## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, March 21, 2017, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Torsten Mütze (TU Berlin)

## Trimming and gluing Gray codes

We consider the algorithmic problem of generating each subset of $[n]:=\{1,2,...,n\}$ whose size is in some interval $[a,b]$, $0\leq a\leq b\leq n$, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For $a=0$ and $b=n$ this is the classical problem of generating all $2^n$ subsets of $[n]$ by element additions/removals, and for $a=b$ this is the classical problem of generating all $\binom{n}{a}$ subsets of $[n]$ by element exchanges. In graph theoretical terms, we ask for the existence of (almost) Hamilton cycles in the subgraph of the $n$-dimensional cube $Q_n$ induced by all levels $[a,b]$. We prove the existence of such cycles for a large range of values $n$, $a$, and $b$, and provide corresponding optimal generation algorithms, improving upon and generalizing several previous results. This is joint work with Petr Gregor (Charles University).

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