Automatic Scheduling of Hypermedia Documents with Elastic Times

Transcription

1 Automatic Scheduling of Hypermedia Documents with Elastic Times Celso C. Ribeiro Joint work with M.T. Medina and L.F.G. Soares Computer Science Department Catholic University of Rio de Janeiro, Brazil Workshop on New Trends in Scheduling Parallel and Distributed Systems Luminy, October 1-5, 2001

2 Automatic Scheduling 2 Contents Contents: Introduction Problem formulation Primal heuristic Lagrangean relaxation Dual heuristic Concluding remarks

3 Automatic Scheduling 3 Introduction Hypermedia systems should have some desirable features, such as: supporting a rich set of capabilities for each media segment (video, audio, text, pictures, animation,... ): alternatives for different presentations presentation duration flexibility behavior changes during presentation allowing the explicit definition of temporal relationships among media segments with non-deterministic times supporting a temporal formatting algorithm that considers non-determinism allows on-the-fly corrections whenever unpredictable events (network delays, user interactions) occur

4 Automatic Scheduling 4 Introduction A presentation event is a media segment presentation d min : minimum allowed presentation time d max : maximum allowed presentation time d ideal :durationwithbestpresentationquality c: cost of shrinking or stretching the presentation duration

5 Automatic Scheduling 5 Introduction Scheduling algorithm: determination of starting times and optimal durations d opt (which will be tried by the formatter) so as to ensure spatial and temporal consistency at compile time: build an execution plan to guide the scheduling of the object presentation at run time: online adjustments to handle unpredictable events (network delays, user interactions) Difficult of performing fast rescheduling in run time led the systems that make time adjustments to make them only at compile time. Other systems: Firefly, Isis, Madeus (linear programming, PERT, constraints,... )

6 Automatic Scheduling 6 Introduction Problem: find (at compile time) the otimal starting times and durations of objects to be presented, to ensure spatial and temporal consistency of the presentation while respecting limits on shrinking and stretching the ideal duration of each object. Objectives: (1) minimize the total cost of shrinking and stretching objects; (2) minimize the number of presentation events that will have their durations different than the ideal ones; (3) spread the changes in object duration proportionally to their duration; and (4) find the shortest and longest presentation durations. Objectives (1), (3), and (4) can be dealt with at compile time by linear programming models.

7 Automatic Scheduling 7 Introduction Although conflicting, objectives (1) minimize the total cost of shrinking and stretching objects and (2) minimize the number of presentation events that will have their durations different than the ideal ones are the most natural. Possible strategy to handle both of them:select the best solution with respect to (2) among all those which optimize (1). The combinatorial nature of the minimization of the number of objects whose ideal duration is modified makes objective (2) the most difficult to be tackled by combinatorial algorithms. Extendable mixed 0-1 LIP formulation using objective (2) Fast and effective primal heuristic based on variable fixations Lagrangean relaxation and dual heuristic

8 Automatic Scheduling 8 Problem formulation Hypermedia document represented by a directed graph G =[N,A]: set Ns = {1,...,n} of synchronization points where presentation events either start or stop set N = {0} {1,...,n} {n +1} of nodes, where 0 and n +1 denote respectively the initial and the final nodes set A of arcs: presentation events are arcs (i, j) with both extremities i, j {1,...,n} and durations d min ij, d ideal ij,andd max ij dummy initial arcs (0,j)withj {1,...,n} and durations d min 0j =0,d ideal 0j =0,andd max 0j = dummy final arcs (i, n +1)withi {1,...,n} and durations d min i,n+1 =0,dideal i,n+1 =0,anddmax i,n+1 =

9 Automatic Scheduling 9 Problem formulation Variables: continuous variable ti 0, for every i N time in which synchronization takes place at i Ns presentation starting time: t0 = 0 end of presentation: tn+1 continuous variable xij [d min ij,d max ij ], for every (i, j) A duration of each presentation (or dummy) event (i, j) A binary variable yij {0, 1}, for every event (i, j) A yij = 1, if the ideal duration of event (i, j) ismodified yij =0,otherwise artificial costs cij associated to every arc (i, j) A cij =1,if(i, j) is a presentation event cij = 0, otherwise (i.e., (i, j) isadummyevent)

10 Automatic Scheduling 10 Problem formulation Formulation: mixed 0-1 integer programming problem ASHD z =min with: (i,j) A cij yij ti tj + xij =0 (i, j) A (1) xij bij yij d ideal ij (i, j) A (2) xij +aij yij d ideal ij (i, j) A (3) yij {0, 1} (i, j) A (4) ti 0 i N \{0} t0 =0 = d min d ideal aij ij ij = d max d ideal bij ij ij

11 Automatic Scheduling 11 Problem formulation Formulation: mixed 0-1 integer programming problem ASHD z =min with: (i,j) A cij yij ti tj + xij =0 (i, j) A (1) xij bij yij d ideal ij (i, j) A (2) xij +aij yij d ideal ij (i, j) A (3) yij {0, 1} (i, j) A (4) ti 0 i N \{0} t0 =0 = d min d ideal aij ij ij = d max d ideal bij ij ij Open question: complexity of ASHD?

12 Automatic Scheduling 12 Primal heuristic ASHD is the LP relaxation of ASHD: yij {0, 1} 0 yij 1 Primal heuristic PH for problem ASHD: procedure PH 1 Solve ASHD and let (y, x, t) be its optimal solution; 2 while (i, j) A :0< y ij < 1do 3 Fix to one in ASHD all variables yij such that y ij =1; 4 Fix to zero in ASHD all variables yij such that y ij =0; 5 Variable selection: select and fix at either zero or one in ASHD a variable yst with a fractional value 0 < y st < 1inASHD; 6 Solve ASHD and let (y, x, t) be its optimal solution; 7 end while; 8 return y; end PH;

13 Automatic Scheduling 13 Primal heuristic procedure Variable selection 1 if (i, j) A, i = 0,j = n +1:0< y ij α then do 2 (s, t) argmin (i,j) A,i =0,j =n+1 {y ij :0< y ij α}; 3 Fixyst to zero in ASHD; 4 if ASHD is not feasible then fix yst to one in ASHD; 5 else if (i, j) A, i = 0,j = n +1:1 α y ij < 1thendo 6 (s, t) argmax (i,j) A,i =0,j =n+1 {y ij :1 α y ij < 1}; 7 Fix yst to one in ASHD; 8 else do 9 (s, t) argmin (i,j) A,i =0,j =n+1 min{y ij, 1 y ij }; 10 if yst 0.5 thendo 11 Fix yst to zero in ASHD; 12 if ASHD is not feasible then fix yst to one in ASHD; 13 else fix yst to one in ASHD; 14 return (s, t); end Variable selection;

14 Automatic Scheduling 14 Computational results PH Primal heuristic UltraSPARC station, CPLEX 5.0, 35 instances, n = Best value Time (s) Duality Instance Nodes Arcs PH Exact PH Exact gap (%) a a a c c > 4hs 52.3 d d e e maximum size of the enumeration tree exceeded

15 Automatic Scheduling 15 Computational results PH Conclusions: PH found good solutions within small computation times for real-size instances. Reasonable computation times, except for very large instances (500 or more nodes). PH found the optimal solutions for two of seven instances for which the exact optimal is known and was off by exactly one unit for two others. Average and maximum gaps are respectively 54.5% and 59.6%. Very large duality gaps even for small instances for which PH found the optimal solution:

16 Automatic Scheduling 16 Computational results PH Conclusions: PH found good solutions within small computation times for real-size instances. Reasonable computation times, except for very large instances (500 or more nodes). PH found the optimal solutions for two of seven instances for which the exact optimal is known and was off by exactly one unit for two others. Average and maximum gaps are 54.5% and 59.6%. Very large duality gaps even for small instances for which PH found the optimal solution: conjecture: solutions are quite good, but bounds are poor search for polyhedral cuts or valid inequalities should be pursued if one wants to develop efficient exact methods

17 Automatic Scheduling 17 Lagrangean relaxation Lagrangean dual of problem ASHD: Associate non-negative dual multipliers λ + ij and λ ij to constraints (2) and (3) for each arc (i, j) A: Lagrangean function: L(x, y, t, λ +, λ )= cij yij+λ + ij (x ij bij yij d ideal ij )+λ ij ( x ij+aij yij+d ideal ij ) (i,j) A Dual function: w(λ +, λ )=min L(x, y, t, λ +, λ ) with: ti tj + xij =0 (i, j) A (1) yi,j {0, 1} (i, j) A (4) d min ij xij d max ij (i, j) A (5) ti 0 i N \{0} t0 =0.

18 Automatic Scheduling 18 Dual heuristic Dual heuristic DH: Combines the primal heuristic and the solution of the dual problem by subgradient optimization. Uses the multipliers obtained along the solution of the dual problem to periodically generate reduced costs c ij associated with the 0-1 variables yij. Periodically applies the primal heuristic PH. Reduced costs c ij replacing the original costs c ij in the primal heuristic are computed according with two different schemes: c ij = c ij + aij λ ij b ij λ + ij (Lagrangean multiplipliers) c ij =(1 ŷ ij) cij (primal aproximation) Stopping criteria: total number of iterations of the subgradient algorithm equals 1200 or the relative difference between two consecutive dual function values becomes less than 10 4.

19 Automatic Scheduling 19 Dual heuristic Dual heuristic DH for problem ASHD: procedure DH 1 Apply PH to compute a feasible solution y; 2 Set the upper bound zbest (i,j) A c ij y ij and ybest y; 3 while stopping criteria not attained do 4 Perform a certain number of subgradient iterations; 5 Recompute reduced costs c using the current dual multipliers (λ +, λ ); 6 Apply PH using c to compute a new feasible solution y; 7 if (i,j) A c ij y ij <zbest 8 then set zbest (i,j) A c ij y ij and ybest y; 9 end; 10 return y; end DH;

20 Automatic Scheduling 20 Computational results DH Dual heuristic (1/2) DH PH Exact Instance Nodes Arcs zbest s zbest s z a a b b b c c c d d d

21 Automatic Scheduling 21 Computational results DH Dual heuristic (2/2) DH PH Exact Instance Nodes Arcs zbest s zbest s z e e f f f g

22 Automatic Scheduling 22 Computational results DH Conclusions: The dual heuristic DH consistently improves upon the solutions found by the primal heuristic PH, finding better solutions for 17 out of the 35 test problems. However, the computational times observed for the dual heuristic DH are much larger. Need for better/faster construction algorithms and local search procedures (work-in-progress) based on variable complementation.

23 Automatic Scheduling 23 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems.

24 Automatic Scheduling 24 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems. Dual heuristic: better solutions, but at the cost of much larger computation times.

25 Automatic Scheduling 25 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems. Dual heuristic: better solutions, but at the cost of much larger computation times. Establish the computational complexity of problem ASHD.

26 Automatic Scheduling 26 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems. Dual heuristic: better solutions, but at the cost of much larger computation times. Establish the computational complexity of problem ASHD. Replace CPLEX by a customized LP algorithm to speedup the solution of the linear relaxation ASHD.

27 Automatic Scheduling 27 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems. Dual heuristic: better solutions, but at the cost of much larger computation times. Establish the computational complexity of problem ASHD. Replace CPLEX by a customized LP algorithm to speedup the solution of the linear relaxation ASHD. Improved construction procedures and local search strategies.

28 Automatic Scheduling 28 Concluding remarks Concluding remarks and extensions: Primal heuristic: effective in finding good solutions in small processing times (real-size instances); quite promising not only in algorithmic terms, but also concerning its integration within existing document formatting systems. Dual heuristic: better solutions, but at the cost of much larger computation times. Establish the computational complexity of problem ASHD. Replace CPLEX by a customized LP algorithm to speedup the solution of the linear relaxation ASHD. Improved construction procedures and local search strategies. Integration of such heuristics and the bicriteria optimization approach within an existing document formatter to perform compile time scheduling and even run time adjustments.

29 Automatic Scheduling 29 Paper and slides Automatic Scheduling of Hypermedia Documents with Elastic Times Maira T. Medina Celso C. Ribeiro Luiz Fernando G. Soares Computer Science Department Catholic University of Rio de Janeiro Paper: //www-di.inf.puc-rio.br/ celso/artigos/ashd.pdf Slides: //www-di.inf.puc-rio.br/ celso/talks/luminy.pdf