Tight Bounds for Online Vector Bin Packing

Ilan Cohen

Joint work with: Yossi Azar ,Bruce Shepherd, Seny Kamara

Jobs Scheduling

Scheduler

Server 1Server N

Jobs Scheduling

CPU

Server

Memory GPU …

Cloud Computing

Vector Bin Packing

On Line Algorithms

Competitive Ratio

Related Results

Our Results

Vector Bin Packing

Our Results

VBP(B=1) Lower Bound

Online- VBP(B=1)

α bins.

Online- Graph coloring α colors

On-line Graph Coloring

aa bb cc dd ee

On-line Graph Coloring

VBP(B=1) Lower Bound

Online- VBP(B=1)

α bins

Online- Graph

coloringα colors

Reduction VBP to Graph Coloring

Reduction VBP to Graph Coloring

1n

...

1

0 1

i j

ij

Ind:

Vec:

Reduction VBP to Graph Coloring

aa bb cc dd ee

00001

000151

001051

01

0

51

51

1

00

5151

Reduction VBP to Graph Coloring

( , )

( , )

1 10 1

i j

j i

i jInd:

Vec:

Reduction VBP to Graph Coloring

0000000000001111

00000000111100

01

0000111100000

00

1

000

0000000

1111

1

1

aa bb cc dd

1,1

1,2

1,3

1,4

2,1

2,2

2,3

2,4

3,1

3,2

3,3

3,4

4,1

4,2

4,3

4,4

VBP(B=1) Lower Bound

Online- VBP(B=1)

α bins

Online- Graph

coloringα colors

VBP(B=2) Lower Bound

Online VBP(B=2)

α binsα classes of

Triangle-free sub graphs

Online- Graph

coloringα·k colors

k colors for each sub graph

Reduction VBP to Graph Coloring

Reduction VBP(B=2) to Graph Coloring

Reduction VBP(B=2) to Graph Coloring

0000000000001111

00000000111100

01

0010001011110000

1

00

0001000

1111

1

1

n

1,1

1,2

1,3

1,4

2,1

2,2

2,3

2,4

3,1

3,2

3,3

3,4

4,1

4,2

4,3

4,4

a b c d

Reduction VBP(B=2) to Graph Coloring

VBP(B=2) Lower Bound

Online VBP(B=2)

α binsα classes of

Triangle-free sub graphs

Online- Graph

coloringα·k colors

Reduction VBP(B=2) to Graph Coloring

On-line Graph Coloring

On-line Graph Coloring

VBP(B=2) Lower Bound

Online VBP(B=2)

α binsα classes of

Triangle-free sub graphs

Online- Graph

coloringα·k colors

a b c d

Coloring vertices of bin j

Correctness

Analysis

VBP Lower Bound

Online VBP (B)α bins

α classes ofB+1 clique free graphs

Online- Graph coloring

α· colors

α·k classes ofB clique

free graphsBk

B Clique Free to B-1 Clique Free

Reduction VBP to Graph Coloring

Reduction VBP to Graph Coloring

Scheme

Analysis

Upper Bound (B > 1)

Pack Into Virtual bins

BcBlog(d)

OPT virtual bins

All Vector Stream:

B

r bins

B

B

B

cBlog(d)

Distribute bin vectors

Virtual Bin i:

The virtual VBP algorithm.

The virtual VBP algorithm.

Proof Sketch

B

r bins

B

B

B

cBlog(d)

Distribute bin vectors

Virtual Bin i:

Distributing the vectors

Distributing the vectors

De-randomize the algorithm

De-randomize the algorithm

De-randomize the algorithm

Conclusions

Open Questions

Thank you!

The {0,1}-VBP Upper Bound

The {0,1}-VBP Upper Bound

Unrelated Machine Model

Unrelated Machine Model