Frechet distance decomposability- Can we do a divide and conquer approach for Frechet?
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Given a dataset of Polygonal Lines which are partitioned using equal-sized grid cells. Suppose two Lines P and Q which are segmented at cell boundaries and each of which is converted to two subsequences residing in the corresponding cells. If the distance between sub-sequences in each cell is less than epsilon, can we conclude that the distance of the whole Lines (not segmented ones) is also less than the epsilon.
In mathematical words, suppose the two P and Q lines each one partitioned into 2 subsequences at the point indicated by star and discrete Frechet Distance function dF.
I'm not good in mathematics. In the figure you can see that the distance dF(P1, Q1)=d1 and dF(P2, Q2)=d2, and dF(P, Q)=d2.
Does dF(P1, Q1) < e and dF(P2, Q2) < e imply that dF(P,Q) < e? If yes, how can I prove this?
I guess it does not imply, because if there is max length coupling that crosses the boundary of grids, it will be missed in the partitioned computation of Discrete Frechet distance.
The figure
geometry hausdorff-distance
asked Dec 19 '18 at 15:13
user302787user302787
11
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add a comment |
$begingroup$
Given a dataset of Polygonal Lines which are partitioned using equal-sized grid cells. Suppose two Lines P and Q which are segmented at cell boundaries and each of which is converted to two subsequences residing in the corresponding cells. If the distance between sub-sequences in each cell is less than epsilon, can we conclude that the distance of the whole Lines (not segmented ones) is also less than the epsilon.
In mathematical words, suppose the two P and Q lines each one partitioned into 2 subsequences at the point indicated by star and discrete Frechet Distance function dF.
I'm not good in mathematics. In the figure you can see that the distance dF(P1, Q1)=d1 and dF(P2, Q2)=d2, and dF(P, Q)=d2.
Does dF(P1, Q1) < e and dF(P2, Q2) < e imply that dF(P,Q) < e? If yes, how can I prove this?
I guess it does not imply, because if there is max length coupling that crosses the boundary of grids, it will be missed in the partitioned computation of Discrete Frechet distance.
The figure
geometry hausdorff-distance
asked Dec 19 '18 at 15:13
user302787user302787
11
$endgroup$
add a comment |
$begingroup$
Given a dataset of Polygonal Lines which are partitioned using equal-sized grid cells. Suppose two Lines P and Q which are segmented at cell boundaries and each of which is converted to two subsequences residing in the corresponding cells. If the distance between sub-sequences in each cell is less than epsilon, can we conclude that the distance of the whole Lines (not segmented ones) is also less than the epsilon.
In mathematical words, suppose the two P and Q lines each one partitioned into 2 subsequences at the point indicated by star and discrete Frechet Distance function dF.
I'm not good in mathematics. In the figure you can see that the distance dF(P1, Q1)=d1 and dF(P2, Q2)=d2, and dF(P, Q)=d2.
Does dF(P1, Q1) < e and dF(P2, Q2) < e imply that dF(P,Q) < e? If yes, how can I prove this?
I guess it does not imply, because if there is max length coupling that crosses the boundary of grids, it will be missed in the partitioned computation of Discrete Frechet distance.
The figure
geometry hausdorff-distance
asked Dec 19 '18 at 15:13
user302787user302787
11
$endgroup$
Given a dataset of Polygonal Lines which are partitioned using equal-sized grid cells. Suppose two Lines P and Q which are segmented at cell boundaries and each of which is converted to two subsequences residing in the corresponding cells. If the distance between sub-sequences in each cell is less than epsilon, can we conclude that the distance of the whole Lines (not segmented ones) is also less than the epsilon.
In mathematical words, suppose the two P and Q lines each one partitioned into 2 subsequences at the point indicated by star and discrete Frechet Distance function dF.
I'm not good in mathematics. In the figure you can see that the distance dF(P1, Q1)=d1 and dF(P2, Q2)=d2, and dF(P, Q)=d2.
Does dF(P1, Q1) < e and dF(P2, Q2) < e imply that dF(P,Q) < e? If yes, how can I prove this?
I guess it does not imply, because if there is max length coupling that crosses the boundary of grids, it will be missed in the partitioned computation of Discrete Frechet distance.
The figure
geometry hausdorff-distance
geometry hausdorff-distance
asked Dec 19 '18 at 15:13
user302787user302787
11
asked Dec 19 '18 at 15:13
user302787user302787
11
asked Dec 19 '18 at 15:13
user302787user302787
11
asked Dec 19 '18 at 15:13
user302787user302787
11
asked Dec 19 '18 at 15:13
user302787user302787
11
11
add a comment |
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