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Optimal Allocation of Electronic Content in Networks. Israel Cidon- Technion Shay Kutten- Technion Ran Soffer- Redux. Bandwidth requirements example. 1 *. 5. 1. 3 *. 2. 4. 3. 2. 0. 5. 6. 1. 0. 7. 5. 7 *. 6. 8. 9. 10. 11. 12 *. 3. 2. 12. 5. 7. 6. 1. 15.

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optimal allocation of electronic content in networks
Optimal Allocation of Electronic Content in Networks

Israel Cidon- Technion

Shay Kutten- Technion

Ran Soffer- Redux

slide2
Bandwidth requirements example

1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

3

2

12

5

7

6

1

15

Users’ requirements

server

*

the problem
The Problem
  • A practical problem ([NS95] Schaffa F. and Nussbaumer J.P. “On Bandwidth and Storage Tradeoffs in Multimedia Distribution Networks”, IEEE 1995)
    • A multimedia delivery to home.
    • Users connected to a Community Access TV (CATV) tree (A directed tree oriented mesh).
    • Servers containing all types of information can be connected at every level of the tree.
model example
users

users

users

users

users

users

users

users

Model example

server

level 1

server

level 2

server

level 3

server

level 4

why tree
Why tree?
  • General graphs: high complexity
  • Trees are in common use for distribution, hierarchy,
  • Trees studied in the related papers
  • See also Vassilakis et al (2000), Buddhikot (1998),
  • Triantafillou and Faloutsos (to appear in Par.
  • Comp), Bisdikian and Patel (ICC95), etc.
ns95 s findings
[NS95]’s Findings

Assumed all servers connected at same level

  • Tradeoffstorage cost communication cost:
  • Then best storage level is near leaves of distribution tree. Otherwise- near root.
related problems
Related problems
  • Related OR problems we mapped here:(see e.g. “Discrete Location Theory” book)
    • The p-Median problem.
    • The p-Center problem.
    • Uncapacitaed facility location problem.Algorithm for the undirected case:Billionnet A. and Costa M.C. “Solving the uncapacited problem on (undirected)trees”, DAM 49 pp. 51-59, 1994.
    • Tamir, 96, locating known# servers on undirected trees.
related problems cont
Related problems (cont.)
  • Krishnan, Raz, and Shavitt
  • IEEE/ACM Transactions on Networking, to appear.
  • Li,Galin, Italiano, Deng, and K. SohrabyINFOCOM'99
  • Optimized delivery time, when #server is known.
our contributions
Our contributions
  • -A more general model:
  • - Not all servers have to be in same level
  • - Cost(servers) on different machines may be different
  • - Cost(bandwidth) on different links may be different
  • Closed solution
  • Unknown number of servers
  • Better complexity
  • Observing: dynamic programming is better for distributed implementation,
  • connecting to OR problems.
slide10
If cost(server)=10 & cost(BW)=1

then cost=40+1+5+1+6+7+5+2+3=70

1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

3

2

12

5

7

6

1

15

Users’ requirements

server

*

slide11
ALGORITHM IDEA

*

v

i

k

i’s subtree

Dynamic programming: how to combine the solutions for i and k to get v?

slide12
ALGORITHM IDEA

*

v

i

k

i’s subtree

But to solve for i we need to know where is the server*

slide13
*

ALGORITHM IDEA

v

i

k

i’s subtree

But to solve for i we need to know where is the server*

slide14
ALGORITHM IDEA

*

v

i

k

i’s subtree

But to solve for i we need to know where is the server*

slide15
ALGORITHM IDEA

v

*

i

k

i’s subtree

But to solve for i we need to know where is the server*

slide16
Dynamic programming: solution

*

Distance j

i

i’s subtree

slide17
Dynamic programming: solution

*

Distance j

i

i’s subtree

slide18
j

i

i

*

Distance j

i

...

Data structure at node i

slide22
Final allocation

Root allocates itself iff

cost(line 0) is min.

A child i now knows the

line j to use

here

*

up

i

Is cost(line 2)

or

slide23
1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

Leaves tables

slide24
Leaves tables

1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

slide26
1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

Parents’ tables

Root’s tables

complexity
Complexity

2

  • Computation: O(dN)=O(N )=(d Children )

d: tree depth

N: nodes

  • Message: O(N)
  • Bit: d log cost per message
  • Time: O(d)

i

i

conclusions open problems
Conclusions & Open problems
  • We found similarity between internet and Operational research problems.
  • Dynamic programming is a more convinient tool for distributed implementation.
  • Try to utilize methods for the application tree solutions in more general networks.
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