Approximation Algorithms (Load Balancing)

Transcription

1 July 6, 204

2 Problem Definition : We are given a set of n jobs {J, J 2,..., J n }. Each job J i has a processing time t i 0. We are given m identical machines.

3 Problem Definition : We are given a set of n jobs {J, J 2,..., J n }. Each job J i has a processing time t i 0. We are given m identical machines. Goal : We want to assign (load) the jobs to machines such that the maximum load is minimized. In other words, we would like to balance the loads.

4 Let A(i) be the set of jobs that are assigned to M i. Then the load of M i denoted by T i = j A(i) t i. We wish to minimize T = max i T i.

5 Let A(i) be the set of jobs that are assigned to M i. Then the load of M i denoted by T i = j A(i) t i. We wish to minimize T = max i T i. The load balancing problem in NP-complete. Even when there are two machines.

6 J J 2 J 3 J 4 J 5 J M M 2 M 3

7 J J 2 J 3 J 4 J 5 J M M 2 M 3

8 J J 2 J 3 J 4 J 5 J M M 2 M 3

9 Greedy-Balance. Set T i = 0 and A(i) = for all machines M i. 2. for j = to n 3. Let M i be a machine with minimum load (min k T k ). 4. Assign job j to machine M i. 5. Set A(i) A(i) {J j } 6. Set T i T i + t j

10 ) The optimal load T is at least T m j t j

11 ) The optimal load T is at least T m j t j 2) T max j t j.

12 ) The optimal load T is at least T m j t j 2) T max j t j. Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T.

13 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i.

14 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore :

15 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore : m(t i t i ) k T k

16 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore : m(t i t i ) k T k Also we know that k T k j t j. Therefore

17 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore : m(t i t i ) k T k Also we know that k T k j t j. Therefore T i t j m j t j T.

18 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore : m(t i t i ) k T k Also we know that k T k j t j. Therefore T i t j m j t j T. Also we know that t j T.

19 Lemma Algorithm Greedy-Balance produces an assignment of jobs to machines with max load T 2T. Proof. Consider the time we add job j into machine M i. The load of machine M i was T i t j before adding J j to M i. Also T i t j was the smallest load. Every other machine has load at least T i t j. Therefore : m(t i t i ) k T k Also we know that k T k j t j. Therefore T i t j m j t j T. Also we know that t j T. Therefore load of M i after adding J j is T i = (T i t j ) + t j 2T.

20 The Greedy-Balance could actually be as close as possible to 2T. J J 2 J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 0 J J 2 J 3 Greedy Algorithm 4 Optimal 4 M M 2 M 3 M 4 M M 2 M 3 M 4

21 The Greedy-Balance could actually be as close as possible to 2T. J J 2 J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 0 J J 2 J 3 Greedy Algorithm 4 Optimal 4 M M 2 M 3 M 4 M M 2 M 3 M 4

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

23 In general suppose there are m machines and n = m(m ) + jobs. The first m(m ) jobs each with time t j = and the last job n = m(m ) + has time t n = m.

24 In general suppose there are m machines and n = m(m ) + jobs. The first m(m ) jobs each with time t j = and the last job n = m(m ) + has time t n = m. The optimal has T = m while the Greedy algorithms has max load 2m.

25 An Improved Approximation Algorithm Sort-Balance. Set T 0 = 0 and A(i) = for all machines M i. 2. Sort the jobs in decreasing order of processing times t j. 3. Assume t t 2... t n. 4. for j = to n 5. Let M i be a machine with minimum load (min k T k ). 6. Assign job j to machine M i. 7. Set A(i) A(i) {J j } 8. Set T i T i + t j

26 If there are more than m jobs, then T 2t m+.

27 If there are more than m jobs, then T 2t m+. Lemma Algorithm Sort-Balance produces an assignment of jobs to machines with max load T 3 2 T. Using similar analysis as in the previous lemma (leave it as exercise)

28 There exists an algorithm that find a solution for Load balancing very close to optimal T

29 There exists an algorithm that find a solution for Load balancing very close to optimal T In fact there is an algorithm that for every ɛ > 0 it finds a solution that is not worse that ( + ɛ)t.

30 There exists an algorithm that find a solution for Load balancing very close to optimal T In fact there is an algorithm that for every ɛ > 0 it finds a solution that is not worse that ( + ɛ)t. But the running time of the algorithm is where n is the number of jobs. O(n ( ɛ ).5 )