### Abstract

Recently we developed a new method in graph theory based on the regularity lemma. The method is applied to find certain spanning subgraphs in dense graphs. The other main general tool of the method, besides the regularity lemma, is the so-called blow-up lemma (Komlós, Sárközy, and Szemerédi [Combinatorica, 17, 109-123 (1997)]. This lemma helps to find bounded degree spanning subgraphs in ε-regular graphs. Our original proof of the lemma is not algorithmic, it applies probabilistic methods. In this paper we provide an algorithmic version of the blow-up lemma. The desired subgraph, for an n-vertex graph, can be found in time O(nM(n)), where M(n) = O(n^{2.376}) is the time needed to multiply two n by n matrices with 0, 1 entires over the integers. We show that the algorithm can be parallelized and implemented in NC^{5}.

Original language | English |
---|---|

Pages (from-to) | 297-312 |

Number of pages | 16 |

Journal | Random Structures and Algorithms |

Volume | 12 |

Issue number | 3 |

Publication status | Published - May 1998 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*12*(3), 297-312.

**An Algorithmic Version of the Blow-Up Lemma.** / Komlós, János; Sarkozy, Gabor N.; Szemerédi, E.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 12, no. 3, pp. 297-312.

}

TY - JOUR

T1 - An Algorithmic Version of the Blow-Up Lemma

AU - Komlós, János

AU - Sarkozy, Gabor N.

AU - Szemerédi, E.

PY - 1998/5

Y1 - 1998/5

N2 - Recently we developed a new method in graph theory based on the regularity lemma. The method is applied to find certain spanning subgraphs in dense graphs. The other main general tool of the method, besides the regularity lemma, is the so-called blow-up lemma (Komlós, Sárközy, and Szemerédi [Combinatorica, 17, 109-123 (1997)]. This lemma helps to find bounded degree spanning subgraphs in ε-regular graphs. Our original proof of the lemma is not algorithmic, it applies probabilistic methods. In this paper we provide an algorithmic version of the blow-up lemma. The desired subgraph, for an n-vertex graph, can be found in time O(nM(n)), where M(n) = O(n2.376) is the time needed to multiply two n by n matrices with 0, 1 entires over the integers. We show that the algorithm can be parallelized and implemented in NC5.

AB - Recently we developed a new method in graph theory based on the regularity lemma. The method is applied to find certain spanning subgraphs in dense graphs. The other main general tool of the method, besides the regularity lemma, is the so-called blow-up lemma (Komlós, Sárközy, and Szemerédi [Combinatorica, 17, 109-123 (1997)]. This lemma helps to find bounded degree spanning subgraphs in ε-regular graphs. Our original proof of the lemma is not algorithmic, it applies probabilistic methods. In this paper we provide an algorithmic version of the blow-up lemma. The desired subgraph, for an n-vertex graph, can be found in time O(nM(n)), where M(n) = O(n2.376) is the time needed to multiply two n by n matrices with 0, 1 entires over the integers. We show that the algorithm can be parallelized and implemented in NC5.

UR - http://www.scopus.com/inward/record.url?scp=0032373312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032373312&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032373312

VL - 12

SP - 297

EP - 312

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -