## A parallel algorithmic version of the local lemma (1991)

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### BibTeX

@INPROCEEDINGS{Alon91aparallel,

author = {Noga Alon},

title = {A parallel algorithmic version of the local lemma},

booktitle = {},

year = {1991},

pages = {367--378},

publisher = {IEEE Computer Society Press}

}

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### Abstract

The Lovász Local Lemma is a tool that enables one to show that certain events hold with pos-itive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.

### Citations

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(Show Context)
Citation Context ...ir distance in G is either 2 or 3. Every 2, 3-tree on a set T of vertices of G must contain a tree on T in this new graph. The new graph has maximum degree smaller than D = d3 . It is well known (see =-=[17]-=-) that an � infinite D-regular rooted tree contains precisely rooted subtrees of size u, and this 1 �Du (D−1)u+1 u easily implies that the number of trees of size u containing one specific given verte... |

270 | Simple construction of almost k-wise independent random variables. Random Struct
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Citation Context ...e that the recoloring step is deterministic even in the version described above, so the only problem is the derandomization of the First Pass. This can be done by applying the techniques from [19] or =-=[6]-=-. (It is also possible to apply the methods of [18] and [13], but this will not supply NC 1 -algorithms, although it would yield algorithms with a smaller number of processors). The basic idea is that... |

256 | Small-bias probability spaces: Efficient constructions and applications
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Citation Context ...s observe that the recoloring step is deterministic even in the version described above, so the only problem is the derandomization of the First Pass. This can be done by applying the techniques from =-=[19]-=- or [6]. (It is also possible to apply the methods of [18] and [13], but this will not supply NC 1 -algorithms, although it would yield algorithms with a smaller number of processors). The basic idea ... |

226 |
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(Show Context)
Citation Context ...nt Ai is mutually independent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in =-=[14]-=-, [2], [4], [8], [7], [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found ... |

216 |
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(Show Context)
Citation Context ...H as follows. For each i ∈ GF (q), the color of the j-th point in the i-th coloring is blue if χ(i − j) = 1 and is red if χ(i − j) = −1. Using the results in [6] (based on the ideas in [19] (see also =-=[3]-=-)), it can be shown that at least one of the q colorings defined above will produce a successful First Pass. All these (deterministically defined) colorings can be checked in parallel, completing the ... |

102 |
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Citation Context ...). We omit the detailed computation. The randomized algorithm above is trivially parallelizable and can be implemented on a standard EREW -PRAM in time O(log N) using N O(1) parallel processors. (See =-=[16]-=- for the basic definitions of an EREW -PRAM and the complexity classes NC and NC 1 .) Moreover, the algorithm can be derandomized maintaining the running time (with some increase in the number of proc... |

79 | Restricted colorings of graphs
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- 1993
(Show Context)
Citation Context ...ent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], [2], [4], [8], [7], =-=[10]-=-, [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such a method that works f... |

74 |
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(Show Context)
Citation Context ...und in [14], [2], [4], [8], [7], [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck =-=[12]-=- found such a method that works for many of these examples with a little loss in the constants. Beck demonstrated his method by considering the problem of hypergraph 2-coloring. A hypergraph H = (V, E... |

51 | The probabilistic method yields deterministic parallel algorithms
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Citation Context ...n the version described above, so the only problem is the derandomization of the First Pass. This can be done by applying the techniques from [19] or [6]. (It is also possible to apply the methods of =-=[18]-=- and [13], but this will not supply NC 1 -algorithms, although it would yield algorithms with a smaller number of processors). The basic idea is that for every constant c there is a constant b = b(c) ... |

50 | The linear arboricity of graphs
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- 1988
(Show Context)
Citation Context ...ependent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], [2], [4], [8], =-=[7]-=-, [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such a method that w... |

47 | A Survey of Parallel Algorithms for Shared Memory Machines - Karp, Ramachandran - 1988 |

22 | Single round simulation on radio networks
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(Show Context)
Citation Context ...um number of steps in which it is possible to complete a transmission along every edge of D. This parameter is of central interest in the study of the task referred to as a Single Round Simulation in =-=[5]-=-. For simplicity let us restrict our attention to symmetric graphs, i.e., (x, y) is an edge of D iff (y, x) is an edge. In [4], [5] it is shown that there exist two positive constants c1 and c2 such t... |

19 |
On the complexity of radio communication
- Alon, Bar-Noy, et al.
- 1989
(Show Context)
Citation Context ...tually independent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], [2], =-=[4]-=-, [8], [7], [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such a met... |

18 |
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- Thomassen
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(Show Context)
Citation Context ...iding whether a given directed graph contains a (simple) even (directed) cycle arises in various contexts, including the study of the problem of sign-solvability of a system of linear equations, (see =-=[21]-=-). There is no known polynomial time algorithm for answering this decision problem, but it is also not known to be NP-complete. (It is known that the similar decison problem of deciding whether a give... |

13 |
Simulating log c nwise independence
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(Show Context)
Citation Context ...sion described above, so the only problem is the derandomization of the First Pass. This can be done by applying the techniques from [19] or [6]. (It is also possible to apply the methods of [18] and =-=[13]-=-, but this will not supply NC 1 -algorithms, although it would yield algorithms with a smaller number of processors). The basic idea is that for every constant c there is a constant b = b(c) such that... |

9 |
Path factors of a graph
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(Show Context)
Citation Context ...l edges of G. For an integer ∆ ≥ 1 define st(∆) = Max{st(G)}, where the maximum is taken over all simple graphs with maximum degree ∆. The star arboricity of graphs was introduced by Akiyama and Kano =-=[1]-=- and has been studied in various papers. The asymptotic behaviour of st(∆) is determined in [11], improving a previous estimate from [2]. In particular it is known that for every γ > 0 there is a ∆ = ... |

9 |
Cycles of length 0 modulo k in directed graphs
- Alon, Linial
(Show Context)
Citation Context ...ut at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], [2], [4], [8], [7], [10], [11], =-=[9]-=-. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such a method that works for many of ... |

8 |
The strong chromatic number of a graph
- Alon
- 1992
(Show Context)
Citation Context ...y independent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], [2], [4], =-=[8]-=-, [7], [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such a method t... |

7 |
Any 7-regular digraph contains an even cycle
- Friedland
- 1989
(Show Context)
Citation Context ...ther a given directed graph contains an even cycle through a given edge is NP-complete). There are several results that supply a sufficient condition for a digraph to contain such a cycle. Friedland (=-=[15]-=-) showed that for every r ≥ 7, any r-regular digraph contains an even cycle. His proof supplies no polynomial time agorithm for finding an even cycle in such a digraph. In [9] it is shown that one can... |

6 |
The star arboricity of graphs, Discrete Mathematics 75
- Algor, Alon
- 1989
(Show Context)
Citation Context ...is mutually independent of all but at most b of the other events Aj. If ep(b + 1) < 1 then with positive probability none of the events Ai holds. Many applications of this lemma can be found in [14], =-=[2]-=-, [4], [8], [7], [10], [11], [9]. For several years there has been no known method of converting the proofs of any of these examples into an efficient algorithm. Very recently J. Beck [12] found such ... |

3 | J.E.Hopcroft and J.Wyllie, The directed subgraph homeomorphism problem - Fortune |