Cyclic scheduling: an introduction

Transcription

1 Cyclic scheduling: an introduction Claire Hanen Université Paris X Nanterre / LIP6 UPMC Cyclic scheduling for EPIT 00 p. /?

2 Summary Definition Uniform precedence constraints Feasibility Periodic schedules Earliest schedule Resource constraints Periodicity and Patterns Complexity approximation ILP formulations Perspectives Cyclic scheduling for EPIT 00 p. /?

3 Definition A set of tasks T to be repeated many times assumed infinite For i T, < i,k > denotes k th occurrence of i An infinite schedule σ defines: k 0 t σ (< i,k >) starting time of < i,k > Resources for each task execution Cyclic scheduling for EPIT 00 p. /?

4 Optimizing Minimizing the average cycle time : A(σ) = max A(σ,i) = limsup t σ (< i,k >) k + i T k Maximizing the throughput: D(σ) = A(σ) For a given average cycle time, minimizing the amount of resources per time units (ex: nb of processors, nb of registers,...) Cyclic scheduling for EPIT 00 p. /?

5 Applications and models Implementing loops on parallel architectures compiling, code generation embedded applications litterature on software pipelining and dataflow computations Mass production Cyclic shop problems Hoist scheduling problem. litterature on cyclic scheduling in production systems Models of parallelism timed Petri Nets Graphs Max + algebra litterature on parametric paths, timed event graphs, Max + Cyclic scheduling for EPIT 00 p. 5/?

6 Infinite schedule? Algorithms-> finite description Static schedule Must be regular /time and resources Dynamic policy examples: earliest schedule, regular priorities on resources assignment Cyclic scheduling for EPIT 00 p. 6/?

7 Loop example Assume arrays A, B, C, D, and that the processor can compute several instructions in parallel : for I = to N do B(I) = A(I ) + task <, k > precedes <, k > C(I) = B(I) + 5 task <, k > precedes <, k > D(I) = B(I ) D(I) task <, k > precedes <, k > A(I) = C(I ) + D(I) task <, k >, <, k > precede <, k > Precedence constraints are uniforms: if < i, k > precedes < j, l > then for all integers δ, < i, k + δ > precedes < j, l + δ > Cyclic scheduling for EPIT 00 p. /?

8 yclic scheduling with uniform precedence A set T of n generic tasks with processing times p,..., p n non reentrance: < i, k > precedes < i, k + > k A multi-graph G = (T, A) of uniform constraints For each arc a A, A value called length L(a) Z also mentioned as latency or delay A value called height H(a) Z also called dependence distance (L(a), H(a)) If i = b(a) et j = e(a), k, t σ (< i, k >) + L(a) t σ (< j, k + H(a) >) b(a) e(a) Find an infinite feasible schedule minimizing the average cycle time. Definition. The Length of path µ, denoted L(µ) = sum of arcs length. Height of µ, denoted by H(µ) = sum of arcs height. Remark. Non-reentrance can be expressed with uniform constraints : arcs (i, i) with length p i and height. Cyclic scheduling for EPIT 00 p. 8/?

9 Example Negative length can be used to model deadlines : t σ (< 5,k >) t σ (<,k >) + 0 Cyclic scheduling for EPIT 00 p. 9/?

10 Questions Feasibility Construction and properties of schedules. Periodic schedules Earliest schedule Performance Cyclic scheduling for EPIT 00 p. 0/?

11 Feasibility For any path µ from i to j, k max(, H(C)), t σ (< i, k >) + L(µ) t σ (< j, k + H(µ) >) For any circuit C of G, and task i in C: k max(, H(C)), t σ (< i, k >) + L(C) t σ (< i, k + H(C) >) Non reentrance Lemma. [Lee 05][Munier 06] If G is feasible then for any circuit C such that H(C) 0, L(C) 0. If H N this condition is sufficient [many authors] [Chretienne 85] If H(C) > 0, k = qh(c) + r then t σ (< i, k >) ql(c) + t σ (< i, r >). If H(C) < 0 and k = qh(c) + r then t σ (< i, k >) ql(c) + t σ (< i, r >) Lemma. If H(C) 0, A(σ, i) = limsup k + If H(C) < 0, A(σ, i) L(C) H(C) t σ (< i, k >) k L(C) H(C). Cyclic scheduling for EPIT 00 p. /?

12 Feasibility and performance bounds For a circuit C the Index of C: α(c) = L(C) H(C) C + (i) set of circuits C s.t. i C and H(C) 0. C (i) set of circuits C s.t. i C and H(C) < 0. C + = i T C + (i) C = i T C (i) Corollary. If G is feasible then for any task i, and for any schedule σ max α(c) A(σ, i) min C C + (i) α(c) C C (i) A critical circuit is a circuit C C + s.t α(c) is maximum : lower bound on A(σ). Cyclic scheduling for EPIT 00 p. /?

13 Periodic schedules Definition. A schedule σ is periodic if each task i has a period w i such that: t σ (< i,k >) = t σ (< i, >) + (k )w i Let a be an arc from b(a) to e(a) k, t σ (< b(a), >) + (k )w b(a) + L(a) t σ (< e(a), >) + (k )w e(a) + H(a)w e(a) t σ (< e(a), >) t σ (< b(a), >) L(a) w e(a) H(a) + (k )(w b(a) w e(a) ) Cyclic scheduling for EPIT 00 p. /?

14 Existence of a periodic schedule Lemma. for any arc a, w b(a) w e(a). We denote by C,...,C q the strong components of G. Let α + (C s ) = Corollary. max α(c) and C C s,h(c)>0 α (C s ) = min α(c) C C s,h(c)<0 s, i,j C s,w i = w j = W s and α + (C s ) W s α (C s ) Moreover, if there is an arc a s.t. b(a) C s,e(a) C s then W s W s Cyclic scheduling for EPIT 00 p. /?

15 Example C = {,,}, C = {,5}, C = {6} α + (C ) = 0, α (C ) =, α + (C ) =, α (C ) =, α + (C ) = 5 0 W, W, 5 W, W W, W W W = 0, W =, W = 0. Notice that W = W = W = 0 is also a solution. Cyclic scheduling for EPIT 00 p. 5/?

16 Feasibility Theorem. [Munier 06] G is feasible if and only if there exists a periodic schedule. Idea : If no periodic schedule exists, build paths µ x between i and j in G such that H(µ x )= h and for some k. lim L(µ x) +, which contradicts x + t σ (< j,k >) t σ (< i,k + h >) L(µ x ) Cyclic scheduling for EPIT 00 p. 6/?

17 Computation of a periodic schedule For a strong connected graph G and a given W : a, t σ (< e(a), >) t σ (< b(a), >) L(a) W.H(a) Let V W (a) = L(a) W.H(a) If (G,V W ) has positive circuits then infeasibility. otherwise t σ (< i, >) = longest path to i in (G,V W ) is a solution. Bellman-Ford algorithm Cyclic scheduling for EPIT 00 p. /?

18 Computation of critical circuits [Dasdan et al 99] Polynomial algorithms : Binary search on W: at each step check if (G, V W ) has positive circuits 0(nm(log n + log max a (L(a), H(a)))) [Lawler 9][Gondran-Minoux 85] Linear programming with primal dual approach O(n m) [Burns 9] An efficient pseudo polynomial algorithm: Howard s algorithm[cochet-terrasson et al 98] W = lower bound At each step check if G, V W ) has positive circuit C with breadth first search. If H(C) <= 0 stop. Otherwise set W = α(c). complexity O(m.X), X product of degrees of nodes. Cyclic scheduling for EPIT 00 p. 8/?

19 mputation of an optimal periodic schedu Compute the strong components of G O(n + m) Check feasibility for each component C s, and the critical circuit value α s.o(nm log(n) + log(v max )) Compute the reduced graph of components.o(n + m) Sort the components by topological order.o(n + m) for each component C s. if C s has no predecessor, set W s = α s. if C x,..., C xr are predecessors of C s. Let β s = max i W xi If α s β s set W s = α s. otherwise check if (C s, V βs ) has some positive circuit.o(nm) if so,infeasibility otherwise set W s = β s. Compute the longest paths from a dummy source node to any node i on G with on each component value V Ws and on each intermediate arc between C s, C s value V Ws. O(nm) t σ (< i, >) = longest path to i. Cyclic scheduling for EPIT 00 p. 9/?

20 Example Graph with L W s H values: A periodic schedule: period of,,,6 is 0, period of,5 is Cyclic scheduling for EPIT 00 p. 0/?

21 Positive heights The problem has been studied earlier [Chretienne 85][many authors] Theorem. For any w max C circuit α(c) there is a periodic schedule, all tasks have period w. No need to compute strong components. Cyclic scheduling for EPIT 00 p. /?

22 K-periodic schedules Definition. In a K-periodic schedule σ, a schedule of K i occurrences of task i is repeated every W i time units: l l 0, t σ (< i,l + K i >) = t σ (< i,l >) + W i Lemma. The average cycle time of task i in a K periodic schedule is A(σ,i) = W i K i. Un ordonnancement -périodique (pour chaque tâche) Cyclic scheduling for EPIT 00 p. /?

23 Properties of the earliest schedule [Romanovskii 6,Chretienne 85, Munier 06] Theorem. If G is feasible, the earliest schedule σ is K-periodic and A(σ ) = max C C + α(c) After a transitory time, the earliest schedule becomes regular and its behavior is ruled by the critical circuit index. Remark. K might be large product of heights of critical circuits. Open questions Length of the transitory time? Exact measure of K. Cyclic scheduling for EPIT 00 p. /?

24 Stability/boundedness cycle time w cycle time w (L(a), H(a)) b(a) e(a) If w < w then b(a) produces pieces or results faster than e(a). Unstability Moreover the iteration delay: D σ (k) = max i T tσ (< i,k >) min i T tσ (< i,k >) k + + This might occur in earliest schedule and in periodic schedules for general uniform graphs. Cyclic scheduling for EPIT 00 p. /?

25 Example if task has processing time : (,0) (,0) (,) (,) (,) 5 (,) (,0) 6 (,0) There are strong components C = {,, }, C = {}, C = {5,6, } α(c ) =, α(c ) =, α(c ) = 5 The earliest schedule of i is periodic with period 6. The earliest schedule of {5, 6,} is periodic with period 5. Unstable schedule Here a static -periodic schedule with period 5 can be built. Cyclic scheduling for EPIT 00 p. 5/?

26 Cyclic problems with resources Complexity of a cyclic problem/ its non cyclic version. Tools for constructing periodic schedules: circuits and patterns Polynomial problems Are periodic schedules optimal? Decomposed scheduling: a general approach Approximation of decomposed scheduling ILP formulations Cyclic scheduling for EPIT 00 p. 6/?

27 Complexity Consider a problem resources prec C max which is NP-hard. A simple Reduction: precedence graph G Add two dummy nodes to G (source s, sink t with unit proce s G t set H(a) = 0 for all arcs in G and arcs from s and arcs to t. Add an arc from t to s with height. Any schedule σ of G is a sequence of schedules of G. Corollary. Its average cycle time A(σ ) is the mean of makespan of G schedules A(σ ) B C max (G) B P uniform prec, p i = A, pre assigned processors uniform prec A, cyclic job-shop with uniform constraints are NP-hard Cyclic scheduling for EPIT 00 p. /?

28 Circuits Consider a uniform graph G =circuit, such that for any arc a L(a) = p b(a) : usual precedence constraints. Lemma 5. In any schedule of G no more than H(G) tasks are performed in parallel. Corollary. [Munier 9] The problem P circuit,l(a) = p b(a) A is solvable in polynomial time. Idea : if H(G) m then any schedule meets the resource constraint. If H(G) > m, then we can reduce the height of some arcs so that H(G) = m without modifiying the lower bound on A(σ) max(max p i T i, p i i T m ) Cyclic scheduling for EPIT 00 p. 8/?

29 Patterns Let σ be a periodic schedule with unique period w. t σ (< i,k >) = t σ (< i, >) + (k )w Definition. The Pattern of σ is defined by: π σ (i) = t σ (< i, >)modw. The iteration setting of σ is η σ (i) Z, s.t. t σ (< i, >) = π σ (i) + η σ (i)w Cyclic scheduling for EPIT 00 p. 9/?

30 Patterns The pattern defines the schedule of tasks in an interval [kw, (k + )w] for enough large k The iteration setting indicates which occurrences of tasks are involved in this interval In the interval [kw, (k + )w], i starts at kw + π σ (i) its occurrence < i,k + η σ (i) > Cyclic scheduling for EPIT 00 p. 0/?

31 Feasibility of a pattern Question: Given a pattern π,and a period w is there an iteration setting η such that the uniform constraints are met by the periodic schedule? For an arc a, t(< e(a), >) t(< b(a), >) L(a) wh(a). Lemma 6. An iteration setting satisfies for any arc a L(a) + π(b(a)) π(e(a)) η(e(a)) η(b(a)) H(a) = E w,π (a) w Lemma. An iteration setting exists iff (G,E w,π ) has no positive circuit. can be checked in polynomial time Cyclic scheduling for EPIT 00 p. /?

32 Polynomial problems Corollary 5. if G has no other circuit than the non-reentrance loops, any pattern has an iteration setting. Theorem. [Munier 9] P acyclic unif orm prec A is sovlable in polynomial time. Idea: Consider tasks as independent, and schedule them on m processors using Mc-Naughton algorithm (preemptive scheduling). Use the schedule as a Pattern of a periodic schedule and compute the iteration setting. Theorem 5. dedicated processors acyclic unif orm prec A is solvable in polynomial time. In particular non-reentrant job-shop or flow-shop. Idea: schedule operations on each machine as independent. Set W = C max (which is a lower bound on A(σ) in this case). Use the schedule as a Pattern of a periodic schedule and compute the iteration setting. [Robert, Legrand] also used similar ideas to build schedules for a broadcasting problem on a heterogeneous platform with net contentions. Cyclic scheduling for EPIT 00 p. /?

33 Example h= 6 5 h= Cyclic scheduling for EPIT 00 p. /?

34 Optimality of periodic schedules? In general, periodic schedules are not optimal schedules. Example: cyclic problem with two processors h= 6 5 h= However, periodic schedules are simple to implement and easier to compute. Cyclic scheduling for EPIT 00 p. /?

35 ecomposed scheduling: a general approac Also known as Decomposed software pipelining[eisenbeis, Darte, Gasperoni, Schwiegelsohn, de dinechin, Munier,...] OR Build a non cyclic schedule S of T that meets the resource constraints and eventually some precedence constraints. Consider this schedule as a pattern, and set W = C max (S) Build a feasible iteration setting for the pattern. (if possible) Build an iteration setting. Deduce from the iteration setting precedence relations between tasks in a pattern Build a non cyclic schedule S of T that meets the resource constraints and these precedence constraints. Consider this schedule as a pattern, and set W = C max (S) Cyclic scheduling for EPIT 00 p. 5/?

36 Gasperoni-Schwiegelsohn algorithm An algorithm for P uniform with L(a) = p b(a) A. Compute an optimal schedule σ on infinitely many processors. Consider the pattern π σ and remove arcs a from G such that π σ (b(a)) + p b(a) > π σ (e(a)) G. Schedule G on the m processors according to a list scheduling algorithm schedule S. Set S as a pattern with w = C max (S) and combine with the iteration setting of σ periodic feasible schedule σ. Cyclic scheduling for EPIT 00 p. 6/?

37 Example (,) (,0) (,0) (,) (,) 5 (,) (,0) 6 (,0) G has arcs 6 5 Schedule on processors periodic schedule with period 6 Cyclic scheduling for EPIT 00 p. /?

38 Bounds Theorem 6. [Gasperoni-Schwiegelsohn] A(σ) ( m )A(σopt ) + max i T p i [Darte et al] generalized the algorithm by choosing a convenient G using retiming. Cyclic scheduling for EPIT 00 p. 8/?

39 Disjunctive constraints Assume that two tasks i and j use the same resource: k, l, either t σ (< i, k >) + p i t σ (< j, l >) or t σ (< j, l >) + p j t σ (< i, k >). If σ is periodic with period w, in time interval [xw, (x + )w)] there is only one occurrence < i, u i (x) >. < j, u i (x) + h ij > next occurrence of j. < j, u i (x) + h ij > < i, u i (x) > < j, u i (x) + h ij > < j, u i (x) + h ij > precedes < i, u i (x) >. Hence setting h ij Z as a variable, the disjunctive constraints can be expressed as: xw (x + )w t σ (< j, >) t σ (< i, >) p i wh ij t σ (< i, >) t σ (< j, >) p j wh ji h ij + h ji = Cyclic scheduling for EPIT 00 p. 9/?

40 ILP model General form of constraints: t σ (< e(a), >) t σ (< b(a), >) L(a) wh(a) Heights might be either fixed or variables. Additional linear constraints on variable heights might be added. Remark. For a fixed w, the constraint is linear. Remark. Once disjunctive variables of this ILP are fixed, it remains a uniform graph scheduling problem. Many problems with disjunctive resources (shop-problems, but also Hoist scheduling problems)[roundy],[brucker], [Levner et al] [Chu et al][lei et al]can be formulated, for a fixed w as an mixed integer linear program. Cyclic scheduling for EPIT 00 p. 0/?

41 Algorithms for disjunctive problems Branch and bound algorithms For cyclic job-shop problems [Roundy,Hanen, Kampmeyer and Brücker] For hoist scheduling[lei and Wang, Chu and Proth] Metaheuristics [Kampmeyer and Brücker] Cyclic scheduling for EPIT 00 p. /?

42 Some Open problems Transitory state of the earliest schedule. Complexity of the cyclic problem with unit lengths an processors. How to use results on scheduling problems in their cyclic version? Efficiency of Jackson algorithm for the -machine problem Specific list schedules for parallel machines problems. Defining regular dynamic policy and study their behaviour. Regularity of the schedule, performance bounds. Cyclic scheduling for EPIT 00 p. /?