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

Importance: Medium ✭✭
Author(s): Regan, Kenneth
Subject: Theoretical Computer Science
» Complexity
Keywords: Circuits
sorting
Recomm. for undergrads: yes
Posted by: KWRegan
on: July 19th, 2007
Problem   Can $ O(n) $-size circuits compute the function $ f $ on $ \{0,1,2\}^* $ defined inductively by $ f(\lambda) = \lambda $, $ f(0x) = 0f(x) $, $ f(1x) = 1f(x) $, and $ f(2x) = f(x)2 $?

This function moves all 2s in $ x $ flush-right, leaving the sequence of 0s and 1s the same, and represents stable topological sort of the partial order $ 0,1 < 2 $. It is linear-time computable in any model that supports the operations of a double-ended queue in $ O(1) $ 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 $ 0 < 1 < 2 $, called the "Dutch National Flag Problem", has $ O(n) $-size circuits by counting. It suffices to compute $ f(x) $ when $ |x| $ is a power of $ 2 $ and exactly half the entries are $ 2 $. For this and more see my Computational Complexity blog item, PDF file here.

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