# Linear-size circuits for stable $0,1 < 2$ sorting?

**Problem**Can -size circuits compute the function on defined inductively by , , , and ?

This function moves all 2s in flush-right, leaving the sequence of 0s and 1s the same, and represents stable topological sort of the partial order . It is linear-time computable in any model that supports the operations of a double-ended queue in time, including multi-tape Turing machines, but is to me the "easiest" function for which I do not know linear-size circuits. By contrast sorting , called the "Dutch National Flag Problem", has -size circuits by counting. It suffices to compute when is a power of and exactly half the entries are . For this and more see my Computational Complexity blog item, PDF file here.

## Bibliography

* indicates original appearance(s) of problem.