Optimal allocation of electronic content in networks
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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

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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


Optimal allocation of electronic content in networks

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.


Optimal allocation of electronic content in networks

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

*


Optimal allocation of electronic content in networks

ALGORITHM IDEA

*

v

i

k

i’s subtree

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


Optimal allocation of electronic content in networks

ALGORITHM IDEA

*

v

i

k

i’s subtree

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


Optimal allocation of electronic content in networks

*

ALGORITHM IDEA

v

i

k

i’s subtree

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


Optimal allocation of electronic content in networks

ALGORITHM IDEA

*

v

i

k

i’s subtree

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


Optimal allocation of electronic content in networks

ALGORITHM IDEA

v

*

i

k

i’s subtree

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


Optimal allocation of electronic content in networks

Dynamic programming: solution

*

Distance j

i

i’s subtree


Optimal allocation of electronic content in networks

Dynamic programming: solution

*

Distance j

i

i’s subtree


Optimal allocation of electronic content in networks

j

i

i

*

Distance j

i

...

Data structure at node i


Optimal allocation of electronic content in networks

Computing the parent’s cost. Example: line 2

v

k

i


Optimal allocation of electronic content in networks

Computing the parent’s cost. Example: line -1

v

k

i


Optimal allocation of electronic content in networks

Computing the parent’s cost. For line 0 add cost of server (e.g. 10)

v

*

k

i


Optimal allocation of electronic content in networks

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) <cost(line 0)?

or <cost(line -1)?


Optimal allocation of electronic content in networks

1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*

Leaves tables


Optimal allocation of electronic content in networks

Leaves tables

1*

5

1

3*

2

4

3

2

0

5

6

1

0

7

5

7*

6

8

9

10

11

12

*


Optimal allocation of electronic content in networks

Leaves tables


Optimal allocation of electronic content in networks

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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