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

2. 2 8 1*1* 2 3*3* 4 12 * 11109 7*7* 6 5 Bandwidth requirements example 1516751223 320 5 7 610 15 Users’ requirements server *

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

5. 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. 6 [NS95]’s Findings Tradeoff storage cost communication cost: –Then best storage level is near leaves of distribution tree. Otherwise- near root. Assumed all servers connected at same level

7. 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. 8 Related problems (cont.) Krishnan, Raz, and Shavitt IEEE/ACM Transactions on Networking, to appear. Li,Galin, Italiano, Deng, and K. Sohraby INFOCOM'99 Optimized delivery time, when #server is known.

9. 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. 10 8 1*1* 2 3*3* 4 12 * 11109 7*7* 6 5 If cost(server)=10 & cost(BW)=1 then cost=40+1+5+1+6+7+5+2+3=70 Users’ requirements 1516751223 320 5 7 610 15 server *

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

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

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

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

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

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

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

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

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

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

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

22. 22 * here i Final allocation Root allocates itself iff cost(line 0) is min. up A child i now knows the line j to use Is cost(line 2) < cost(line 0)? or < cost(line -1)?

23. 23 8 1*1* 2 3*3* 4 12 * 11109 7*7* 6 5 320 5 7 610 15 Leaves tables

24. 24 8 1*1* 2 3*3* 4 12 * 11109 7*7* 6 5 320 5 7 610 15 Leaves tables

25. 25 Leaves tables

26. 26 Parents’ tables Root’s tables 8 1*1* 2 3*3* 4 12 * 11109 7*7* 6 5 320 5 7 610 15

27. 27 Complexity Computation: O(dN)=O(N )=(  d Children ) d: tree depth N: nodes Message: O(N) Bit: d log cost per message Time: O(d) ii 2

28. 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.