## 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, August 18, 2015, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Andrew Winslow (Université Libre de Bruxelles)

## Tiling Isohedrally with a Polyomino

Tetris pieces are a special case of a general class of digital shapes called polyominoes. A covering of a region by disjoint copies of a polyomino is called a tiling of the region. Questions about which regions admit tilings by which polyominoes have been studied for many decades, popularized in the 1950s and 60s by Martin Gardner and Solomon Golomb. In this work, we consider tilings of the plane restricted to the special class of isohedral tilings. For any pair of tiles in an isohedral tiling, there exists a rigid mapping of the plane to itself that sends every tile to another, including one tile of the pair to the other. Despite such regularity, the fastest previous algorithm for determining if a polyomino admits an isohedral tiling is that of Keating and Vince (Discrete & Computational Geometry, 1999) running in $O(n^{18})$ time, where n is the number of edges of the polyomino. Here we give a $O(n\log^2(n))$-time algorithm using a number of old and new structural results on the palindromic, anti-palindromic, and repeating substrings of a string. This is joint work with Stefan Langerman.

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