Arnaud Legrand, Olivier Beaumont, Loris Marchal, Yves Robert

Transcription

1 Arnaud Legrand, Olivier Beaumont, Loris Marchal, Yves Robert École Nor 46 Allée d I Télé Téléc Adresse é

2 when using a general gr comes from the decompo are used concurrently to that a concrete scheduli asymptotically optimal, solutions). Nous nous intéressons ic d une application comple grille de calcul. De tel de données à travers le le débit d une telle diffu nombre de messages do pour le parallélisme de d modélisée par un graphe ont des vitesses différent avec des topologies restr gés), et montrons que ce Nous montrons commen construire un ordonnanc Keywords: Scheduling Mots-clés: Ordonnanceme

3 For the first two problems, th For the third problem, the obje steady-state operation, i.e. the a In the case of the atomic br the source) would receive the m implemented using a spanning t complex: the idea is to use severa fractions of the total message. A packets, which are sent in a pipe for an illustration with two-dime The series of broadcasts prob Desprez et al. [24], but with a sources successively broadcast o communications. Here, we assum this is closer to a master-slave pa slaves in a pipelined fashion, for in problem instances. The series of broadcasts rese the latter using an algorithm for of the atomic messages (packets) to the pipelined broadcast proble function of the total message len In this paper, we re-visit the in the context of heterogeneous broadcasting with processors com capacities, and/or different start

4 in Section 7. Finally, we state so The target architectural platform (V,E,c), as illustrated in Figure paths. Let p = V be the number role: it initially holds all the dat are destination nodes which mus There are several scenarios for this paper, we concentrate on the receive data from one of its neigh any given time-step, there are at in emission and the other in rece Each edge e j,k : P j P k is communicate one unit-size mess the pipelined broadcast problem) reverse direction, from P k to P j, communication link between P j and P k are neighbors in the comm precisely: if P j sends a unit-size 2 Framework P k cannot initiate another r a send operation), P j cannot initiate another

5 Series of broadcasts In the se (potentially infinite) sequence of of the link capacities c j,k. The o to maximize the throughput, i.e We work out a little example in Pipelined broadcast In the p a large message of total size L. T The time to send a packet of s start-up costs in the definition o optimization problem, which we to find the number and size of th so that the total execution time 3 Comparing topolog We start this section with an exam We compare the best throughpu (DAG), or the full topology wit broadcast tree is a NP-hard prob 3.1 Working out an exam Optimal solution Consider the simple example of t can be achieved on this network

6 at t = 1, and message m 2 faster (c 1,3 = 1/2), one tim every odd-numbered time-s P 2 operates in a similar fash to P 4. Overall, the steady-state is by the source processor eve to 1. Period 1 Time 1 2 Link P s P 1 m 1 P s P 2 m P 1 P 2 m P 1 P 3 P 2 P 1 P 2 P 4 Figure 2: An op We further use the example and of DAGs over spanning trees

7 and P 2 are children of the source In the first case, where P 1 and tree of Figure 3. The labels on represent the (average) number unit. The value 1/2 means that o because of the one-port constrain can be achieved using this tree. In the second case, one of t Without loss of generality, we ass leads to the spanning tree of Fig on one hand, the one-port constr transfer a message to its childre every 3 time-steps. We can inde the figure. Overall, this is the be this network. We choose a less restrictive assum instead of a broadcast tree, out entry vertex, namely the source tree? The answer is positive. The spanning trees: the DAG shown (P 1,P 2 ) is replaced by the edge of Figure 5. Because the first br achieve a throughput at least 1/ Broadcast DAGs

8 As a conclusion, we point ou strongly depends upon the graph example shows, restricting to tre of 2 3 ), and restricting to DAGs is of 1 instead of 4 5 ). It turns out that computing t when restricting to DAGs than w fore, we give the solution for DA general graphs (Section 5). Befor result: we show that finding the b for the Series problem is NP-Co 3.2 Finding the best broa The problem of determining th follows: Definition 1 (BEST-BROAD G = (G,V,c) representing a netw throughput TP. The associated decision prob Definition 2 (BEST-BROAD G = (V,E,c) representing a ne broadcast tree T such that TP Theorem 1. BEST-BROADCA

9 TREE, which is known to be NP TRAINT-SPANNING-TREE is integer K V d, does there exist than K? We construct the follow we let G = (V,E,c) where V = V the un-oriented edge (j,k) is in source at random, i.e. any node all edges, except for the edges o the bound D = 1. Clearly, the in of I 1. We show that I 2 admits a Assume first that I 1 admits We orient T d so as to trans T is a broadcast tree in G, all nodes in T have an outat most K (because s has n its children in T is j V,(P s,p j ) E T 1 The time spent by any oth j V,(P i,p j ) E T 1 c i,j = Therefore TP D, and I 2

10 j V,(P s,p j ) E T TP where δ + (i) is number of c δ + (s) K. Similarly for from i has weight 1/(K 1 degree of i is at most K (o arborescence T rooted at s than or equal to K, hence 4 Series of broadcast In this section we provide an algo optimization problem, in the rest 4.1 Linear program for D In this section, we assume the n that all nodes are reachable from n j,k is the (fractional) num P k during one time-unit, t j,k is the fraction of time time-unit.

11 knowns): SSBPDAG(G), Steady-S Maximize TP, subject to P j, P k t j,k = n j, P j, P k 0 t j,k P j, P k,(p j,p P j, P j with j s, P k,(p k,p P k,(p k,p We make a digression to pro optimal solution of the Series p 4.2 The edge coloring lem Given a set of processors operat set of communications within a p is that Equations 4 and 5 are sat P j, P j, P k,(p j P k,(p k However, it is not obvious that t because only independent comm scheduled simultaneously. Fortun

12 Proof. Because G is a DAG, we each vertex j, such that h(j) < each node is reachable from s an H be the maximal height of any We define q as the least comm TP) in the solution of the linea broadcast of q TP messages f schedule can be pipelined to reac We build a schedule by indu messages at time q h(i) and th We state the following result, wh can be found in polynomial time Theorem 2. The solution of the to the Series problem on a DAG ber of broadcasts that can be init the corresponding optimal period The source already has the Consider a node P j of heig than or equal to h 1 have to Equation (6), in the solu P j

13 constraints are satisfied for Each sender P k will have t a node P j receives during t period [(h(p j ) 1) q,h(p can be performed in time q P l,(p k,p l ) E and h( Since the one-port constrai sult presented above, we ca these communications in ti We now prove that this sche done by pipelining the schedule constraints are still satisfied, dur the time spent by node P j r P k,(p j,p k ) E

14 Link P P P P P Once pipelined, it gives the f Period 0 Link P s P 1 [0,3 P s P 2 P 1 P 2 P 1 P 3 P 2 P 4 The last step is to use the receptions or emissions never ove Period 1 Time Link P s P P s P 2 P 1 P 2 P 1 P 3 P 2 P 4

15 size of the initial data. The sketc In what follows, we give a set o solution at steady-state. We norm to each processor every T timethis is the dual problem of Section broadcast per time-unit. Howeve of messages that transit along ed But things get more complicated For any node P j, we denote such that (P j,p k ) E; similarly P k such that (P k,p j ) E. Since we deal with broadcas But because of the pipelining, sev 1. We express the conditions t to the Series problem by m provides a lower bound for 2. From the solution of the li used to broadcast the diffe enables us to reach the low 3. From the set of trees, we to write the code of the br initial data. 5.2 Lower bound

16 This constraint reads: for number of messages destin the number of same type m in steady-state operation, i Link occupation The following sages that are transferred of messages that transit on for each i, the fraction x j,k i main difficulty is that the m be partly the same, since t fore, the constraint n j,k = be too pessimistic. Since o consider that all the messag set, namely the largest one occupation time t j,k of the We also need to write down incoming and outgoing com be the time spent by P j f

17 Steady-State Broa Minimize T, subject to i, i s, j, P j P s and P (P j,p k ) E, (P j,p k ) E, j, j, j,k, j, j, 5.3 Weighted broadcast t The solution of the linear prog needed to broadcast one unit-siz be achieved, because of the assu edge are all sub-sets of the large is possible to find a set of broadc Branching theorem. Unfortunat exponential in the problem size. Branching theorem, that produc

18 Therefore, by the Max-flow, Min G between P s and P i is at least disconnect P s and P i. Since the a Lemma 3. κ(g,p 0 ) N Proof. Suppose that κ(g,p 0 ) = and Fulkerson, for each P i, ther denote the value of this flow y j,k i construction), and let us denote between P s and P i. Then, i, i, j, P j P s and P j (P j,p k ) E, (P j,p k ) E, j, j, Therefore, there would exist a N N T < T, which is a contradic Finally we have the following Theorem 4. κ(g,p 0 ) = N.

19 where χ T j,k (T l) = 1 if (P j,p k ) Moreover, the trees can be found m Lemma 4. l λ l = κ(g,p 0 ) = Proof. The proof of this lemma formation of G into a multi-grap Lemma 5. The set of trees can Proof. The number of trees is b be encoded in size of order V ( and maxn j,k, can be encoded in above. Therefore, the weighted versi time a set of weighted trees, wh use these trees in order to broad overall number of messages that Moreover, since the overall w platform, we have:

20 V = V out j,k where p = V is the number of and Pk in is weighted by the qua overall amount of data transitin the communications, we use the chapter 20]): Theorem 6. Let G M = (V,E,m ings M 1,...,M km, with integer w where χ M j,k (M i) = 1 if (P j,p k µ i = m Moreover, the matchings can be i Lemma 6. i µ i NT. Proof. By Equation(17), m j,k m j,k c j,k j and m j,k c j,k k

21 P b P e (c) Graph of the Nn j,k = max i P s P a P c P d P b P e (e) Second broadcast tree, λ Figure 6: Example where m j,k < message is 5 time-units, due to by the least common multiple N on each edge. Figures 6(d) and 6 figure, each of them with a weigh trees. On the edge (P c,p e ), we tree, so m c,e = 1, whereas Nn c,e through this edge in the optimal made when we use trees.

22 by (19), and by ( Let us denote by In the following, we exhibit NsT, and Ns messages are bro throughput 1/T. Let m l j (q) be the set of messa the q-th period. The sketch of the Step i (P j,p k ), i M (j,k) l, (P j,p k ) T l end end We prove the correctness of t

23 M(r Total number of messages re sages are sent along the ed different, and are different overall number of messages Therefore, during one period new different messages. Therefo one period is 1 T, hence its optim sum of the duration of the differe as claimed in the proof of Lemm 5.5 Asymptotic optimalit In this section, we prove that the cally, no scheduling algorithm (e a given time-frame than ours, up to the formal statement of this r

24 P j, P k,(p k,p j ) E P i, P j P i,p s, T k,j K ( P k,(p j,p k ) forwarded by P j to P i ) P i,p i P s,opt(g,k) = the source to each node) P i,p i P s,opt(g,k) = node) Let x i,j k = Nj,k i opt(g,k), n j,k = m P k N out (P j ) t j,k. Let T(K) = op All the equations of the linear pr value. This concludes the proof. Again, this lemma states tha There remains to bound the lo periodic solution, to come up w state operation. Consider the fol Solve the linear program fo the period T such that eve

25 Clean-up during the J last final destination. No proce be evenly divisible by T). The number of messages b steady(g,k) = (r + 1) T Clearly, the initialization and tation, using parallel routing and do not need to refine the previou Theorem 7. The previous sched totically optimal: Proof. Using the previous lemm algorithm, we have steady(g, K) result because I, J, and TP(G) 6 Pipelined broadcas In the pipelined broadcast proble of total size L, which can be spl model must include start-up ove linear in the packet size, the bes

26 T pipe (L) β Proof. Let w = max j,k j,k c j,k so tha units to broadcast 1 TP T unit-s SSB(G), and T the integer period messages of length ν +w. As sho and to quit the steady-state mo time-units to broadcast ν messag Conversely, Lemma 7 shows into account, so that we have a fo 7 Experiments In this section, we work out a random generator of topology [6 topology is represented on Figur Figure 7(b) shows the result represent communications, and t in its list, it means that Nx i,j k = through edge (i,j) to reach P k. 152 time-units. From these communications, Figure 8, where both the logical

27 2 2 Figure 7 2 2(5) 2(6)

28 the top of tree. Some theoretica have been developed for the prob A more complex model is int message into account, but also t time needed to receive the messa proportional to the length of the Yet another model of commu the message between any process T i,j and a part depending on the m the two processors, B i,j. Since th total communication time betwe proposed for the broadcast and t All previous models assume t a given processor can send dat overlapping this operation with o Other collective communicati have been studied in the context others. 9 Conclusion In this paper, we have studied Our major objective was to max mode, when a large number of sa single large message is split into p the best throughput may well re

29 (a) First (b) Secon logical tree

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