Editing Open Problems:48
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'''Question:''' Is there a sketching scheme for $P_c$ where $c=O(1)$ and $s=O(1)$? | '''Question:''' Is there a sketching scheme for $P_c$ where $c=O(1)$ and $s=O(1)$? | ||
β | '''Background:''' If the shift metric is replaced by Hamming metric, one can achieve $s=O(1)$ using random sampling {{cite|KushilevitzOR-00}}. The actual problem can be solved for $c=O(\log^2 n)$ and $s=O(1)$ {{cite|AndoniIK-08}}. The algorithm proceeds by embedding the shift metric into Hamming metrics, and it is known that this step must induce $\Omega(\log n)$ distortion {{cite|KhotN-06 | + | '''Background:''' If the shift metric is replaced by Hamming metric, one can achieve $s=O(1)$ using random sampling {{cite|KushilevitzOR-00}}. The actual problem can be solved for $c=O(\log^2 n)$ and $s=O(1)$ {{cite|AndoniIK-08}}. The algorithm proceeds by embedding the shift metric into Hamming metrics, and it is known that this step must induce $\Omega(\log n)$ distortion {{cite|KhotN-06}}. |