Combinatorial generation via permutation languages

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10419955" target="_blank" >RIV/00216208:11320/20:10419955 - isvavai.cz</a>

Result on the web

<a href="https://dl.acm.org/doi/abs/10.5555/3381089.3381163" target="_blank" >https://dl.acm.org/doi/abs/10.5555/3381089.3381163</a>

DOI - Digital Object Identifier

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Result language

angličtina

Original language name

Combinatorial generation via permutation languages

Original language description

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group S(n. )Recently, Pilaud and Santos realized all those lattice congruences as (n - 1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

<a href="/en/project/GA19-08554S" target="_blank" >GA19-08554S: Structures and algorithms in highly symmetric graphs</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2020)

ISBN

978-1-61197-599-4

ISSN

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e-ISSN

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Number of pages

12

Pages from-to

1214-1225

Publisher name

Association for Computing Machinery

Place of publication

New York

Event location

Salt Lake City

Event date

Jan 5, 2020

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

000554408101018