UET Multiprocessor Scheduling Problems Nan Zang

Nan Zang Overview of the paper  Introduction  Classification  Complexity results for Pm | prec, p j = 1 | C max  Algorithms for Pk | prec, p j = 1 | C max  Future research directions and conclusions

Nan Zang Scheduling  Scheduling concerns optimal allocation of scarce resources to activities.  For example Class scheduling problems  Courses  Classrooms  Teachers

Nan Zang Problem notation (1)  A set of n jobs J = { J 1, J 2, ……, J n } The execution time of the job J j is p(J j).  A set of processors P = {P 1, P 2, ……}  Schedule Specify which job should be executed by which processor at what time.  Objective Optimize one or more performance criteria

Nan Zang Problem Notation (2)  α describes the processor environment Number of processors, speed, …  β provides the job characteristics release time, precedence constraints, preemption,…  γ represents the objective function to be optimized The finishing time of the last job (makespan) The total waiting time of the jobs 3-field notation α|β|γ (Graham et al.)

Nan Zang Job Characteristics  Release time (r j ) - earliest time at which job J j can start processing. - available job  Preemption (prmp) - jobs can be interrupted during processing.  Precedence constraints (prec) - before certain jobs are allowed to start processing, one or more jobs first have to be completed. - a ready job

Nan Zang Precedence constraints (prec) Definition  Successor  Predecessor  Immediate successor  Immediate predecessor  Transitive Reduction Before certain jobs are allowed to start processing, one or more jobs first have to be completed.

Nan Zang Precedence constraints (prec) Definition  Successor  Predecessor  Immediate successor  Immediate predecessor  Transitive Reduction One or more job have to be completed before another job is allowed to start processing.

Nan Zang Precedence constraints (prec) Definition  Successor  Predecessor  Immediate successor  Immediate predecessor  Transitive Reduction One or more job have to be completed before another job is allowed to start processing.

Nan Zang Special precedence constraints (1)  In-tree (Out-tree)  In-forest (Out-forest)  Opposing forest  Interval orders  Quasi-interval orders  Over-interval orders  Series-parallel orders  Level orders

Nan Zang Special precedence constraints (2) In-forest Out-tree In-tree Out-forest Opposing forest

Nan Zang UET scheduling problem formal definition Pm| prec, p j = 1 | C max (m≥1)  Processor Environment  m identical processors are in the system.  Job characteristics  Precedence constraints are given by a precedence graph;  Preemption is not allowed;  The release time of all the jobs is 0.  Objective function  C max : the time the last job finishes execution. ( If c j denotes the finishing time of J j in a schedule S, )

Nan Zang Gantt Chart A Gantt chart indicates the time each job spends in execution, as well as the processor on which it executes Time axis P1P1 P2P2 P3P3 Slot 1Slot 2

Nan Zang Overview of the paper  Introduction  Classification  Complexity results for Pm | prec, p j = 1 | C max  Algorithms for Pk | prec, p j = 1 | C max  Future research directions and conclusions

Nan Zang Classification Due to the number of processors  Number of processors is a variable (m) Pm | prec, p j = 1 | C max  Number of processors is a constant (k) Pk | prec, p j = 1 | C max

Nan Zang Classification Due to the number of processors  Number of processors is a variable (m) Pm| prec, p j = 1 | C max  Number of processors is a constant (k) Pk| prec, p j = 1 | C max

Nan Zang Pm| prec, p j = 1 | C max (1) Theorem 1 Pm| prec, p j = 1 | C max is NP-complete.  Ullman (1976) 3SAT ≤ Pm | prec, p j = 1 | C max 2. Lenstra and Rinooy Kan (1978) k-clique ≤ Pm| prec, p j = 1 | C max Corollary 1.1 The problem of determining the existence of a schedule with C max ≤3 for the problem Pm| prec, pj = 1 | C max is NP-complete.

Nan Zang Pm| prec, p j = 1 | C max (2) Mayr (1985)  Theorem 2 Pm | p j =1, SP | C max is NP-complete. SP: Series - parallel  Theorem 3 Pm | p j =1, OF | C max is NP-complete. OF: Opposing - forest

Nan Zang SP and OF  Series-parallel orders Does NOT have a substructure isomorphic to Fig 1.  Opposing-forest orders Is a disjoint union of in-tree orders and out-tree orders. Fig 2: Opposing forest a c b d Fig 1

Nan Zang Conclusion on Pm| prec, p j = 1 | C max  3SAT is reducible to the corresponding scheduling problem.  m is a function of the number of clauses in the 3SAT problem.  Results and techniques do not hold for the case Pk | prec, p j = 1 | C max

Nan Zang Classification  Number of processors is a variable (m) Pm | prec, p j = 1 | C max  Number of processors is a constant (k) Pk | prec, p j = 1 | C max

Nan Zang Optimal Schedule for Pk | prec, p j = 1 | C max  The complexity of Pk | prec, p j = 1 | C max is open.  8 th problem in Garey and Johnson’s open problems list.(1979)  One of the three problems remaining unsolved in that list  If k = 2, P2 | prec, p j = 1 | C max is solvable in polynomial time.  Fujii, Kasami and Ninomiya (1969)  Coffman and Graham (1972)  For any fixed k, when the precedence graph is restricted to certain special forms, Pk | prec, p j = 1 | C max turns out to be solvable in polynomial time.  In-tree, Out-tree, Opposing-forest, Interval orders…

Nan Zang Special precedence constraints  In-tree (Out-tree)  In-forest (Out-forest)  Opposing forest  Interval orders  Quasi-interval orders  Over-interval orders  Level orders

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang List scheduling policies (1)  Set up a priority list L of jobs.  When a processor is idle, assign the first ready job to the processor and remove it from the list L.

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang Graham ’ s list algorithm  Graham first analyzed the performance of the simplest list scheduling algorithm.  List scheduling algorithm with an arbitrary job list is called Graham’s list algorithm.  Approximation ratio for Pk | prec, p j = 1 | C max δ = 2-1/k. (Tight!) Approximation ratio is δ if for each input instance, the makespan produced by the algorithm is at most δ times of the optimal makespan.

Nan Zang Algorithms for Pk| prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang HLF algorithm (1)  T. C. Hu (1961)  Critical Path algorithm or Hu’s algorithm  Algorithm  Assign a level h to each job.  If job has no successors, h(j) equals 1.  Otherwise, h(j) equals one plus the maximum level of its immediate successors.  Set up a priority list L by nonincreasing order of the jobs’ levels.  Execute the list scheduling policy on this level based priority list L.

Nan Zang HLF algorithm (2) Level 3Level 2Level

Nan Zang HLF algorithm (3)  Time complexity O(|V|+|E|) (|V| is the number of jobs and |E| is the number of edges in the precedence graph)  Theorem 4 (Hu, 1961) The HLF algorithm is optimal for Pk | p j = 1, in-tree (out-tree) | C max.  Corollary 4.1 The HLF algorithm is optimal for Pk | p j = 1, in-forest (out-forest) | C max.

Nan Zang HLF algorithm (4)  N.F. Chen & C.L. Liu (1975) The approximation ratio of HLF algorithm for the problem with general precedence constraints: If k = 2, δ HLF ≤ 4/3. If k ≥ 3, δ HLF ≤ 2 – 1/(k-1). Tight!

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang MSF algorithm (1)  Algorithm:  Set up a priority list L by nonincreasing order of the jobs’ successors numbers. (i.e. the job having more successors should have a higher priority in L than the job having fewer successors)  Execute the list scheduling policy based on this priority list L.

Nan Zang MSF algorithm (2)

Nan Zang MSF algorithm (3)  Time complexity O(|V|+|E|)  Theorem 5 ( Papadimitriou and Yannakakis, 1979 ) The MSF algorithm is optimal for Pk | p j = 1, interval | C max.  Theorem 6 (Moukrim, 1999) The MSF algorithm is optimal for Pk | p j = 1, quasi-interval | C max.

Nan Zang Special precedence constraints  Interval orders Does NOT have a substructure isomorphic to Fig 1.  Quasi – interval orders Does NOT have a substructure isomorphic to TYPE I, II or III. Type I Type II Type III ed c ab d ba e c d b a e c a c b d Fig 1

Nan Zang MSF algorithm (4)  Ibarra & Kim (1976) The performance of MSF algorithm: If k = 2, δ MSF ≤ 4/3, and this bound is tight.  If k ≥ 3, no tight bound is known. δ MSF is at least 2-1/(k+1).

Nan Zang MSF algorithm (5)

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang CG algorithm (1)  Coffman and Graham (1972)  An optimal algorithm for P2 | prec, p j = 1 | C max.  Best approximation algorithm known for Pk | prec, p j = 1 | C max, where k ≥ 3.

Nan Zang  Definitions –Let IS(J j ) denote the immediate successors set of J j. –A job is ready to label, if all its immediate successors are labeled and it has not been labeled yet. –N(J j ) denotes the decreasing sequence of integers formed by ordering of the set { label (J i ) | J i  IS(J j ) }. –Let label(J j ) be an integer label assigned to J j. CG algorithm (2)

Nan Zang CG algorithm (3)  Assign a label to each job:  Choose an arbitrary task J k such that IS(J k ) = 0, and define label(J k ) to be 1  for i  2 to n do  R be the set of jobs that are ready to label.  Let J* be the task in R such that N(J*) is lexicographically smaller than N(J) for all J in R  Let label(J*)  i end for 2. Construct a list of jobs L = {J n, J n-1,…, J 2, J 1 } according to the decreasing order of the labels of the jobs. 3. Execute the list scheduling policy on this priority list L.

Nan Zang CG algorithm (4) N(J 9 )=(1) N(J 8 )=(7,6,5,4,3,2) N(J 10 )= N(J 11 )=N(J 12 )=(8)

Nan Zang CG algorithm (4)

Nan Zang CG algorithm (5)  Time complexity O(|V|+|E|)  Theorem 5 (Coffman and Graham, 1972) The CG algorithm is optimal for P2 | prec, p j = 1| C max.  Theorem 6 (Moukrim, 2005) The CG algorithm is optimal for Pk | p j = 1, over-interval | C max.

Nan Zang Special precedence constraints  Quasi – interval orders Does NOT have a substructure isomorphic to TYPE I, II or III.  Over – interval orders Does NOT have a substructure isomorphic to TYPE I or II. Type I Type II Type III ed c ab d ba e c d b a e c

Nan Zang CG algorithm (6) The performance of CG algorithm when k≥3  Lam and Sethi (1978) δ CG ≤ 2 – 2/k  Braschi and Trystram (1994) C max (S) ≤ (2 – 2/k) C max (S*) – (k – 2 – odd(k))/k (tight!) Note: S is a CG schedule. S* is an optimal schedule. If k is an odd, odd(k):=1; otherwise, odd(k):=0.

Nan Zang List Scheduling Policy Conclusions  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  Property  easy to implement  extended to the problem Pm | prec, pj = 1| C max  Research directions:  Allow priority lists to depend on the number k of processors.

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policy algorithm  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang FKN algorithm (1)  Fujii, Kasami and Ninomiya (1969)  First optimal algorithm for P2 | prec, p j = 1 | C max.  Basic Idea  Find a minimum partition of the jobs There are at most two jobs in each set. The pair of jobs in the same set can be executed together.  Make a valid schedule according to a particular order of the partition. (Some clever swap work needed!)  The length of the result schedule = value of the min partition  Can be solved by some maximum matching algorithm.

Nan Zang FKN algorithm (2)  Hard to extend!  FKN algorithm cannot be extended to k = 3 directly. Minimal partition is {J 1, J 5, J 6 } {J 4, J 2, J 3 } and |P|=2. However, The optimal C max corresponds partition {J 1, J 4 } {J 2,J 3,J 5 } {J 6 }

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang Merge Algorithm (1)  Dolev and Marmuth (1985)  Required input: An optimal schedule S for a high-graph H(G).  Merge algorithms show how to “ Merge” the known optimal schedule S with the remaining jobs.  Produce an optimal schedule for the whole job set G.

Nan Zang Merge Algorithm (2) Definitions height h(G) highest level of the vertices in G. median μ(G) height of kth highest component +1 high-graph H(G) a subgraph of G, made up of all the components which are strictly higher than the median. low-graph L(G) remaining subgraph of G, except for H(G)

Nan Zang Merge Algorithm (3) C1C1 C2C2 C3C3 C4C4 H(C 1 )=5H(C 2 )=3H(C 3 )=3H(C 4 )=2 K=3 3rd highest μ(G)=4 H(G)L(G) J3J3 J2J2 J 10 J4J4 J 5 J 6 J 7 J1J1 J9J9 J 12 J8J8 J 11 J 13 J 14 J 15 J 16

Nan Zang Merge Algorithm (4) Idea of the Merge Algorithm  If there is an idle period in S, then fill it with a highest initial vertices in L(G), and remove it from L(G)  Similar to HLF Algorithm

Nan Zang Merging Algorithm (5) J 10 J9J9 J8J8 J 11 J 12 J 13 J 14 J 15 J 16 S: Optimal schedule for H(G) L(G) S’: from merge algorithm

Nan Zang Merge Algorithm (6) Theorem 10 (Reduction theorem) Let G be a precedence graph and S be an optimal schedule for H(G). Then, the Merge algorithm finds an optimal schedule for the whole graph G in time and space O(|V|+|E|). Corollary 10 If H(G) is empty, then HLF is optimal for G.

Nan Zang Merge Algorithm (7) Why Merge algorithm is useful? 1. Find an optimal schedule for a subgraph H(G). 2. H(G) contains fewer than k- 1 components. Precedence Constraints Time complexity Level ordersO(n k-1 ) Opposing forestO(n 2k-2 logn) Bounded heightO(n h(k-1) ) Dolev and Marmuth How to use it? 1. If H(G) is easy to solve. 2. If every closed subgraph of G can be classified into polynomial number of classes.

Nan Zang Algorithms for Pk | prec, p j = 1 | C max  List scheduling policies  Graham’s list algorithm  HLF algorithm  MSF algorithm  CG algorithm  FKN algorithm (Matching algorithm)  Merge algorithm

Nan Zang Main results known LISTLIST Approxi mation ratio K=2 K≥3 IntreeOFInterval Quasi- interval Over- interval Arbitrary List3/22-1/k HLF4/3opt2-1/(k-1) MSF4/3opt ≥2-1/(k+1) CGopt ≈2-2/k FKNopt Mergeopt (Opt: We can get optimal solution in polynomial time. )

Nan Zang Overview of the paper  Introduction  Classification  Complexity result for Pm | prec, p j = 1 | C max  Algorithms for Pk | prec, p j = 1 | C max  Future research directions and conclusions

Nan Zang Future research directions (1)  Finding a new class of orders which can be solved by known algorithms or their generalizations.  over-interval (Moukrim, 2005)  Find an algorithm with a better approximation ratio for the UET scheduling problem.  CG algorithm (1972)  Ranade (2003) special precedence constraints (loosely connected task graphs ) δ = 1.875

Nan Zang Future research directions (2)  Use the UET multiprocessor scheduling algorithms to solve other related scheduling problems  CG algorithm is optimal for P2 | r j, p j = 1| C max.  HLF algorithm is optimal for P | r j, p j = 1, outtree| C max. (Huo and Leung, 2005)  MSF algorithm is optimal for P | p j = 1, cm=1, quasi-interval| C max. (Moukrim, 2003)  CG algorithm is optimal for P2| prmp, p j = 1| C max and ∑C j (Coffman, Sethuraman and Timkovsky, 2003)  The most challenging research task:  Solve the famous open problem Pk |p j = 1| C max (k≥3).

Thank You!