Trimming and gluing Gray codes

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10384203" target="_blank" >RIV/00216208:11320/18:10384203 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.tcs.2017.12.003" target="_blank" >https://doi.org/10.1016/j.tcs.2017.12.003</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.tcs.2017.12.003" target="_blank" >10.1016/j.tcs.2017.12.003</a>

Result language

angličtina

Original language name

Trimming and gluing Gray codes

Original language description

We consider the algorithmic problem of generating each subset of [n] := {1, 2, ..., n} whose size is in some interval [k,1], 0 &lt;= k &lt;= l &lt; n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k = 0 and l = n this is the classical problem of generating all 2(n) subsets of [n] by element additions/removals, and for k = l this is the classical problem of generating all ((n)(k)) subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant O(1) time per generated set and O(n) space. Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Qt, induced by all levels [k, I]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q(2k+1) induced by all levels [k - c, k + 1 + c], c epsilon {0, 1 ,..., k} has a Hamilton cycle. We also prove an approximate version of this generalized conjecture, showing that this graph has a cycle that visits a (1 - o(1))-fraction of all vertices.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

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/GA14-10799S" target="_blank" >GA14-10799S: Hybercubic, graph and hypergraph structures</a><br>

Continuities

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

Publication year

2018

Confidentiality

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

Name of the periodical

Theoretical Computer Science

ISSN

0304-3975

e-ISSN

—

Volume of the periodical

714

Issue of the periodical within the volume

1. března

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

22

Pages from-to

74-95

UT code for WoS article

000424959300006

EID of the result in the Scopus database

2-s2.0-85037594257