Unioning Sketches with Different values of (k)

One of the benefits of the Theta Sketch algorithms is that they support the union of sketches that have been created with different values of \(k\) or Nominal Entries. More specifically, it is possible to create a Union operation with a \(k_U\) and then update the union with sketches created with different \(k_i\) that can be either larger or smaller than \(k_U\).

The interesting case, of course, is where \(k_U > k_i\), and, it turns out that it is possible that the Relative Standard Error, \(\color{black}{RSE = 1/{\sqrt{k}}}\), of the resulting Union, can be improved, i.e., \(RSE_U < min(RSE_i)\).

This is in contrast to the HLL algorithm, where unioning is only possible with the same \(k\) or with values that are powers-of-2 smaller. In this case the \(RSE_U = min(RSE_i)\).