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Such applications Acomprise parallel acquisition and distribution of multiple video 3streams [2], [14], switching of simultaneous voice @communication sessions [6], [3], [13], and high energy physics, >where particle collision events need to be transmitted from a @large number of detectors and filters to clusters of processing @ nodes [1]. !r 7The aggregate throughput of a collective communication >pattern (traffic) depends on the underlying network topology. AThe amount of data that has to pass across the most loaded links Aof the network, called bottleneck links, gives their utilization Etime. 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Assuming in this example a single link �� 7throughput �^, the liquid throughput offered by the �� "network is �_. Under identical ����@packet size and link throughputs (kept all along this paper for ����Xthe sake of simplicity) the �{liquid throughput�] of a traffic �{X�] is the �� 6ratio �` multiplied by the single link throughput, �� >where �c is the total number of transfers and �d is the ���@8number of transfers carried out by one bottleneck link. !d��� ?Now let us see if the order in which the transfers are carried ����=out in this wormhole network has an impact on the collective ����:communication performance. A straight forward schedule to ����:carry out these 25 transfers is the round-robin schedule, ����Baccording to which at first each transmitting processor sends the ����Hmessage to the receiving processor staying in front of it, �Dthen to "����Bthe receiving processor staying at the next position, etc. Such a ���@+round robin schedule consists of 5 phases. X�� (MThe�] transfers of the first �D�,�] �Dsecond �" and fifth 0ߪ�� <�, phases of the round-robin schedule may be carried out �� :simultaneously, but the third phase {�?, �A, �B, �� :�F, �H} �contains transfers congesting with the G�� Jforth phase�D {�]�I, �J, �K, �L, �M}�D, e.g. link "�� ?�N (marked thick) can not be simultaneously used by the two �� ;transfers �O and �P. None of these two phases can be ����Acarried out in less than two time frames and therefore the whole ����@schedule lasts 7 time frames, instead of seemingly 5. Therefore ����<the performance of our collective communication carried out ����9according to the round-robin schedule corresponds to the �� 9throughput of �]�Q �Dmessages per time frame or �� *�S, �Dwhich is less than the liquid ���@ throughput. #m��� =Nevertheless, a solution exists to schedule the 25 transfers �� :within 6 time frames. The sequence of time frames {�V, �� 8�W, �X, �Y, �Z, �]} is an example of the ����=liquid schedule for the 25-transfer collective communication ���@ request. 4���`3. Definitions )ު��� <The method we propose allows us to efficiently build liquid 0ꪅ����?schedules for non-trivial network topologies. Thanks to liquid ����;schedules we may considerably increase the collective data ����9exchange throughputs, compared with traditional topology ����;unaware schedules such as round-robin or random schedules. ����<The present section introduces the definitions that will be ����=further used for describing the liquid schedule construction ���@method. !r��� @A single point-to-point transfer is represented by the set of ����7communication links forming the network path between a ����<transmitting and a receiving processor according to a given ����Frouting schema. A �{transfer�] is a set of links (i.e. the path ����:between a sending processor and a receiving processor). A ;���@Ftraffic �]is a set of transfers (i.e. a collective data exchange). c5��� >Fig. 2 shows a traffic for a collective data exchange carried ����:out on the network of Fig. 1. The bottleneck links of the ٲ۶ !š$A��8������ٲ۶ !š$AVFA ���,X�l���.8)cX���D:����fXJJ�E ������SFI��:A���������FF �������� ٲ۶:Œ,��:B�D� ���ٲ۶:Œ,€,�1���1CHHF��������Ga combination of non-congesting transfers taken from �{X�], such 0����Kthat its complement in �{X�] contains only transfers congesting with ���@that simultaneity. !w��� 7We can categorize full simultaneities according to the ����Lpresence or absence of a given transfer �{x�]. A full simultaneity is ";����Ux�]-positive if it contains transfer �{x�]. If it does not contain transfer �� Qx�], it is �{x�]-negative. Thus the set of full simultaneities �# is ����Jpartitioned into two non-overlapping subsets: an �{x�]-positive and ;�� Ix�]-negative subset of �@. For example, if�{ y�] is another "����Mtransfer, the set of �{x�]-positive full simultaneities may be further ����Vpartitioned into �{y�]-positive and �{y�]-negative subsets. Iteration of ����@this concept allows us to recursively traverse the whole set of �� H@all full simultaneities �u, one by one, without repetitions. !i��� OLet us define a �{category�] of full simultaneities of �{X�] as an ����Wordered triplet (�{excluder�], �{depot, includer�]), where the includer is ����La simultaneity of �{X�] (not necessarily full), the excluder contains ����Gsome transfers of �{X�] non-congesting with the includer and the ����?depot contains all the remaining transfers non-congesting with ���@the includer. !}��� 9A category, defined by the transfers of its includer and ����Cexcluder, constrains a subset of full simultaneities. We therefore ����Ysay that a full simultaneity is �{covered�] by a category �{R�], if the full ����Csimultaneity contains all the transfers of the categorys includer ����>and does not contain any transfer of the categorys excluder. ����AConsequently, any full simultaneity covered by a category is the ����@categorys includer together with some transfers taken from the ����?categorys depot. The collection of all full simultaneities of ;����aX�]covered by a category �{R�] is defined as the �{coverage�] of �{R�]. We �� H-denote the coverage of �{R�] as �$. !1��� <Transfers of a categorys includer form a simultaneity (not ����Cfull). By adding different variations of transfers from the depot, ����>we may obtain all possible full simultaneities covered by the ���@ category. !2�� (DThe category �a is a �{prim-category�] since it covers all �� H0full simultaneities of �{X�], i.e. �r. h��� IBy taking an arbitrary transfer �{x�] from the depot of a category ;����dR�],we partition the coverage of �{R�]into �{x�]-positive and �{x�]-negative "����Ssubsets. The respective �{x�]-positive and �{x�]-negative subsets of a ����Tcoverage of �{R�] are coverages of two categories derived from �{R�]: a ���@@positive subcategory and a negative subcategory of �{R�]. a. �� (GThe positive subcategory �  is formed from the category �{R�] ����Iby adding transfer �{x�] to its includer, and by removing from its ��Gdepot and excluder�  all transfers congesting with �{x�]. The �� Fnegative subcategory �* is formed from the category �{R�] by ����@moving transfer �{x�] from its depot to its excluder. The �� Jreplacement of a category �{R�] by its two sub categories � and �� K� is defined as a �{fission�] of the category. Fig. 3 and Fig. 4 ����;show an example of fission of a category into positive and ���@negative sub categories. ٲ۶ !š$A��:D�D�����ٲ۶ !š$AVB E  ���,X�l���7�e۲\O��Q��"����r ^^� +�����~ٲº:���0�����ٲº:1���W4MUTUT���`FThe duration of a traffic �eX� is the load of its bottlenecks. Hӑ}`��W�#:@����@�L?�A�����)cX���C�����)cX)cX��'��/indexes[0,1,char[R],times[string["+"],char[x]]]/N~"��C"�_������Q�-����b���W�3I�M7�M��b��b͛<�ݢ��W�3ILN7LN��ݢ�ݢ/ߩ��W�3IMO7MO��/ߩ�/;x5@]��W�3INP7NP��5@]�zN5q֌��W�3IOQ7OQ��q֌�{q֐E6��W�3IPn7Pn��E6�E6.}[<zŕe6h��K�p�����[r�.����� 71,��T�S����71,�MDNumber of transfers (and number of nodes) for 362 different traffics/N[ۨ�=D��N� Q������rd�e����/N`kX��\��0�Y����fXVV�����kX��\�U�����kXkX5��'��id[(*i2i*)comma[indexes[0,1,char[l],times[char[a],char[b]]],indexes[0,1,char[l],times[char[b],char[c]]],indexes[0,1,char[l],times[char[c],char[a]]]]]E1JO��\�0*�����dށX`�1t����`,X,B��\�W�l`�l��`,X,B�`,X@nj͚/NWpaX��P�0Uf����fX``� 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_(��@ˈ,Ǝ,X�`tB@ˈ,G,͚,X��\� _(��G,͚,X�`tBG,N#,n,X��\� _(��N#,n,X�`tBN#, /N۶: ��S�����/N۶: ������������������SFI��@����������������� /N۶: ��T�����/N۶: ���������������� /N:6E��U�����/N:6E0�%���%}0������r����;of transfers allocated to the same time frame use a common ���@resource (link, wavelength). !]��� 2The liquid scheduling problem cannot be solved in ����=polynomial time. Solving the problem by applying a heuristic ����<graph colouring algorithm provides in short time suboptimal ����=solutions, whose throughputs are often 10% to 20% lower than ����:the liquid throughput [4]. In the present contribution we ����>propose an exact method for computing liquid schedules, which ����Bis fast enough for real time scheduling of traffics on small size ���@ networks. O"���`!2. The liquid scheduling problem #��� >Let us consider a network topology (Fig. 1) consisting of ten 0�� 4end nodes �! (henceforth called processors), two �� /wormhole cut-through switches �5 and twelve �� ,unidirectional links �9 having identical �� Cthroughputs. The processors �o�{ �]only transmit data and �� @�p only receive data. Its easy to guess the routing, e.g. a �� dmessage from �{t4�] to �{r3�] traverse links�{ �q�] and �{�s�], and a �� H_message from �{t1�] to �{r2�] uses only links �{�w�] and �{�x�]. Ee6hUU[6V�� h�. F&UTg��`,Fig. 1.�Example of a network topology. uy6S��� :We denote transfers symbolically to mark out the occupied 06R����Mnetwork links. For example the transfer from �{t4�] to �{r3�] is �� Zsymbolically represented as �D�<, the transfer from �}t1�D to �}r2�D as "�� <�C. We may also represent a set of transfers carried out ����5simultaneously, e.g. a traffic transferring messages �� Hfsimultaneously from �{t4�] to �{r3�] �Dand from �}t1�D to �}r2�D by �D. c} ��� ;Let each sending processor have messages to be transmitted ����@to each receiving processor and let all messages have identical ����Bsizes [11]. Thus, in the present example, we have 25 transfers to �� 1carry out. Each of the ten links �E carries 5 �� ?transfers and the two links �G must each carry 6 transfers. �� :Therefore the links �U are the network bottlenecks and ����;have the longest active time. If the duration of the whole ����>collective communication is as long as the active time of the ����;bottleneck links, we say that the collective communication ����Areaches its liquid throughput. In that case the bottleneck links /N۶:–PC��V�A����/N۶:–PC”�0���0GT��B����5����>network are marked in bold. The exchange shown in Fig. 2 is a 0����Aparticular case of a traffic. Any collective exchange comprising ����;transfers between possibly overlapping sets of sending and ���@#receiving processors is a traffic. ,.ʦr Y.�� h�+ >&UT:�� PFig. 2.�Example of a traffic composed of 25 transfers carried out over the M8���@network shown on Fig. 1. 0K�� (lA link �{l�] is �{utilized�]by a transfer �{x�]if �. A link �{l�] is utilized 0K����dby a traffic �{X�] if �{l�] is utilized by a transfer of �{X�]. Two transfers are ����Jin �{congestion�] if they share a common link. Note that we will be ����=limiting ourselves to data exchanges consisting of identical ���@packet sizes. !^��� [A �{simultaneity�] of a traffic �{X�] is a subset of �{X�] consisting of ����?mutually non-congesting transfers. A transfer is in congestion ����Cwith a simultaneity if the transfer is in congestion with at least ����?one member of the simultaneity. A simultaneity of a traffic is ;����Hfull�] if all transfers in the complement of the simultaneity in the "����Atraffic are in congestion with that simultaneity. A simultaneity ����@of a traffic obviously can be carried out within one time frame �� L(the time to carry out a single transfer). The �{load�] �  of link ";����fl�] in a traffic �{X�] is the number of transfers in �{X�] using link �{l�]. The �� Nduration�] � of a traffic �{X�] is the maximal value of the load ���@)among all links involved in the traffic. ![�� (2The links having maximal load values, i.e. �k, ����bare called �{bottlenecks. �]The �{liquid throughput�] of a traffic �{X�] is the �� 6ratio � multiplied by the single link throughput, �� HCwhere � is the number of transfers in the traffic �{X�]. !p��� ZWe define a simultaneity of �{X�] as a �{team�] of �{X�] if it uses all ����dbottlenecks of �{X�]. A team of �{X�] is �{full�] if it is a full simultaneity of ;�� RX.�] Let  be the set of all full simultaneities of �{X�]. Let �h "�� Cand F be respectively the sets of all teams and the set of all �� H6full teams of �{X�]. Obviously �j and �l. #o ��� @In order to form liquid schedules, we try to schedule transfers ����>in such a way that all bottleneck links are always kept busy. ����@Therefore we search for a liquid schedule by trying to assemble ����>non-overlapping teams carrying out all transfers of the given ����Ftraffic, i.e. we partition the traffic into teams. To cover the whole ����Asolution space we need to generate all possible teams of a given ����Ctraffic. This is an exponentially complex problem. It is therefore ����>important that the team traversing technique be non-redundant ����Band efficient, i.e. each configuration is evaluated once and only ���@once, without repetitions. K"M���`!4. Obtaining full simultaneities 7aKi��� ATo obtain all full teams, we first optimize the retrieval of all 0mKh����@simultaneities and then use that algorithm to retrieve all full ���@teams. a��� BRecall that in a traffic �{X�], any mutually non-congesting ����Ccombination of transfers is a simultaneity. A full simultaneity is /N۶:Œ,��W�DE����/N۶:Œ,„�+���+tmffF����Ru폹Cu�� h�4 S8UT� (TFig. 3.�yAn initial category before fission, where symbol �{, represents any 29�� Ptransfer that is in congestion with �zx�y and symbol �z represents any ���@1transfer which is simultaneous with �zx�y. WM��� GFig. 3 shows an example of a category �{R�] and Fig. 4 shows the 0Y����@resulting two sub categories obtained from the initial category ���@Hby a fission relatively to a transfer �{x�] taken from the depot. *.ʦr ]�� h� U8UT%e�� RFig. 4.�yFission of the category of Fig. 3 into its positive and negative sub 9%c���@ categories. Vz��� GThe coverage of �{R�] is partitioned by the coverages of its sub 0z�� Ccategories �  and �, i.e. the coverage of a category is the ����*union of coverages of its sub categories: �� $�, and the coverages of the sub �� H,categories have no common transfers, �. !��� FA �{singular�] category is a category that covers only one full ����Asimultaneity. That full simultaneity is equal to the includer of ����<the singular category. The depot and excluder of a singular ���@category are empty. ! ��� >We apply the binary fission to the prim-category and split it ����Ainto two categories. Then, we apply the fission to each of these ����@categories. Repeated fission increases the number of categories ����;and narrows the coverage of each category. Eventually, the ����?fission will lead to singular categories only, i.e. categories ����@whose coverage consists of a single full simultaneity. Since at ����5each stage we have been partitioning the set of full ����:simultaneities, at the final stage we know that each full ���@?simultaneity is covered by one and only one singular category. !��� @The algorithm recursively carries out the fission of categories ���@8and yields all full simultaneities without repetitions. !|��� 9There is a further optimization to be considered. Take a ����Acategory. A full simultaneity must contain no transfer from that ����@categorys excluder in order to be covered by that category. In ����Caddition, since the full simultaneity is full, it is in congestion ����@with all transfers that it does not contain. Obviously any full ����=simultaneity covered by some category must congest with each ����9member of that categorys excluder. Therefore, transfers ����@congesting with the transfers of the excluder must be available ULUU/%=��Ein the depot of the category� . If the excluder contains at least t;%<����>one transfer, for which the depot has no congesting transfer, ����Ithen this category is �{blank�]. The includer of a blank category, ����>cannot be further extended by the transfers of the depot to a ����>simultaneity which is full (and congests with every remaining ��SFI��;Y��������� �������� ٲ۶:™z��;Z� ����ٲ۶:™z˜G�)���)�� ����|����?transfer of the excluder). The coverage of a blank category is ���@<therefore empty and there is no need to pursue its fission. !3��� BLet us now instead of retrieving all full simultaneities retrieve ����Aall full teams (i.e. those full simultaneities, which ensure the ���@&utilization of all bottleneck links). ! ��� NA category within �{X�] is �{idle�] if its includer and its depot ����Itogether dont use all bottlenecks of �{X�]. This mean that we can ����<not grow the current simultaneity (i.e. the includer of the ����Dcategory) into a full simultaneity, which will use all bottlenecks. ����>The coverage of an idle category does therefore not contain a ����@full simultaneity, which is a team. Idle categories allow us to ���@prune the search tree. !\��� 9Carrying out successive fissions, starting from the prim-����:category and continuously removing all the blank and idle ���@/categories ultimately leads to all full teams. %"���`)5. Speeding up the search for full teams {ت��� ;This section presents an additional method for speeding up 䪚�� HIthe search for all full teams � of an arbitrary traffic �{X�]. !��� KLet us consider from the original traffic �{X�] only those transfers ����Gthat use bottlenecks of �{X�] and call this set of transfers the ;�� Yskeleton�] of �{X�]. We denote the skeleton of �{X�]as �. Obviously, �� H�&. !H��� >Fig. 5 shows the relative size of skeletons compared with the ����=size of the corresponding traffic, for 362 different traffic ����Apatterns within the T1 32 node cluster computer (see Fig. 10, in ����;section 7). The skeleton sizes are on average 31.5% of the ���@corresponding traffic sizes. TUU�� h�' Y8UTs�� TFig. 5.�yProportion of the number of transfers within a skeleton, compared with 9 s���@6the number of transfers of the corresponding traffic.  ��� BWhen considering the skeleton of a traffic �{X�] as another 0(����Ctraffic, the bottlenecks of the skeleton of a traffic are the same ����=as the bottlenecks of the traffic. Consequently, a team of a ���@1skeleton is also a team of the original traffic. !P��� @We may first obtain all full teams of the traffics skeleton by ����Biteratively applying the fission algorithm and by eliminating the ����Bidle categories. Then, a full team of the original traffic may be ����=obtained by adding a combination of non-congesting transfers ���@%to a team of the traffics skeleton. a�� (?We therefore obtain the set of a traffics full teams �( by ���@"carrying out the following steps: ٲ۶ !š$A��;\� �����ٲ۶ !š$AVF2 ���,X�l��� /N۶:™z��X� ����/N۶:™zƒ�&���&QT�� ����)� (A1.Obtain the set of the skeletons full teams �m by applying ���@the fission algorithm. ��`72.Create for each skeletons full team a category by: #B�� D2.1.Initializing the includer with the transfers of the skeletons ���@ full team; ��`)2.3.Initializing the excluder as empty; !�� L2.2.And putting into the depot all transfers of �X� non-congesting ���@with the includer. #()�� E3.Apply the fission to each category, discarding the check for idle ����Ccategories, since the includer is already a team, i.e. it uses all ���@ bottlenecks. /��� 7By first applying the fission to the skeleton and then 0����Aexpanding the skeletons full teams to the traffics full teams, ����?we strongly reduce the processing time and at the same time we ���@Dobtain all full teams of the original traffic without repetitions. ! ��� ;We measured the reduction in search space according to the ����8different search space reduction methods we propose. We ����=consider 23 different traffic patterns within the T1 cluster ����;computer (see section 7). The search space is given by the ����Anumber of categories that are being iteratively traversed by the ����:fission algorithm. Fig. 6 shows the obtained search space ����<reductions compared with a naive algorithm that would build ����Bfull teams according to a coverage partitioning strategy, i.e. by ����=constructing categories thanks to the fission algorithm, but ���@,without any of the proposed optimizations. ~UU݀�� h�e 8UT+*�� QFig. 6.�ySearch space reduction obtained by idle+skeleton+blank optimization 9+(���@steps. c|��� 8The skeleton algorithm together with the idle and blank 0{����Aoptimizations reduces on average the search space to 10.6%, i.e. ����>full teams are computed 9.43 times faster than without search ����>space reduction techniques. Note that in the above comparison ����=even the naive algorithm is smart enough to avoid repeatedly ���@#exploring the full simultaneities. P"T���`$6. Construction of liquid schedules thv��� ;Having the capability of building full teams, this section ptu����=presents the general method for building liquid schedules on ����?irregular topologies for any collective communication pattern. /N۶:Ž��Y�01����/N۶:Ž~�p����U&��2 ����f�� 8�, this traffic has no team and therefore no liquid ���@ schedule. %;���`,6.1. Liquid schedule naive search algorithm &��� >We first propose a simple technique for the construction of a 0D����=liquid schedule and then introduce an optimization improving ���@0the efficiency of liquid schedule construction. #- ��� ?Our strategy for finding a liquid schedule relies on partition����Aing the traffic into a set of teams forming the sequence of time ����Nframes�D. �]Associate to the traffic �{X�] all its possible teams �� 6�  which could be selected as the schedules first �� 5time frame. �/ is the variety of possible subtraf����Efics remaining after the choice of the first time frame. Each of the �� Fpossible subtraffics �0 remaining after the selection of the first ����@time frame has its own set of possibilities for the second time �� #frame �1. The choice of the sec����@ond team for the second time frame yields a further reduced sub���@traffic�D (see Fig. 9). 5^^�� h�3 8&UTWU� (ZFig. 9.�Liquid schedule search tree. �6�z �ydenotes a reduced subtraffic at r9cU�� Lthe layer �7 of the tree and ��8 �ydenotes a candidate for the �� Jtime frame �:. The operator �; applied to a subtraffic �= yields ���@5the set of all possible candidates for a time frame. /N۶:•��Z�����/N۶:•“7����_K�� ����C����Aa K-ring [9] and has a static routing scheme. The throughputs of 0����Gall links are identical and equal to 86�{MB/s �][8]. The cluster ����@consists of 32 nodes, each one comprising 2 processors, i.e. 64 ���@processors, [12], [5]. +hr+UUr'�� h�n 9KUT�� TFig. 10.�y�Architecture of the T1 cluster computer interconnected by a high L���@$performance wormhole switch fabric. ;r"��� 8The sample traffic patterns are selected from different 0r!����9configurations of half-to-half collective data exchanges ����<between a set of sending and a set of receiving processors, ����@where each sending processor carries out a transmission to each ����;receiving processor. We identified for T1 architecture 362 ���@1different collective communication patterns [4]. !=��� >The 362 different traffic patterns were scheduled both by our ����7liquid scheduling algorithms and according to topology-����=unaware round-robin schedule. Overall throughput results for ����6each method are measured and presented in Fig. 11 for ����@comparison. The values of the theoretical liquid throughput are ���@ also given. AUU/�� h�- J&UT:3�� SFig. 11.�Theoretical liquid throughputs and measured throughputs for traffics MD3���@9scheduled according to round-robin and liquid schedules. <T��� =Each black dot represents the median of 7 overall throughput p`����8measurements carried out according to liquid schedules. ����CProcessor to processor transfers have a size of 5�{MB�]. The ����>measured aggregate throughputs (black dots) are very close to ����Bthe theoretically expected values of the liquid throughput (light ����=gray area). For many topologies, liquid scheduling allows to /N۶:E��[�45����/N۶:E<�2���2����6����_����>liquid scheduling may considerably improve the utilization of 0����4transmission resources such as communication links, ����5wavelengths and orthogonal frequency spectra. Liquid ����5schedules avoid congestions and minimize the overall ���@1transmission time for collective communications. !d��� 6In the future, we intend to develop multipath routing ����@solutions, which increase the traffics fault-tolerance against ���@?link failures and at the same time keep the throughput liquid. B"n���`9. References H7�� D[1] Large Hadron Collider, Computer Grid project, CERN, 2004�k, +���@http://lcg.web.cern.ch/LCG/ #.7�� E[2] S.-H.G. Chan, Operation and cost optimization of a distributed ����<server architecture for on-demand video services, IEEE Com���@8munications Letters, Vol. 5, No. 9, Sept. 2001, 384-386 !D�� >[3] Siemens Carrier Networks, EWSD Digital Switching System, ����DApril 2004, �khttp://www.icn.siemens.com/carrier/products/switch+���@ing/ewsdsw.html #07�� >[4] Emin Gabrielyan, Roger D. Hersch, Network Topology Aware ����>Scheduling of Collective Communications, ICT03 - 10th Interna����;tional Conference on Telecommunications - 2003, 1051-1058, +���@http://ieeexplore.ieee.org/ ],�� F[5] �wRalf Gruber, Commodity computing results from the Swiss-Tx 7���@5project Swiss-Tx Team, Grid Computing Meeting, 2000 ��`B[6] H.323 Standards, �khttp://www.openh323.org/standards.html ! �� J[7] Paul R. Halmos, �kNaive Set Theory�w, Springer-Verlag New York ���@%Inc, ISBN 0-387-90092-6, 1974, 26-29 !A�� <[8] R. Horst, TNet: A Reliable System Area Network, IEEE ���@1Micro, vol. 15, no. 1, February 1995, pp. 37-45. #[,�� A[9] P. Kuonen, The K-Ring: a versatile model for the design of ����7MIMD computer topology, Proc. of the High-Performance ����8Computing Conference (HPC'99), San Diego, USA, 381-385, ���@ April 1999 #a7�� ?[10] Benjamin Melamed, Khosrow Sohraby, Yorai Wardi, Measure����9ment-Based Hybrid Fluid-Flow Models for Fast Multi-Scale ����/Simulation, DARPA/NMS BAA 00-18 AGREEMENT No. ����=F30602-00-2-0556, Sept. 2000, �khttp://204.194.72.101/pub/+���@nms2000sep/UMissouri-KC.pdf #&7�� G[11] M. Naghshineh, R. Guerin, Fixed versus variable packet sizes in ����>fast packet-switched networks, Proc.Twelfth Annual Joint Con����:ference of the IEEE Computer and Communications Societies ����:INFOCOM '93., Networking: Foundation for the Future, IEEE ���@Press, Vol. 1, 1993, 217-226. #\,�� G[12] Pierre Kuonen, Ralf Gruber, Parallel computer architectures for ����:commodity computing and the Swiss-T1 machine, EPFL Super����@computing Review, Nov 1999, pp. 3-11,�m http://sawww.epfl.ch/-���@+SIC/SA/publications/SCR99/scr11-page3.html '7��`*[13] SIP Forum, http://www.sipforum.org/ a/�� J[14] Dinkar Sitaram, Asit Dan, �kMultimedia Servers�w, Morgan Kauf����?mann Publishers, San Francisco California, 2000, 69-73, ISBN 1-���@ 55860-430-8 ٲ۶:•��@�����ٲ۶:••Q�4���4aB�� �������� @Dead ends are possible if there are no choice for the next time 0����Cframe, i.e. no team of the original traffic may be formed from the ����Btransfers of the reduced traffic. A dead end situation may occur, ����Bfor example, when the remaining subtraffic appears to be like the ����?one shown in Fig. 8. Once a dead end is faced, backtracking oc���@curs. !`��� =The construction recursively advances and backtracks until a ����?valid liquid schedule is formed. A valid liquid schedule is ob����Atained, when the transfers remaining in the reduced traffic form ���@@one single team for the last time frame of the liquid schedule. # ��� >We use the search tree shown in Fig. 9 and assume that at any �� <stage the choice �[ for the next time frame is among the �� /set of the original trafics teams �}, i.e. �� �~. In the next subsections ����=we reduce the search space by considering newly emerging bot���@$tlenecks at successive time frames. ;ͪ��� :6.2. Search space reduction by considering newly emerging ٪���@/bottlenecks and by considering only full teams 6쪘��� ;We observe in Fig. 7 that when we step from one time frame ����Dto the next, additional new bottleneck links emerge�], e.g. from �� HBtime frame 3 on, links � and �% appear as new bottlenecks. #D��� >In the construction strategy presented in the previous subsec����Ation we considered as a possible time frame any team of the orig����Minal traffic �}X�D that can be built from the transfers of the reduced ����Nsubtraffic�]. �DWe have shown [4] that for the liquidity of a schedule, ����Ait is necessary for each time frame to be not only a team of the �� Horiginal traffic but also a team of the reduced subtraffic. If �v is �� `a liquid schedule on �}X�D and �}A�D is a �}time frame�D of �, then � �� His a liquid schedule on �. #6��� =Thus a liquid schedule may not contain a time frame which is ����Aa team of the original traffic but is not a team of a subtraffic ����>obtained by removing some of the other time frames. Therefore ����?we can limit at each iteration our choice to the collection of ����Aonly those teams of the original traffic which are also teams of ����Bthe current reduced subtraffic. �DSince the reduced subtraffic "����Bcontains additional bottleneck links, there are less teams in the ����:reduced subtraffic than teams remaining from the original ���@traffic�]. #e��� =By considering in each time frame all occurring bottlenecks, ����>we considerably reduce the search space without affecting the �� H!solution space�D, i.e. �b. #I��� >A further optimization consists in building a liquid schedule ����Aby limiting the choice of teams of the reduced subtraffic to its ����Ffull teams, see [4]. �]The choice of the teams in the construction "����=may be narrowed from the set of all teams to the set of full �� H"teams only, i.e. �D�f�]. !L��� ;The expression below �Dsummarizes the �]search space ���@ reduction. N cFXY 3�� h�g O"rQ���`7. Experimental verification C…5��� <As basic network topology for our testbed, we use the Swiss-P‘4����AT1 cluster (called T1, see Fig. 10). 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