Approximation Algorithms (Load Balancing)arashr/lecture24.pdf · Approximation Algorithms (Load Balancing) July 16, 2014 Approximation Algorithms (Load Balancing)

Approximation Algorithms (Load Balancing)

July 16, 2014


Problem Definition :We are given a set of n jobs {J1, J2, . . . , Jn}.Each job Ji has a processing time ti ≥ 0.

We are given m identical machines.

Goal :

We want to assign (load) the jobs to machines suchthat the maximum load is minimized.In other words, we would like to balance the loads.


Problem Definition :We are given a set of n jobs {J1, J2, . . . , Jn}.Each job Ji has a processing time ti ≥ 0.

We are given m identical machines.

Goal :

We want to assign (load) the jobs to machines suchthat the maximum load is minimized.In other words, we would like to balance the loads.


Let A(i) be the set of jobs that are assigned to Mi .Then the load of Mi denoted by Ti =

∑j∈A(i) ti .

We wish to minimize T = maxi Ti .

The load balancing problem in NP-complete. Evenwhen there are two machines.


Let A(i) be the set of jobs that are assigned to Mi .Then the load of Mi denoted by Ti =

∑j∈A(i) ti .

We wish to minimize T = maxi Ti .

The load balancing problem in NP-complete. Evenwhen there are two machines.


2

J1 J2 J3 J4 J5 J6

2 6 3 4 2

M1 M2 M3

2

6

3

2

4

2


2

J1 J2 J3 J4 J5 J6

2 6 3 4 2

M1 M2 M3

2

6

3

2

4

2


2

J1 J2 J3 J4 J5 J6

2 6 3 4 2

M1 M2 M3

2

6

3

2

4

2


Greedy-Balance

1. Set Ti = 0 and A(i) = ∅ for all machines Mi .

2. for j = 1 to n

3. Let Mi be a machine with minimum load (mink Tk).

4. Assign job j to machine Mi .

5. Set A(i)← A(i) ∪ {Jj}

6. Set Ti ← Ti + tj


1) The optimal load T ∗ is at least

T ∗ ≥ 1

m

∑j

tj

2) T ∗ ≥ maxj tj .

Lemma

Algorithm Greedy-Balance produces an assignment of jobs tomachines with max load T ≤ 2T ∗.



T ∗ ≥ 1

m

∑j

tj

2) T ∗ ≥ maxj tj .

Lemma




T ∗ ≥ 1

m

∑j

tj

2) T ∗ ≥ maxj tj .

Lemma



Lemma


Proof.

Consider the time we add job j into machine Mi . The load ofmachine Mi was Ti − tj before adding Jj to Mi .

Also Ti − tj was

the smallest load. Every other machine has load at least Ti − tj .Therefore :

m(Ti − ti ) ≤∑k

Tk

Also we know that∑

k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m

∑j tj ≤ T ∗. Also we know that tj ≤ T ∗.

Therefore load of Mi after adding Jj is Ti = (Ti − tj) + tj ≤ 2T ∗.


Lemma


Proof.

Consider the time we add job j into machine Mi . The load ofmachine Mi was Ti − tj before adding Jj to Mi . Also Ti − tj was



Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m




Lemma


Proof.




Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m




Lemma


Proof.




Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m




Lemma


Proof.




Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m

∑j tj ≤ T ∗.

Also we know that tj ≤ T ∗.



Lemma


Proof.




Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m




Lemma


Proof.




Tk


k Tk ≤∑

j tj . Therefore

Ti − tj ≤ 1m




The Greedy-Balance could actually be as close as possible to 2T ∗.

1

J1 J2

1

M1 M2 M4

4

1

J3

1

J4

1

J5

1

J6

1

J7

1

J8

1

J9 J10

1

J11

1

J12

1

J13

4

M3 M1 M2 M3 M4

Greedy Algorithm Optimal


The Greedy-Balance could actually be as close as possible to 2T ∗.

1

J1 J2

1

M1 M2 M4

4

1

J3

1

J4

1

J5

1

J6

1

J7

1

J8

1

J9 J10

1

J11

1

J12

1

J13

4

M3 M1 M2 M3 M4

Greedy Algorithm Optimal


In general suppose there are m machines and n = m(m − 1) + 1jobs.

The first m(m − 1) jobs each with time tj = 1 and the last jobn = m(m − 1) + 1 has time tn = m.

The optimal has T ∗ = m while the Greedy algorithms has maxload 2m − 1.










An Improved Approximation Algorithm

Sort-Balance

1. Set T0 = 0 and A(i) = ∅ for all machines Mi .

2. Sort the jobs in decreasing order of processing times tj .

3. Assume t1 ≥ t2 ≥ ... ≥ tn.

4. for j = 1 to n

5. Let Mi be a machine with minimum load (mink Tk).

6. Assign job j to machine Mi .

7. Set A(i)← A(i) ∪ {Jj}

8. Set Ti ← Ti + tj


If there are more than m jobs, then T ∗ ≥ 2tm+1.

Lemma

Algorithm Sort-Balance produces an assignment of jobs tomachines with max load T ≤ 3

2T∗.

Using similar analysis as in the previous lemma (leave it as exercise)


If there are more than m jobs, then T ∗ ≥ 2tm+1.

Lemma

Algorithm Sort-Balance produces an assignment of jobs tomachines with max load T ≤ 3

2T∗.

Using similar analysis as in the previous lemma (leave it as exercise)


There exists an algorithm that find a solution for Load balancingvery close to optimal T ∗

In fact there is an algorithm that for every ε > 0 it finds a solutionthat is not worse that (1 + ε)T ∗.

But the running time of the algorithm is

O(n(1ε)1.5)

where n is the number of jobs.





O(n(1ε)1.5)






O(n(1ε)1.5)



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Approximation Algorithms (Load Balancing)arashr/lecture24.pdf · Approximation Algorithms (Load Balancing) July 16, 2014 Approximation Algorithms (Load Balancing)