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|title=Matchings in the Turnstile Model
 
|source=bertinoro14
 
|source=bertinoro14
 
|who=Andrew McGregor
 
|who=Andrew McGregor
 
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Consider an unweighted graph on $n$ nodes defined by a stream of edge insertions and deletions. Is it possible to approximate the size of the maximum cardinality matching up to constant factor given a single pass and $o(n^2)$ space? Recall that a factor 2 approximation is easy in $O(n \log n)$ space if there are no edge deletions.
 
  
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== Updates ==
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Suppose we are in a turnstile model for graph streaming: this means that the stream consists of a sequence of edge insertions and deletions (with no edge being deleted before it's inserted). In this setting, can we design an algorithm for computing a matching that takes space $o(n^2)$ and gets a constant approximation (or better ) ?
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The question is fully settled when the goal is to output the edges of an approximate maximum matching: to obtain an $\alpha$-approximation to maximum matching in dynamic streams, $\Omega(n^2/\alpha^3)$ space is necessary {{cite|AssadiKLY-16}} and $\widetilde{O}(n^2/\alpha^3)$ space is sufficient {{cite|AssadiKLY-16|ChitnisCEHMMV-16}}. When the goal is only to estimate the value of maximum matching (as opposed to finding the edges), $\Omega(n/\alpha^2)$ space is necessary and $\widetilde{O}(n^2/\alpha^4)$ space is sufficient {{cite|AssadiKL-17}}.
 

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