## Lower Bounds for On-line Graph Coloring (1994)

Citations: | 23 - 6 self |

### BibTeX

@MISC{Halldórsson94lowerbounds,

author = {Magnús M. Halldórsson and Mario Szegedy},

title = {Lower Bounds for On-line Graph Coloring },

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

An algorithm for vertex-coloring graphs is said to be on-line if each vertex is irrevocably assigned a color before later vertices are considered. We show that for every such algorithm, there exists a log n-colorable graph for which the algorithm uses at least 2n = log n colors. This also holds for randomized algorithms, to within a constant factor, against an oblivious adversary. We then show that various means of relaxing the constraints of the on-line model do not reduce these lower bounds. The features include presenting the input in blocks of up to log 2 n vertices, recoloring any fraction of the vertices, presorting vertices by degree, and disclosing the adversary's previous coloring.

### Citations

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Competitive algorithms for on-line problems
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(Show Context)
Citation Context ...for other problems. For the 3-coloring problem, exactly 7 bins are needed, a far cry from the (nevertheless weak) (log2 n) lower bound for the standard model [15]. Similarly, for the k-server problem =-=[12]-=-, we can give a simple algorithm with a performance ratio of 3, for any k 3, compared to the lower bound and conjectured upper bound of k for the standard model. 85 Conclusions. We have presented str... |

133 |
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(Show Context)
Citation Context ... amount of repacking per item is allowed [3]. It is also well known that inputs sorted by non-increasing item size { a relaxation of the \unknown ordering" principle { require signi cantly fewer bins =-=[8]-=-. We consider the above variations in the context of the graph coloring problem, and show them all to yield limited or no improvement. Our lower bound holds to within a constant factor even if the alg... |

62 |
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(Show Context)
Citation Context ...degrees of the vertices. Such strategies have been extensively evaluated both experimentally and analytically in association with the ubiquitous, inherently on-line First-Fit coloring algorithm (e.g. =-=[13]-=-). We can easily circumvent any such attempt by padding the input so that all vertices will be of the same degree. With n=(k01) extra vertices for each of the k colors, each original vertex can then b... |

45 |
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(Show Context)
Citation Context ... the number of bins used to the chromatic number (the minimum number of colors required), ranging over all input graphs. This problem has been much studied, particularly for speci c classes of graphs =-=[10, 4, 7]-=-. Lovasz, Saks and Trotter [11] gave an algorithm for general graphs that obtains a performance ratio of O(n= log 3 n), slightly improving the trivial bound of n. Vishwanathan [15] gave a randomized a... |

43 | On-line and first fit colorings of graphs - Gyárfás, Lehel - 1988 |

37 | An on-line graph coloring algorithm with sublinear performance ratio - Lovász, Saks, et al. - 1989 |

25 |
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(Show Context)
Citation Context ...ses of graphs [10, 4, 7]. Lovasz, Saks and Trotter [11] gave an algorithm for general graphs that obtains a performance ratio of O(n= log 3 n), slightly improving the trivial bound of n. Vishwanathan =-=[15]-=- gave a randomized algorithm which attains a performance ratio of O(n= p log n) against an oblivious adversary. His algorithm was modi ed in [5] to improve the performance ratio to O(n= log n). In thi... |

20 |
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(Show Context)
Citation Context ...rove a 2n= log 2 n lower bound for deterministic on-line graph coloring, which holds to within a constant factor for randomized algorithms. The previous best lower bounds known were (log n) for trees =-=[1, 4, 10]-=-, and (logk n) for k-colorable graphs, where k is xed [15]. 1.3 Variations of the on-line models. On-line computing places strong constraints on the algorithm; we would like to extend our lower bounds... |

12 |
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(Show Context)
Citation Context ...show in this section that, for any randomized on-line coloring algorithm there exists a kcolorable graph on which the algorithm will use at least expected n=k bins, where k = O(log n). By Yao's lemma =-=[16]-=-, it su ces to show that there exists a distribution of k-colorable graphs for which the average number of colors used by any deterministic algorithm is at least n=k. We construct a distribution of k-... |

10 | Online matching with blocked input
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(Show Context)
Citation Context ...refore, in this problem, the e ect of blocked input on the performance guarantee is a threshold function. Similar threshold-like behavior has been shown for the on-line problems of bipartite matching =-=[9]-=-, and coloring inductive graphs [7]. 4.2 Lookahead and bu ering. We can treat two interesting variations of blocked input similarly. An algorithm is on-line with lookahead ` if it bases its answer to ... |

8 |
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(Show Context)
Citation Context ... the number of bins used to the chromatic number (the minimum number of colors required), ranging over all input graphs. This problem has been much studied, particularly for speci c classes of graphs =-=[10, 4, 7]-=-. Lovasz, Saks and Trotter [11] gave an algorithm for general graphs that obtains a performance ratio of O(n= log 3 n), slightly improving the trivial bound of n. Vishwanathan [15] gave a randomized a... |

8 |
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(Show Context)
Citation Context ... single-pass algorithm may have enough memory to keep a limited number of requests in a queue for later processing, which has, for example, been shown to be useful in a server problem with excursions =-=[2]-=-. The \irrevocability" condition can also often be made more exible. Decisions may be reversible as long as the changes are infrequent and localized. In the case of the bin packing problem, fewer bins... |

5 | Parallel and on-line graph coloring
- Halldorsson
- 1992
(Show Context)
Citation Context ...lightly improving the trivial bound of n. Vishwanathan [15] gave a randomized algorithm which attains a performance ratio of O(n= p log n) against an oblivious adversary. His algorithm was modi ed in =-=[5]-=- to improve the performance ratio to O(n= log n). In this paper, we prove a 2n= log 2 n lower bound for deterministic on-line graph coloring, which holds to within a constant factor for randomized alg... |

3 |
New algorithms for on–line bin packing
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(Show Context)
Citation Context ...ble. Decisions may be reversible as long as the changes are infrequent and localized. In the case of the bin packing problem, fewer bins are needed if constant amount of repacking per item is allowed =-=[3]-=-. It is also well known that inputs sorted by non-increasing item size { a relaxation of the \unknown ordering" principle { require signi cantly fewer bins [8]. We consider the above variations in the... |

2 | Lectures on the Probabilistic Method, volume 52 - Ten - 1987 |

2 |
On-line and rst- t colorings of graphs
- Gyárfás, Lehel
- 1988
(Show Context)
Citation Context ... the number of bins used to the chromatic number (the minimum number of colors required), ranging over all input graphs. This problem has been much studied, particularly for speci c classes of graphs =-=[10, 4, 7]-=-. Lovasz, Saks and Trotter [11] gave an algorithm for general graphs that obtains a performance ratio of O(n= log 3 n), slightly improving the trivial bound of n. Vishwanathan [15] gave a randomized a... |