### Theta Sketch Merging (Union) Algorithm Notes

There is a fairly simple argument for the correctness of our space-limited multiway merging algorithm.

1. Let R be the result of the theta union algorithm as described in the Theta Sketch Framework paper. It is the set of all hashes less than the minimum of the thetas of all of the input sketches. Recall that there is no size limit on this “full-size” answer R.

2. We will spell out the official definition of “theta union with cutback”, which establishes the 1-Goodness of its implicit TCF, and hence its unbiasedness.

3. We will prove that the cut-back answer according to this official definition can be computed from the X smallest elements of the full-sized result R, for some value of X that doesn’t need to be spelled out in this high-level overview of the argument.

4. We assume the existence of a gadget G that keeps at least the X smallest elements of any set that you feed into it.

5. We point out that if A and B are any two sets such that each element of B is bigger than any element of A, and if (A U B) is fed into G, then G’s output, which by definition contains at least the X smallest elements of (A U B), also contains at least the X smallest elements of A.

6. Finally, we point out that our main merging loop feeds all elements of R into G, and also some other elements, each bigger than any element of R. [These larger elements are inserted because for a while we only have an upper bound on the final value of theta.] By (5), the output of G contains at least the X smallest elements of R. By (3), we are able to construct the answer conforming to (2).

Note: to avoid extreme verbosity in the above, the phrase “the X smallest elements of S” is used as an abbreviation for the more accurate phrase “the X smallest elements of S if |S| >= X, otherwise all of S”.