{. Soit, [l n , r n [} l'ensemble des intervalles déjà ordonnancés 5 sur la machine j tels que l < r 1

B. Remarquealgorithme and . La-suivante, Pour trouver s'il existe ou non une machine j sur laquelle il existe un intervalle pouvant accueillir la nouvelle tâche soumise (soit parce que la machine est libre, soit parce que la condition de remplacement est satisfaite, voir cond j (?)), il n'est pas nécessaire de tester tous les intervalles possibles (ce qui ne serait de toute façon pas possible puisqu'il en existe un nombre infini indénombrable, notre modèle temporel n'´ etant pas discrétisé). L'algorithme BI choisit donc un sous-ensemble de taille polynomial parmi tous les intervallesàintervallesà tester pour voir si un intervalle ? satisfaisant cond j (?) existe. Chaque intervalle effectivement testé par BI est alors le représentant d'une zone (bornée) pouvant contenir une infinité d'intervalleséquivalentsintervalleséquivalents du point de vue de la condition cond j (?) Le théorème suivant prouve la correction de l'algorithme BI

. Preuve, Par définition, si l'algorithme BI trouve un intervalle [d, d + p[ ? [l, r[, celui-ci est de longueur p et satisfait bien

. G. Présentation-de-l-'algorithme-lr-k.-l-'g, I. Richard, and L. Scwiebert, algorithme LR k (pour Left Right sur k ? 3 machines) est une adaptation de l'algorithme Distributed multicast tree generation with dynamic group membership, Computer communications, pp.26-1105, 2003.

R. Adler and Y. Azar, Beating the logarithmic lower bound : Randomized preemptive disjoint paths and call control algorithms, Journal of Scheduling, vol.6, issue.2, pp.113-129, 2003.
DOI : 10.1023/A:1022933824889

N. Alon and Y. Azar, On-line steiner trees in the euclidean plane, Symposium on Computational Geometry, pp.337-343, 1992.

C. Ambuhl and M. Mastrolilli, On-line scheduling to minimize max flow time: an optimal preemptive algorithm, Operations Research Letters, vol.33, issue.6, pp.597-602, 2005.
DOI : 10.1016/j.orl.2004.10.006

E. Angelelli, M. G. Speranza, and Z. Tuza, Semi-On-line Scheduling on Two Parallel Processors with an Upper Bound on the Items, Algorithmica, vol.29, issue.4, pp.37-243, 2003.
DOI : 10.1007/s00453-003-1037-2

E. Arkin and B. Silverberg, Scheduling jobs with fixed start and end times, Discrete Applied Mathematics, vol.18, issue.1, pp.1-8, 1987.
DOI : 10.1016/0166-218X(87)90037-0

B. Awerbuch, Y. Azar, and Y. Bartal, On-line generalized Steiner problem, ACM- SIAM Symposium on Discrete Algorithms (SODA), pp.68-74, 1996.
DOI : 10.1016/j.tcs.2004.05.021
URL : http://doi.org/10.1016/j.tcs.2004.05.021

F. Baille, Algorithmes d'approximation pour desprobì emes d'ordonnancement bicritères : applicationàapplication`applicationà unprobì eme d, Manuscrit de thèse, 2005.

F. Baille, E. Bampis, and C. Laforest, A note on bicriteria schedules with optimal approximation ratios, Parallel Processing Letters, pp.315-323, 2004.

F. Baille, E. Bampis, C. Laforest, and N. Thibault, Algorithmes d'ordonnancements bicritères en-lignes (résumé), in AlgoTel, pp.71-74, 2005.

A. Bar-noy, R. Canetti, S. Kutten, Y. Mansour, and B. Schieber, Bandwidth Allocation with Preemption, SIAM Journal on Computing, vol.28, issue.5, pp.1806-1828, 1999.
DOI : 10.1137/S0097539797321237
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.8834

A. Bar-noy, S. Guha, J. Naor, and B. Schieber, Approximating the Throughput of Multiple Machines in Real-Time Scheduling, SIAM Journal on Computing, vol.31, issue.2, pp.31-331, 2001.
DOI : 10.1137/S0097539799354138

B. Biì-o, M. Flammini, and L. Moscardelli, Pareto approximations for the bicriteria scheduling problem, 18th International Parallel and Distributed Processing Symposium (IPDPS), 2004.

A. Borodin and R. El-yaniv, Online computation and competitive analysis, 1998.

M. C. Carlisle and E. Lloyd, On the k-coloring of intervals, pp.225-235, 1995.

T. Erlebach and F. C. Spieksma, Interval selection: Applications, algorithms, and lower bounds, Journal of Algorithms, vol.46, issue.1, pp.27-53, 2003.
DOI : 10.1016/S0196-6774(02)00291-2

U. Faigle, W. Kern, and G. Turan, On the performance of on-line algorithms for particular problems, Acta Cybernetica, vol.9, pp.107-119, 1989.

U. Faigle and W. Nawijn, Note on scheduling intervals on-line, Discrete Applied Mathematics, vol.58, issue.1, pp.13-17, 1994.
DOI : 10.1016/0166-218X(95)00112-5
URL : http://doi.org/10.1016/0166-218x(95)00112-5

A. Feldmann, M. Y. Kao, J. Sgall, and S. H. Teng, Optimal online scheduling of parallel jobs with dependencies, Proceedings of the twenty-fifth annual ACM symposium on Theory of computing , STOC '93, pp.393-411, 1998.
DOI : 10.1145/167088.167254

A. Fiat and G. J. Woeginger, Online algorithmes : The state of the art, LNCS no. 1442, 1998.

S. P. Fung, C. F. , and H. Shen, ONLINE SCHEDULING OF UNIT JOBS WITH BOUNDED IMPORTANCE RATIO, International Journal of Foundations of Computer Science, vol.16, issue.03, pp.581-598, 2005.
DOI : 10.1142/S0129054105003170

J. Garay, I. S. Gopal, S. Kutten, Y. Mansour, and M. Yung, Efficient On-Line Call Control Algorithms, Journal of Algorithms, vol.23, issue.1, pp.180-194, 1997.
DOI : 10.1006/jagm.1996.0821
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.2760

J. Garay, J. Naor, B. Yener, and P. Zhao, On-line admission control and packet scheduling with interleaving, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, pp.94-103, 2002.
DOI : 10.1109/INFCOM.2002.1019250

M. Garey and D. Johnson, Computers and intractability, in Freeman and compagny, 1979.

A. Goel and K. Munagala, Extending Greedy Multicast Routing to Delay Sensitive Applications, Algorithmica, vol.33, issue.3, pp.33-335, 2002.
DOI : 10.1007/s00453-001-0122-7
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.1562

S. A. Goldman, J. Parwatikar, and S. Suri, On-line scheduling with hard deadlines, 5th Workshop on Algorithms and Data Structures (WADS), pp.258-271, 1997.
DOI : 10.1007/3-540-63307-3_65

M. H. Goldwasser, Patience is a virtue : the effect of slack on competitiveness for admission control, 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pp.396-405, 1999.

M. H. Goldwasser and B. Kerbikov, Admission control with immediate notification, Journal of Scheduling, vol.6, issue.3, pp.269-285, 2003.
DOI : 10.1023/A:1022956425198

R. L. Graham, Bounds for Certain Multiprocessing Anomalies, Bell System Technical Journal, vol.45, issue.9, pp.1563-1581, 1966.
DOI : 10.1002/j.1538-7305.1966.tb01709.x

D. Hochbaum, Approximation algorithms for NP-hard problems, PWS publishing compagny, 1997.

J. Hromkovic, Algorithmics for Hard Problems -Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics, Texts in Theoretical Computer Science, An EATCS Series, 2004.

T. C. Hu, Optimum Communication Spanning Trees, SIAM Journal on Computing, vol.3, issue.3, pp.188-195, 1974.
DOI : 10.1137/0203015

M. Imase and B. Waxman, Dynamic Steiner Tree Problem, SIAM Journal on Discrete Mathematics, vol.4, issue.3, pp.369-384, 1991.
DOI : 10.1137/0404033
URL : http://openscholarship.wustl.edu/cgi/viewcontent.cgi?article=1725&context=cse_research

Y. Jiang and Y. He, Preemptive online algorihms for scheduling with machine cost, Acta Informatica, pp.41-315, 2005.

D. S. Johnson, J. K. Lenstra, and A. R. Kan, The complexity of the network design problem, Networks, vol.3, issue.4, pp.279-285, 1978.
DOI : 10.1002/net.3230080402

B. Kalyanasunaram and K. Pruhs, Speed is as powerful as clairvoyance, Journal of the ACM, vol.47, issue.4, pp.617-643, 2000.
DOI : 10.1145/347476.347479

J. H. Kim and C. K. , Scheduling broadcasts with deadlines, Theoretical Computer Science, vol.325, issue.3, pp.479-488, 2004.
DOI : 10.1016/j.tcs.2004.02.047
URL : http://doi.org/10.1016/j.tcs.2004.02.047

R. Klasing, C. Laforest, J. Peters, and N. Thibault, Constructing incremental sequences in graphs, Algorithmic Operations Research, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00070361

C. F. and S. P. Fung, Improved competitive algorithms for online scheduling with partial job values, Theorical Computer Science, vol.325, pp.467-478, 2004.

C. Laforest, A good balance between weight and distances for multipoint trees, International Conference On Principles Of DIstributed Systems, pp.195-204, 2002.

L. Layuan and L. Chunlin, A qos multicast routing protocol for dynamic group topology, Euro-Par, pp.980-988, 2003.

R. J. Lipton and A. Tomkins, Online interval scheduling, 5th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, pp.1806-1828, 1994.

Z. Lui and T. Cheng, Scheduling with job release dates, delivery times and preemption penalties, Information Process, Lett, vol.82, pp.107-111, 2002.

J. Naor, A. Rosen, and G. Scalosub, Online time-constrained scheduling in linear networks, Proceedings of INFOCOM, pp.855-865, 2005.

E. Naroska and U. Schwiegelshohn, On an on-line scheduling problem for parallel jobs, Information Processing Letters, vol.81, issue.6, pp.297-304, 2002.
DOI : 10.1016/S0020-0190(01)00241-1

C. A. Philips, C. Stein, E. Torng, and J. Wein, Optimal Time-Critical Scheduling via Resource Augmentation, 29th ACM Symposium on Theory of Computing, pp.140-149, 1997.
DOI : 10.1007/s00453-001-0068-9

R. Sriram, G. Manimaran, and C. Murthy, A rearrangeable algorithm for the construction of delay-constrained dynamic multicast trees, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320), pp.514-529, 1999.
DOI : 10.1109/INFCOM.1999.751662

N. Thibault, Médian-ajout : un algorithme incrémental pour le maintien d'un arbre de connexion, 2003.

N. Thibault and C. Laforest, Algorithme incrémental pour le maintien d'un arbre de connexion, pp.27-34, 2003.

V. V. Vazirani, Approximation algorithms, 2002.
DOI : 10.1007/978-3-662-04565-7

B. Waxman, Routing of multipoint connections, IEEE Journal on Selected Areas in Communications, vol.6, issue.9, pp.1617-1622, 1988.
DOI : 10.1109/49.12889

J. Westbrook and D. Yan, Linear bounds for on-line Steiner problems, Information Processing Letters, vol.55, issue.2, pp.59-63, 1995.
DOI : 10.1016/0020-0190(95)00000-3

G. Woeginger, On-line scheduling of jobs with fixed start and end times, Theoretical Computer Science, vol.130, issue.1, pp.5-16, 1994.
DOI : 10.1016/0304-3975(94)90150-3

R. Wong, Worst-Case Analysis of Network Design Problem Heuristics, SIAM Journal on Algebraic Discrete Methods, vol.1, issue.1, pp.51-63, 1980.
DOI : 10.1137/0601008

B. Y. Wu, K. M. Chao, and C. Y. Tang, Approximation algorithms for the shortest total path length spanning tree problem, Discrete Applied Mathematics, vol.105, issue.1-3, pp.273-289, 2000.
DOI : 10.1016/S0166-218X(00)00185-2

D. Ye and G. Zhang, On-Line Scheduling of Parallel Jobs, 11th Colloquium on Structural Information and Communication Complexity (SIROCCO), pp.279-290, 2004.
DOI : 10.1007/978-3-540-27796-5_25