Parallel Algorithms for Two Processors Precedence Constraint Scheduling

In the general form of multiprocessor precedence scheduling problems a set of n tasks to be executed on m processors is given. Each task requires exactly one unit of execution time and can run on any processor. A directed acyclic graph specifies the precedence constraints where an edge from task x to task y means task x must be completed before task y begins. A solution to the problem is a schedule of shortest length indicating when each task is started. The work of Jung, Serna, and Spirakis provides a parallel algorithm (on a PRAM machine) that solves the above problem for the particular case that m = 2, that is where there are two parallel processors. The two processor precedence constraint scheduling problem is defined by a directed acyclic graph (dag) G = (V,E). The vertices of the graph represent unit time tasks, and the edges specify precedence constraints among the tasks. If there is an edge from node x to node y then x is an immediate predecessor of y. Predecessor is the transitive closure of the relation immediate predecessor, and successor is its symmetric counterpart. A two processor schedule is an assignment of the tasks to time units 1, . . . , t so that each task is assigned exactly one time unit, at most two tasks are assigned to the same time unit, and if x is a predecessor of y then x is assigned to a lower time unit than y. The length of the schedule is t. A schedule having minimum length is an optimal schedule. Thus the problem is the following: