lambda ) the number of arcs coloured \lambda entering v and out(v, \lambda ) the number of labels l such that there is at least one arc of label l coloured \lambda.
Edit intent: clarity
Original:  One likes to find the minimum number of colours \lambda_n(D) such that an m-labbelled digraph D has an n-fiber colouring.
Revised:  One likes to find the minimum number of colours \lambda_n(D) such that the m-labelled digraph D has an n-fiber colouring.
Edit intent: fluency
Original:  One likes to find the minimum number of colours \lambda_n(D) such that an m-labbelled digraph D has an n-fiber colouring.
Revised:  One likes to find the minimum number of colours \lambda_n(D) such that an m-labbelled digraph D has an n-fibre colouring.
Edit intent: fluency
Original:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Revised:  In the particular case when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Edit intent: fluency
Original:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Revised:  In the particular case , when D is 1-labelled then \lambda_n (D) is %DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Edit intent: fluency
Original:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Revised:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% called the directed star arboricity of D, denoted dst(D).
Edit intent: clarity
Original:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, denoted dst(D).
Revised:  In the particular case , when D is 1-labelled then \lambda_n (D) is the%DIFDELCMD < {\it %%% directed star arboricty of D, and is denoted by dst(D).
Edit intent: coherence
Original:  We first show that dst(D)\leq 2\Delta^-(D)+1  and conjecture that if \Delta^-(D)\geq 2  then dst(D)\leq 2\Delta^-(D).
Revised:  We first show that dst(D)\leq 2\Delta^-(D)+1 , and conjecture that if \Delta^-(D)\geq 2 then dst(D)\leq 2\Delta^-(D).
Edit intent: fluency
Original:  We first show that dst(D)\leq 2\Delta^-(D)+1  and conjecture that if \Delta^-(D)\geq 2  then dst(D)\leq 2\Delta^-(D).
Revised:  We first show that dst(D)\leq 2\Delta^-(D)+1 and conjecture that if \Delta^-(D)\geq 2 , then dst(D)\leq 2\Delta^-(D).
Edit intent: fluency
Original:  We also prove that if D is subcubic then dst(D)\leq 3  and that if \Delta^+(D), \Delta^-(D)\leq 2  then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Revised:  We also prove that for a subcubic digraph D, then dst(D)\leq 3 and that if \Delta^+(D), \Delta^-(D)\leq 2 then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Edit intent: clarity
Original:  We also prove that if D is subcubic then dst(D)\leq 3  and that if \Delta^+(D), \Delta^-(D)\leq 2  then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Revised:  We also prove that if D is subcubic then dst(D)\leq 3 , and that if \Delta^+(D), \Delta^-(D)\leq 2 then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Edit intent: fluency
Original:  We also prove that if D is subcubic then dst(D)\leq 3  and that if \Delta^+(D), \Delta^-(D)\leq 2  then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Revised:  We also prove that if D is subcubic then dst(D)\leq 3 and that if \Delta^+(D), \Delta^-(D)\leq 2 , then dst(D)\leq 4. Finally, we study \lambda_n(m,k)=\max\{\lambda_n(D) \tq D %DIFDELCMD < {%%% is m-labelled \et \Delta^-(D)\leq k\}.
Edit intent: fluency
Original:  We show that if m\geq n  then \ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq \lambda_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n} \mbox for some constant C. %DIFDELCMD < }%%% 
Revised:  We show that if m\geq n , then \ds \left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil\leq \lambda_n(m,k) \leq\left\lceil\frac{m}{n}\left\lceil \frac{k}{n}\right\rceil + \frac{k}{n} \right\rceil + C \frac{m^2\log k}{n} \mbox for some constant C. %DIFDELCMD < }%%% 
Edit intent: fluency
Original:  For block size B  and cache size M  , the mesh update cost is O(1+|G| =B) , assuming the tall cache assumption M= \Omega (B^d) , where d is the dimensionality of the mesh 's geometric domain.
Revised:  For block size B , and cache size M , the mesh update cost is O(1+|G| =B) , assuming the tall cache assumption M= \Omega (B^d) , where d is the dimensionality of the mesh 's geometric domain.
Edit intent: clarity
Original:  For block size B  and cache size M  , the mesh update cost is O(1+|G| =B) , assuming the tall cache assumption M= \Omega (B^d) , where d is the dimensionality of the mesh 's geometric domain.
Revised:  For block size B and cache size M , the mesh update cost is O(1+|G| =B) , assuming the tall cache assumption M= o (B^d) , where d is the dimensionality of the mesh 's geometric domain.
Edit intent: clarity
Original:  We then introduce the relaxed-balanced decomposition tree and show that with M = \Omega(B^d), mesh update cost is O(1+  |G|/B) . We give a layout algorithm for relaxed-balanced decomposition trees, which runs cache-obliviously in O( |G| log |G| loglog |G |)operations and O(1+|G | log |G| log log |G|) memory transfers with high probability  .
Revised:  We then introduce the relaxed-balanced decomposition tree and show that with M = \Omega(B^d), mesh update cost is O(1+ ( |G|/B) . We give a layout algorithm for relaxed-balanced decomposition trees, which runs cache-obliviously in O( |G| log |G| loglog |G |)operations and O(1+|G | log |G| log log |G|) memory transfers with high probability .
Edit intent: fluency
Original:  We then introduce the relaxed-balanced decomposition tree and show that with M = \Omega(B^d), mesh update cost is O(1+  |G|/B) . We give a layout algorithm for relaxed-balanced decomposition trees, which runs cache-obliviously in O( |G| log |G| loglog |G |)operations and O(1+|G | log |G| log log |G|) memory transfers with high probability  .
Revised:  We then introduce the relaxed-balanced decomposition tree and show that with M = \Omega(B^d), mesh update cost is O(1+ |G|/B) . We give a layout algorithm for relaxed-balanced decomposition trees, which runs cache-obliviously in O( |G| log |G| loglog |G | log |G| log log |G|) memory transfers with high probability .
Edit intent: clarity
Original: Hayashi and Carthew (Nature 431 [2004], 647) have shown that the packing of cone cells in the Drosophila retina resembles soap bubble packing, and that changing cadherin expression can change this packing and cell shape.
Revised: Hayashi and Carthew (Nature 431 [2004], 647) have shown that the packing of cone cells in the Drosophila retina resembles soap bubble packing, and that changing cadherin expression can change this packing , as well as cell shape.
Edit intent: clarity
Original:  The efficiencies of the mechanisms and the nature of the induced, time-dependent flow fields are found to differ widely among swimmers .
Revised:  The swimming efficiency and the nature of the induced, time-dependent flow fields are found to differ widely among swimmers .
Edit intent: clarity
Original:  The efficiencies of the mechanisms and the nature of the induced, time-dependent flow fields are found to differ widely among swimmers .
Revised:  The efficiencies of the mechanisms and the nature of the induced, time-dependent flow fields are found to differ widely among body designs and propulsion mechanisms .
Edit intent: clarity
Original:  Similarity-measure based clustering is a crucial problem appearing throughout scientific data analysis.
Revised: Motivation: Similarity-measure based clustering is a crucial problem appearing throughout scientific data analysis.
Edit intent: coherence
Original:  In AP, each cluster is identified by a common exemplar all other data points of the same cluster refer to . Since the exemplars in AP are itselves data points, they have to refer to themselves.
Revised:  In AP, each cluster is identified by a common exemplar all other data points of the same cluster refer to , and exemplars have to refer to themselves.
Edit intent: coherence
Original:  The hard constraint of having exactly one exemplar in each cluster restricts the applicability of AP to classes of regularly shaped clusters, and leads to suboptimal performance, {\it e.g. , in analyzing gene expression data.
Revised:  The hard constraint of having exactly one exemplar per cluster restricts AP to classes of regularly shaped clusters, and leads to suboptimal performance, {\it e.g. , in analyzing gene expression data.
Edit intent: clarity
Original:   This limitation can be overcome by a straight-forward generalization of AP.
Revised:  Results: This limitation can be overcome by a straight-forward generalization of AP.
Edit intent: fluency
Original:  It is easier to interpret the statistical meaning of the tuning process.
Edit intent: coherence