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Given <i>n</i> unit execution time (UET) tasks whose precedence constraints form a directed acyclic graph, the arcs are associated with unit communication time (UCT) delays. The problem is to schedule the tasks on two identical processors in order to minimize the makespan. Several polynomial algorithms in the literature are proposed for special classes of digraphs, but the complexity of solving this problem in general case is still a challenging open question. We present in this paper an <i>O</i>(<i>n</i>) time algorithm to compute an optimal schedule for the class of bipartite digraphs of depth one.

The problem of scheduling a set of tasks on a set of identical processors under a precedence relation has been studied for a long time. A general description of the problem is the following. There are n tasks that have to be executed by m identical processors subject to precedence constraints and (may be without) communication delays. The objective is to schedule all the tasks on the processors such that the makespan is the minimum. Generally, this problem can be represented by a directed acyclic graph

According to the three field notation scheme introduced in [

A large amount of work in the literature studies this problem with a restriction on its structure: the time of execution of every task is one unit execution time (UET), the number of processors m is fixed, the communication delays are neglected, constant or one unit (UCT), or special classes of task graph are considered. We find In this context, the problem

The problem of two processors scheduling with communication delays is extensively studied [

A challenging open problem is the two processors scheduling with UET-UCT, i.e. the problem

A schedule UET-UCT on two processors for a general directed acyclic digraph

a)

b) If

A time t of a schedule

A schedule

Let

Definition 1 Let

The definition of a natural schedule

1) The number of sources executed on

2) If

a)

b) If

3) If

a)

b) If

4)

Without loss of generality, we can suppose that both idle times

The idea of solving the problem

Lemma 2 Assume that

1) The two processors

2) The processor

a) Every vertex of B is universal except exactly one.

b) Every vertex of W is universal except exactly one.

Proof. 1) If G is a bipartite complete then obviously

2) Assume that

The inverse, suppose that every vertex of B is universal except exactly one. Since

Notice that if B (or W) contains exactly one non-universal vertex b then the vertex of W which is independent of b is also non-universal but it is not necessary unique (see

Algorithm Schedule_|B|_is_even (G)

If

Schedule

Schedule

Else if

Let

Schedule b on

Schedule

Schedule

Else let

Schedule

Schedule

Schedule

Schedule

Lemma 3 Assume that

1) The processor

2) The processor

a) There is

b) For every

3) The processor

a) There is

b) For every

Proof. 1) If G is a bipartite complete then obviously

2) Assume that

The inverse, by 1, the processor

3) Assume that

To construct a natural schedule for G when

Procedure Two_Vertices (G)

For

For

If

Return 1.

Algorithm Schedule_|B|_is_odd (G) constructs a natural schedule for

Algorithm Schedule_|B|_is_odd (G)

If

Schedule a vertex

Schedule a vertex

Schedule

Schedule

Else if

Let

Schedule b on

Schedule w on

Schedule

Schedule

Else If Two_Vertices (G) = 1 then

Let

Schedule

Schedule w on

Schedule

Schedule a vertex

Schedule

Else let

Let

Schedule

Schedule

Schedule

Schedule

We assume that

The worst case of this Procedure occurs when its result is 1. In this case, for any

We have presented an

This research is funded by the Deanship of Research and Graduate Studies in Zarqa University/Jordan. The author is grateful to anonymous referee’s suggestion and improvement of the presentation of this paper.

RuzaynQuaddoura, (2016) An O(n) Time Algorithm for Scheduling UET-UCT of Bipartite Digraphs of Depth One on Two Processors. American Journal of Operations Research,06,75-80. doi: 10.4236/ajor.2016.61010