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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. 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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  1. Optimal Allocation of Electronic Content in Networks Israel Cidon- Technion Shay Kutten- Technion Ran Soffer- Redux

  2. 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 *

  3. 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.

  4. users users users users users users users users Model example server level 1 server level 2 server level 3 server level 4

  5. 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.

  6. [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.

  7. 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.

  8. 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.

  9. 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.

  10. 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 *

  11. ALGORITHM IDEA * v i k i’s subtree Dynamic programming: how to combine the solutions for i and k to get v?

  12. ALGORITHM IDEA * v i k i’s subtree But to solve for i we need to know where is the server*

  13. * ALGORITHM IDEA v i k i’s subtree But to solve for i we need to know where is the server*

  14. ALGORITHM IDEA * v i k i’s subtree But to solve for i we need to know where is the server*

  15. ALGORITHM IDEA v * i k i’s subtree But to solve for i we need to know where is the server*

  16. Dynamic programming: solution * Distance j i i’s subtree

  17. Dynamic programming: solution * Distance j i i’s subtree

  18. j i i * Distance j i ... Data structure at node i

  19. Computing the parent’s cost. Example: line 2 v k i

  20. Computing the parent’s cost. Example: line -1 v k i

  21. Computing the parent’s cost. For line 0 add cost of server (e.g. 10) v * k i

  22. 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)?

  23. 1* 5 1 3* 2 4 3 2 0 5 6 1 0 7 5 7* 6 8 9 10 11 12 * Leaves tables

  24. Leaves tables 1* 5 1 3* 2 4 3 2 0 5 6 1 0 7 5 7* 6 8 9 10 11 12 *

  25. Leaves tables

  26. 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

  27. 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

  28. 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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