## Two-processor scheduling with start-times and deadlines (1977)

Venue: | SIAM Journal on Computing |

Citations: | 66 - 0 self |

### BibTeX

@ARTICLE{Garey77two-processorscheduling,

author = {M. R. Garey and D. S. Johnson},

title = {Two-processor scheduling with start-times and deadlines},

journal = {SIAM Journal on Computing},

year = {1977},

volume = {6},

pages = {416--426}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Given a set 3- = {T1, T2, , T,} of tasks, each T/having execution time 1, an integer start-time si->0 and adeadlinedi> 0, alongwithprecedence constraintsamongthe tasks,weexamine the problem of determining whether there exists a schedule on two identical processors that executes each task in the time intervalbetween its start-timeand deadline.We present an O(n3) algorithm that constructs such a schedule whenever one exists. The algorithm may also be used in a binary search mode to find the shortest such schedule or to find a schedule that minimizesmaximum "tardiness".A number of natural extensions of this problem are seen to be NP-complete and hence probably intractable. Key words, multiprocessing systems, scheduling algorithms, NP-complete problems 1. Introduction. Since publication of the book Theory ofScheduling [4] by Conway, Maxwell, andMiller in 1967, considerableprogresshasbeenmade inthe mathematical analysis of abstract multiprocessing systems. One combinatorial model which is central to much of this work consists of a numberm of identical, independent processors, a finite set- {T1, T2, Tn} of tasks to be executed,

### Citations

2531 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1975
(Show Context)
Citation Context ...of minimizing the number of tardy tasks, even if we have only one processor and all start-times (or all deadlines) are the same, is NP-complete [9] and hence probably computationally intractable (see =-=[1]-=-, [11], [12] for comprehensive treatments of "NP-completeness"). A number of other - simple generalizations of our scheduling problem are also NP-complete. First let us consider relaxing the constrain... |

196 |
NP-complete scheduling problems
- Ullman
- 1975
(Show Context)
Citation Context ...nsystem(maximum finishing time). Unfortunately these goals canbe quite difficult to achieve and, in fact, most classes of scheduling problems appear to be computationally intractable [8], [10], [12], =-=[15]-=-.A notable exception to this state of affairs is the case of minimizingmaximum finishing timewhenm 2 and each 7" 1, 1 =< =< n. Efficient scheduling algorithms for this case have been described in [3],... |

114 |
Complexity results for multiprocessor scheduling under resource constraints
- Garey, Johnson
- 1975
(Show Context)
Citation Context ...he maximum-time-insystem(maximum finishing time). Unfortunately these goals canbe quite difficult to achieve and, in fact, most classes of scheduling problems appear to be computationally intractable =-=[8]-=-, [10], [12], [15].A notable exception to this state of affairs is the case of minimizingmaximum finishing timewhenm 2 and each 7" 1, 1 =< =< n. Efficient scheduling algorithms for this case have been... |

84 |
Optimal scheduling for two-processor systems
- Coffman, Graham
- 1972
(Show Context)
Citation Context ...[15].A notable exception to this state of affairs is the case of minimizingmaximum finishing timewhenm 2 and each 7" 1, 1 =< =< n. Efficient scheduling algorithms for this case have been described in =-=[3]-=-, [6], [7], [13]. These results have been extended in [9], which presents an efficientscheduling algorithm for themore complicated version of thisproblem in which each taskT/-also has a deadline di>0 ... |

32 |
Boolean matrix multiplication and transitive closure
- Fischer, Meyer
- 1971
(Show Context)
Citation Context ...n). Nextwecompute the transitive closure ofthe partialorderso that, in constant time, we can determine whether or not T precedes T., 1-< i,/"-< n. This canbe accomplishedwith O(n2.81) operationsusing =-=[5]-=- or O(n3) operationsusing any of [2], [14], [16]. Thenwe perform the preliminary deadline modifications to ensure that di -< djwheneverTprecedes T.. Thiscanbedone inO(n2) operations by working "backwa... |

30 |
Optimal sequencing of two equivalent processors
- Fujii, Kasami, et al.
- 1969
(Show Context)
Citation Context ...A notable exception to this state of affairs is the case of minimizingmaximum finishing timewhenm 2 and each 7" 1, 1 =< =< n. Efficient scheduling algorithms for this case have been described in [3], =-=[6]-=-, [7], [13]. These results have been extended in [9], which presents an efficientscheduling algorithm for themore complicated version of thisproblem in which each taskT/-also has a deadline di>0 bywhi... |

19 |
A transitive closure algorithm
- Purdom
- 1970
(Show Context)
Citation Context ...ofthe partialorderso that, in constant time, we can determine whether or not T precedes T., 1-< i,/"-< n. This canbe accomplishedwith O(n2.81) operationsusing [5] or O(n3) operationsusing any of [2], =-=[14]-=-, [16]. Thenwe perform the preliminary deadline modifications to ensure that di -< djwheneverTprecedes T.. Thiscanbedone inO(n2) operations by working "backwards" in the partial order, examining a tas... |

1 |
On economical construction ofthe transitiveclosureofan orientedgraph, Dokl
- ARLAZAROV, DINIC, et al.
- 1970
(Show Context)
Citation Context ...sure ofthe partialorderso that, in constant time, we can determine whether or not T precedes T., 1-< i,/"-< n. This canbe accomplishedwith O(n2.81) operationsusing [5] or O(n3) operationsusing any of =-=[2]-=-, [14], [16]. Thenwe perform the preliminary deadline modifications to ensure that di -< djwheneverTprecedes T.. Thiscanbedone inO(n2) operations by working "backwards" in the partial order, examining... |

1 | On the complexity ofcombinatorialproblems, Networks - MURAOKA, Parallelism - 1975 |