Parallel Algorithms for Two Processors Precedence Constraint Scheduling

Abstract

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: Name Two processor precedence constraint scheduling Input A directed acyclic graph Output A minimum length schedule preserving the precedence constraints.Preliminaries The algorithm assume that tasks are partitioned into levels as follows: (i) Every task will be assigned to only one level (ii) Tasks having no successors will be assigned to level 1 and (iii) For each level i, all tasks which are immediate predecessors of tasks in level i will be assigned to level i + 1.Clearly topological sort will accomplish the above partition, and this can be done by an NC algorithm that uses O(n 3 ) processors and O(log n) time, see [3]. Thus, from now on, it is assumed that a level partition is given as part of the input. For sake of convenience two special tasks, t 0 and t * are added, in such a way that the original graph could be taught as the graph formed by all tasks that are successors of t 0 and predecessors of t * . Thus t 0 is a predecessor of all tasks in the system

The sorted nodes in leveled DAG division algorithm and critical path on processor

“…SLNDD is calculated with a parallel correlative study among SLNDD and the HEFT algorithm for execution. Based on these results, the SNLDD algorithmic approach excels the HEFT algorithm in conditions of schedule distance, time, performance, and characteristics of system behavior (Serna, 2008).…”