Modeling and Optimization of Traffic Flow in Urban Areas - Czech ...

4.3 Implemented Algorithm 49unit j on the processor k then x ijk = true, otherwise x ijk = false. Foreach task T i , where i = 1 . . . n, there are S × R Boolean variables, where Sdenotes the maximum number of time units and R denotes the total numberof processors.The Boolean variables are constrained by the three following rules (modestadaptation of [Memi 02]):1. For each task, exactly one of the S × R variables has to be equalto true. Therefore two clauses are generated for each task T i . The firstguarantees having at most one variable equal to true:(¯x i11 ∨ ¯x i21 ) ∧ · · · ∧ (¯x i11 ∨ ¯x iSR ) ∧ · · · ∧ (¯x i(S−∞)R ∨ ¯x iSR ).The second guarantees having at least one variable equal to true: (¯x i11 ∨¯x i21 ∨ · · · ∨ ¯x i(S−∞)R ∨ ¯x iSR ).2. If there are a precedence constraints such that T u is the predecessorof T v , then T v cannot start before the execution of T u is finished. Therefore,x ujk → (¯x v1l ∧ · · · ∧ ¯x vjl ∧ ¯x v(j+1)l ∧ · · · ∧ ¯x v(j+pu−1)l) for all possible combinationsof processors k and l, where p u denotes the processing time of taskT u .3. At any time unit, there is at most one task executed on a givenprocessor. For the couple of tasks with a precedence constrain this rule isensured already by the clauses in the rule number 2. Otherwise the set ofclauses is generated for each processor k and each time unit j for all couplesT u , T v without precedence constrains in the following form:(x ujk → ¯x vjk ) ∧ (x ujk → ¯x v(j+1)k ) ∧ · · · ∧ (x ujk → ¯x v(j+pu−1)k).The toolbox cooperates with the zChaff solver [Mosk 01] to decidewhether the set of clauses is satisfiable. If it is, the schedule within S timeunits is feasible. An optimal schedule is found in an iterative manner. First,the List Scheduling algorithm (see 4.3.1) is used to find the initial value of S.Then the algorithm iteratively decreases the value of S by one and tests thefeasibility of the solution. The iterative algorithm finishes when the solutionis not feasible.An example of the P |prec|C max problem can be taken from the digitalsignal processing area. A typical scheduling problem is to optimize thespeed of a computation loop, e.g constituting the Jaumann wave digital filter[Groo 92]. The goal is to minimize the computation time of the filterloop, shown as a directed acyclic graph in Fig. 4.8. The nodes in the graphrepresent the tasks (i.e. operations of the loop) and the edges represent the