Workload-Minimizing Structures

We now consider the optimization of workload alone in the P2F and F2F models. We compare the workloads of optimal DAGs and trees with the omniscient algorithm, and show that in the F2F model there always exists an efficiently computed DAG with maximum workload equal toMW,P. We show that that the same is not possible for trees. For the P2F

model, we show that even the class of DAGs cannot always achieve maximum workload equal toMW,P. We then proceed to consider workload-depth tradeoffs in Section 2.1.5.

The F2F Model

We first show that in the F2F model, there is a DAG that achieves maximum workload equal to the omniscient optimal MW,P.

Theorem 4 In the F2F model, there always exists a DAG Gwhose maximum workload is equal to MW,P. In addition, there is an efficient algorithm for constructing G.

Proof Consider the smallest index, sayj, for which we have (Aj+...+An)

n−j+1 = maxi

(Ai+...+An)

n−i+1 .

If there exists a DAG with maximum workload equal toMW,P, then in that DAG,wj, ..., wn

should never get any task with difficultyx≤wj−1, otherwise at least one of them will have

workload larger than (Aj+...+An)

n−j+1 = maxi

(Ai+...+An)

n−i+1 , resulting in max workload larger than

MW,P. An immediate consequence is that the initial probability of workers j, ..., n, i.e.

Ij, ..., In must be zero and the only node that can have edges with non-zero probability to

them is worker (j−1).

That being said, ifj >1, in order to build the optimal DAG, we do the following: first we build the optimal DAG G1 forw1, ..., wj−1 andhA1, .., Aj−1i, then build the optimal DAG

G2 for wj, ..., wn and hAj, ..., Ani. To combine G1 and G2, we use the initial probability of

vertex i ≥ j in G2 as edge weight on the edge from worker (j−1) to i, and we set Ii to

0. It is easy to see that combining the two DAGs this way results in the optimal DAG for

w1, ..., wn andP.

This suggests a recursive procedure for building the optimal DAG when j > 1. However, we still need to deal with the case where j= 1 and the workload must be divided equally among all workers.

For this remaining case w1, ..., wn and hA1, ..., Ani are such that the omniscient algorithm

gives every worker the same workload, equal to (A1+...+An)

n . Algorithm 1 (which we call

WED) presents an efficient procedure with running timeO(n2) that generates a DAG with max workload (A1+...+An)

n for this case.

This algorithm is the recursive form of the following inductive construction:

Induction statement: for k ∈ _N workers, if P is such that MW,P = (A1+...+A_k k), then

ALGORITHM 1:Algorithm WED for Computing the Workload Equalizing DAG in the F2F Model

Data: A task distributionP =hA1, A2, ..., Ani with MW,P = (A1+...+A_n n) Result: A weighted DAGG

I1 ← (A1+A_nA2+...+A₁ n); V(G)← {w1}; E(G)← ∅; if n=1 then return; else P0 ← h(1−I1)A1+A2, A3, ..., Ani;

H ←WED(P0) i.e. the workload optimal DAG for P0;

for i←2 tondo

I_i0 ←the initial probability of workeriin H;

end

V(G)←V(G)∪V(H);

E(G)←E(H);

Add edges fromw1 to every vertex in V(H); for i←2 tondo

p1,i←Ii0;

Ii ←(1−I1)Ii0; end

Induction basis: For k = 1 workers, the optimal DAG is trivial. It is a single node and we have to give all the workloadA1 to the only worker available i.e. I1 = 1.

Induction step: Suppose for a set of k < n workers, we know how to build the optimal DAG to have the workload of every worker equal to (A1+...+Ak)

k . Now suppose we are given

a set ofnworkers with abilityw1, w2, ..., wn, and the task distribution hA1, .., Anisuch that

MW,P = (A1+...+A_n n).

Note that in order to have a DAG in which everyone has workload equal to (A1+...+An)

n ,

this must hold for w1 as well. Sincew1’s workload is equal to I1×A1,I1 must be equal to (A1+...+An)

nA1 then.

To build the rest of the optimal DAG, observe that for workers w2, ..., wn, there is no

difference between a task in [0, w1] and one in (w1, w2] i.e. all of them can solve both.

Therefore we can assume the remaining mass on [0, w1] is actually on (w1, w2] and find the

optimal DAG for the remaining workers and task distribution.

According to the induction hypothesis, for the task distributionhA1(1−I1)+A2, A3, ..., Ani,

we can find the optimal DAG say H, that gives equal workloads tow2, ..., wn. Suppose the

initial probability on node iinH is I_i0. Now we construct G, in the following way: Add a vertex forw1 toH and set the initial probability of this vertex to (A1+...+A_nA1 n). Then set the

probability on the edge fromw1 towi, to I_i0. Also set the initial probability of vertexi >1

inGtoIi=Ii0(1−I1).

We claim that inG,w1, ..., wn all get equal workloads. For w1 this obviously holds due to

the way we chose I1. For the nodes in H let’s look at the task distribution they receive

from outside of H, i.e. either initially or from w1. With probability (1−I1), a task is

initially forwarded to a node in H; we know that such task, has distributionhA1, ..., Ani.

With probability I1 the task is forwarded initially to w1, and if w1 forwards the task to

some one inH, all we know about the task is that it is from the distributionh0, A2, ..., Ani.

get the distribution I1h0, A2, ..., Ani+ (1−I1)hA1, A2, ..., Ani=h(1−I1)A1, A2, ..., Ani.

Now according to the induction hypothesis, H is optimal for this task distribution, giv- ing w2, ..., wn equal workload. Therefore we can conclude G is giving equal workload to

everyone.

We next show that in the F2F model, the weaker class of trees doesnot suffice to achieve maximum workloadMW,P.

Theorem 5 In the F2F model, forn >2 workers, there are worker abilitiesW and a task distribution P for which the maximum workload in any tree is strictly larger than 6₅MW,P.

Proof Let P = hA1, A2, A3i where A1 = ₃1, and A2 = (1₃ +) and A3 = (1₃ −) with

0 < ≤ 1₃. We claim that for this task distribution the optimal DAG is unique, and because this unique DAG is not a tree, we conclude that there exists no tree with the same (optimal) max workload.

Obviously MW,P = 1₃ in this case. To have a DAG with the same max workload, we must

initially give every task to worker 1. So the workload of worker 2 isp1,2(1₃ +) which must

be equal to 1₃, thereforep1,2 = _1/3+1/3 <1, and as a resultp1,3 >0, meaning that the optimal

DAG is not a tree. It is easy to see that forP1

3, the maximum workload of the optimal tree is 2₅, proving the claimed lower bound.

The above theorem confirms that in the F2F model, the class of trees is not as powerful as the class of DAGs in terms of the maximum workload. One question we would like to address is how large this gap can be. Later in Section 2.1.5, we will see that for any task distributionP, we can find a treeT such that the maximum workload in T is less than or equal to 4MW,P, yielding a constant-factor approximation to the optimal DAG.

The P2F Model

We next see that for the P2F model, even the optimal DAG cannot always achieve the omniscient optimal maximum workload. This already holds for a very simple case of two workers, and the task distribution P =h1

2, 1 2i.

Theorem 6 In the P2F model, there are worker values W and a distribution P for which the maximum workload of the optimal DAG is strictly larger than 4₃MW,P.

In Section 2.1.5 we will see that in the P2F model, for any task distributionP, the maximum workload in the optimal tree (and as a result optimal DAG), is less than or equal to 4MW,P.