An algorithmic version of the blow-up lemma

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 Ž w Ž .x lemma Komlos, Sarkozy, and Szemeredi Combinatorica, 17, 109]123 1997 . This lemma ́ ́ ̈ ́ helps to find bounded degree spanning subgraphs in «-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 Ž Ž .. Ž . Ž 2.376. be found in time O nM n , where M n sO n 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. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 297]312, 1998 Correspondence to: Gabor N. Sarkozy *Part of this paper was written while Sarkozy was visiting MSRI Berkeley, as part of the Combinatorics Program. Research at MSRI is supported in part by NSF grant DMS-9022140. Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030297-16 297 ́ ́ KOMLOS, SARKOZY, AND SZEMEREDI 298