wanyu/IteraTeR_full_sent huggingface dataset:
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \Reals and a real \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k, report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Edit intent: clarity
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals^d, and a real \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k, report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Edit intent: meaning-changed
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real fixed \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k, report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Edit intent: meaning-changed
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% \P into a data structure so that for any given query curve Q of size k, report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Edit intent: fluency
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k, we can efficiently report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Edit intent: clarity
Original:  In this paper, we revisit the problem in the presence of curves under the  Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real  \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k,  report all curves in %DIFDELCMD < {\cal %%% P whose Fr\'echet distances to Q are at most \delta.
Revised:  In this paper, we revisit the problem in the presence of curves under the Fr\'echet distance \Reals . Given a set %DIFDELCMD < {\cal %%% P of n curves of size at most m each in \mathbb{R \Reals and a real \delta>0, we aim to preprocess %DIFDELCMD < {\cal %%% P into a data structure so that for any given query curve Q of size k, report all curves in %DIFDELCMD < {\cal %%% \P whose Fr\'echet distances to Q are at most \delta.
Edit intent: meaning-changed
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In the case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Edit intent: fluency
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is given in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Edit intent: clarity
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage , for any\eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Edit intent: meaning-changed
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage \eps>0 we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Edit intent: meaning-changed
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage \eps we propose a deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Edit intent: clarity
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( n \cdot O\big(\big\{\big(}{\eps}}\big)
Edit intent: meaning-changed
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\max\big\{\big(}{\eps}}\big)
Edit intent: meaning-changed
Original:  In  case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(}{\eps}}\big)
Revised:  In case that k is known in the preprocessing stage \eps we propose a fully deterministic data structure whose space is O(n ( 32d\big(\big\{\big(\frac{\sqrt{d}{\eps}}\big)
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ kd } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: fluency
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: coherence
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 ,\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: fluency
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(\frac{\D\sqrt{d}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ kd } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: coherence
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: coherence
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) that can answer \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: fluency
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: fluency
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time , where\D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D is the diameter of \P . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( /\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% n \cdot O\big({\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big(\frac{1{\eps}}\big) with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big)^{md with O(kd) query time within 5 (1 + \varepsilon) approximation factor .
Edit intent: fluency
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within an approximation factor of 5 (1 + \varepsilon) approximation factor .
Edit intent: clarity
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor .
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 + \varepsilon) approximation factor .
Edit intent: meaning-changed
Original:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big)  \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D  . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within  5 (1 + \varepsilon) approximation factor . We also \eps show that our data structure could give an alternative treatment of the approximate subtrajectory range counting (\textsc{ASRC ) problem studied by de Berg et al.
Revised:  ^{ 1/2 } /\varepsilon^3 )\big(}{\eps^2}}\big) ^{ d(k+1) } ) and can answer the \big\}\big) \textsc{(1+ \varepsilon \eps )\delta-ANNS} queries in O(kd) query time \D . Considering k as part of the query slightly changes the space to O( n ( 64d^{1/2/\varepsilon^3}%DIFDELCMD < }%%% )^{md\big({\eps}}\big) with O(kd) query time within 5 (1 + \eps show that our data structure could give an alternative treatment of the approximate subtrajectory range counting (\textsc{ASRC ) problem studied by de Berg et al.
Edit intent: clarity
Original:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps\eps .
Revised:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [8 [ \eps\eps .
Edit intent: meaning-changed
Original:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps\eps .
Revised:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ . We also revisit the time-window data structure for spatial density maps in[ \eps\eps .
Edit intent: meaning-changed
Original:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps\eps .
Revised:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [6 \eps\eps .
Edit intent: fluency
Original:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps\eps .
Revised:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ . Given \theta>0, and n time-stamped points spread over m regions in a map, for any query window W, we propose a data structure of size O(n/\eps\eps .
Edit intent: meaning-changed
Original:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps\eps .
Revised:  ~\mbox{%DIFAUXCMD \cite{bcg-ffq-13}\hspace{0pt}%DIFAUXCMD } [ [ \eps^2) and construction time O((n+m)/\eps .
Edit intent: meaning-changed