We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let \alpha be the cardinality of the largest independent set. Our goal is to estimate \alpha in a small amount of space, given that the input is received as a one-pass turnstile stream. We also consider a generalization of this problem by assigning weights to each object and estimating \beta, the largest value of a weighted independent set. We  provide the first algorithms for estimating \alpha and \beta in turnstile streams . For unit-length intervals, we obtain a (2+\epsilon)-approximation to \alpha and \beta in poly %DIFDELCMD < \left(%%% \frac{\log(n){\epsilon}}%DIFDELCMD < \right) %%% {\epsilon}) } space. We also show a matching lower bound. For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega ] \left( \frac{n^{1/c}{2^c}}%DIFDELCMD < \right) %%% space. To this end, we introduce a new communication problem and lower bound its information complexity. In light of the lower bound we}{\pi}}\right) provide algorithms for estimating \alpha for arbitrary-length intervals under a bounded intersection assumption . We also study the parameterized space complexity of estimating \alpha and \beta, where the parameter is the ratio of maximum to minimum interval length . For unit-radius disks, we obtain a%DIFDELCMD < \left(%%% \frac{8\sqrt{3}{\pi}}%DIFDELCMD < \right)%%% -approximation to \alpha and \beta in\textrm{poly}%DIFAUXCMD %DIFDELCMD < \left(%%% \frac{\log(n){\epsilon}}%DIFDELCMD < \right) %%% space, which is closely related to the hexagonal circle packing constant .
We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let \alpha be the cardinality of the largest independent set. Our goal is to estimate \alpha in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating \beta, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating \alpha and \beta . For unit-length intervals, we obtain a (2+\epsilon)-approximation to \alpha and \beta in poly %DIFDELCMD < \left(%%% {\epsilon}}%DIFDELCMD < \right) %%% (\frac{\log(n){\epsilon}) } space. We also show a matching lower bound. Combined with the 3/2-approximation for insertion-only streams by Cabello and Perez-Lanterno CP15], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a \left( }{2^c}}%DIFDELCMD < \right) %%% \frac{8\sqrt{3}{\pi}}\right)-approximation to \alpha and \beta in poly(\log(n), \epsilon^{-1 provide algorithms for estimating \alpha for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating \alpha and \beta, where the parameter is the ratio of maximum to minimum interval length %DIFDELCMD < \left(%%% }{\pi}}%DIFDELCMD < \right)%%% \textrm{}%DIFAUXCMD %DIFDELCMD < \left(%%% {\epsilon}}%DIFDELCMD < \right) %%% .
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initialize the study of this problem in the turnstile streaming model (insertions and deletions) and
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For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega
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, our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a
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space. To this end, we introduce a new communication problem and lower bound its information complexity. In light of the lower bound we
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