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The Analysis of Optimal Stream Merging Solutions for Media-on-Demand

The Analysis of Optimal Stream Merging Solutions for Media-on-Demand. Amotz Bar-Noy CUNY and Brooklyn College Richard Ladner University of Washington. Media-On-Demand. Type of Media Two hour movie 5 minute song 2 minute news video clip Demand on Media Broadcast for very popular media

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The Analysis of Optimal Stream Merging Solutions for Media-on-Demand

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  1. The Analysis of Optimal Stream Merging Solutions for Media-on-Demand Amotz Bar-Noy CUNY and Brooklyn College Richard Ladner University of Washington

  2. Media-On-Demand • Type of Media • Two hour movie • 5 minute song • 2 minute news video clip • Demand on Media • Broadcast for very popular media • Unicast for the heavy tail AofA 2004

  3. Stream MergingEager, Vernon, Zahorjan (1999) • Goal to minimize cost = total server bandwidth • Uses multicast • Many clients receive the same transmission simultaneously. • Clients receive multiple channels • Clients receive different parts of the media simultaneously. We focus on receive-2. • Clients use local buffers • Some data received is played back immediately while other data is stored for future play back. AofA 2004

  4. Stream Merging System -Receive Two Multicast Channels Server old stream new stream Client Buffer Player AofA 2004

  5. Stream Merging System Multicast Channels Server Client Buffer Player AofA 2004

  6. Stream Merging System Multicast Channels Server Client Buffer Player AofA 2004

  7. Stream Merging System Multicast Channels Server Client Buffer Player AofA 2004

  8. Stream Merging System Multicast Channels Server Client Buffer Player AofA 2004

  9. Batching client startup delay guarantee stream 7 7 6 6 5 5 4 4 3 3 Cost = 14 2 2 1 1 AofA 2004

  10. Stream Merging with Batching client startup delay guarantee stream 7 6 5 4 3 3 Cost = 10 2 2 1 1 A B AofA 2004

  11. Delay vs. Bandwidth Tradeoff 2 hour movie 720 arrivals per movie AofA 2004

  12. Off-line vs. On-line • Off-line • Arrival sequence is known in advance • Used as a basis to compare with on-line algorithms • On-line • Future arrivals not known • Server must add new streams and lengthen old streams to accommodate new arrivals AofA 2004

  13. Outline • Model • Merge cost vs. Full cost • Optimal merge cost • Optimal full cost • Fully loaded arrivals AofA 2004

  14. Merge Cost vs. Full Cost • Optimal Merge cost • Minimum total bandwidth (sum of the lengths of the streams) required to merge to a full stream • M(t1,…,tn) • Optimal Full cost • Minimum total bandwidth required for a media of length L. • F(L,t1,…,tn) AofA 2004

  15. Length of Streams • The length of streams p(x) = A x = B z(x) = D A B C D AofA 2004

  16. Optimality Principle for Merge Cost • Given arrivals t1,…,tn there is k such that in optimal stream merging • tk is the last stream to merge to t1 • Streams t1,…,tk-1 optimally merge eventually to t1 • Streams tk+1,…,tm optimally merge eventually to tk AofA 2004

  17. Dynamic Program for Optimal Merge Cost • Setup • Arrivals t1,t2, … ,tn • M(i,j) = minimum merge cost of a merge tree for the arrivals ti,t2, … ,tj • M(1,n) is the optimal merge cost • Recurrence • M(i,i) = 0 • M(i,j) = mini < k < j{M(i,k-1) + M(k,j) + (tk - ti) + 2(tj - tk)} • O(n3) time, O(n2) storage • Aggarwal, Wolf, Yu (1996), Eager,Vernon, Zahorjan (1999) AofA 2004

  18. O(n2) Optimal Algorithm • Setup Arrivals t1,t2, … ,tn r(i,j) = the right most stream that merges to the root of an optimal merge tree for the arrivals ti,…,tj. • Monotonicity(Knuth ‘71, F. Yao ’80, Borchers, Gupta ‘94) r(i,j - 1) < r(i,j) < r(i + 1,j) • M(i,j) = minr(i,j-1) < k < r(i+1,j){M(i,k-1) + M(k,j) + (tk - ti) + 2(tj - tk)} optimal AofA 2004

  19. New Algorithm Behavior r(i,j-1) r(i+1,j) M(i,j) r(i,j-1) O(n2) time r(i+1,j) AofA 2004

  20. New Algorithm Behavior O(n2) time AofA 2004

  21. Optimal Full Cost L = 8 Optimal Merge Cost + L Optimal Full Cost AofA 2004

  22. Optimal Full Cost Algorithm • Setup • Arrivals t1,t2, … ,tn • G(i) = the optimal full cost for the last n-i+1 arrivals ti,ti+1, … ,tn • G(1) is the optimal full cost • Recurrence • G(n+1) = 0 • G(i) = L + min{ M(i,k-1) + G(k) : i < k < mi }where mi = maximum m such that tm-1 - ti< L-1. AofA 2004

  23. Complexity of Full Cost Algorithm Phase 1: Precompute M(i,i) to M(i,mi) for all i. time using monotonicity Phase 2: Compute G(n+1) to G(1) time from the recurrence for G where m is the average number of arrivals in an interval of length L that starts with an arrival AofA 2004

  24. Fully Loaded Arrivals Merge Cost Streams at 0,1,2,3,4,…,n-1 AofA 2004

  25. Recurrence • Merge recurrence for fully loaded arrivals • Examples AofA 2004

  26. Fibonacci Rules! • Recurrence • Fibonacci numbers • 0,1,1, 2, 3, 5, 8,13, 21, 34, 55, 89, … • F0 = 0, F1 = 1, Fk = Fk-1 + Fk-2 • Solution AofA 2004

  27. Induction Hypothesis Design 55 {34}, 56 {34, 35}, 57 {34, 35, 36}, 58 {34, 35, 36, 37}, 59 {34, 35, 36, 37, 38}, 60 {34, 35, 36, 37, 38, 39}, 61 {34, 35, 36, 37, 38, 39, 40}, 62 {34, 35, 36, 37, 38, 39, 40, 41}, 63 {34, 35, 36, 37, 38, 39, 40, 41, 42}, 64 {34, 35, 36, 37, 38, 39, 40, 41, 42, 43}, 65 {34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44}, 66 {34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45}, 67 {34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46}, 68 {34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47}, 69 {35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48}, 70 {36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49}, 71 {37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50}, 72 {38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51}, 73 {39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52}, 74 {40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53}, 75 {41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54}, 76 {42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55}, 77 {43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55}, 78 {44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55}, 79 {45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55}, 80 {46, 47, 48, 49, 50, 51, 52, 53, 54, 55}, 81 {47, 48, 49, 50, 51, 52, 53, 54, 55}, 82 {48, 49, 50, 51, 52, 53, 54, 55}, 83 {49, 50, 51, 52, 53, 54, 55}, 84 {50, 51, 52, 53, 54, 55}, 85 {51, 52, 53, 54, 55}, 86 {52, 53, 54, 55}, 87 {53, 54, 55}, 88 {54, 55}, 89 {55}, 21 {13}, 22 {13, 14}, 23 {13, 14, 15}, 24 {13, 14, 15, 16}, 25 {13, 14, 15, 16, 17}, 26 {13, 14, 15, 16, 17, 18}, 27 {14, 15, 16, 17, 18, 19}, 28 {15, 16, 17, 18, 19, 20}, 29 {16, 17, 18, 19, 20, 21}, 30 {17, 18, 19, 20, 21}, 31 {18, 19, 20, 21}, 32 {19, 20, 21}, 33 {20, 21}, 34 {21}, 35 {21, 22}, 36 {21, 22, 23}, 37 {21, 22, 23, 24}, 38 {21, 22, 23, 24, 25}, 39 {21, 22, 23, 24, 25, 26}, 40 {21, 22, 23, 24, 25, 26, 27}, 41 {21, 22, 23, 24, 25, 26, 27, 28}, 42 {21, 22, 23, 24, 25, 26, 27, 28, 29}, 43 {22, 23, 24, 25, 26, 27, 28, 29, 30}, 44 {23, 24, 25, 26, 27, 28, 29, 30, 31}, 45 {24, 25, 26, 27, 28, 29, 30, 31, 32}, 46 {25, 26, 27, 28, 29, 30, 31, 32, 33}, 47 {26, 27, 28, 29, 30, 31, 32, 33, 34}, 48 {27, 28, 29, 30, 31, 32, 33, 34}, 49 {28, 29, 30, 31, 32, 33, 34}, 50 {29, 30, 31, 32, 33, 34}, 51 {30, 31, 32, 33, 34}, 52 {31, 32, 33, 34}, 53 {32, 33, 34}, 54 {33, 34}, 5 13 2 {1}, 3 {2}, 4 {2, 3}, 5 {3}, 6 {3, 4}, 7 {4, 5}, 8 {5}, 9 {5, 6}, 10 {5, 6, 7}, 11 {6, 7, 8}, 12 {7, 8}, 13 {8}, 14 {8, 9}, 15 {8, 9, 10}, 16 {8, 9, 10, 11}, 17 {9, 10, 11, 12}, 18 {10, 11, 12, 13}, 19 {11, 12, 13}, 20 {12, 13}, 3 5 8 8 3 2 5 3 13 8 AofA 2004

  28. Optimal Full Cost • L = 24 and n = 39 • Cost = 210 Full steam merged steam AofA 2004

  29. Optimal Full Cost • Define F(L,n,s) to be the optimal full cost when there are exactly s full streams. sopt minimizes F(L,n,s) F(L,n) = F(L,n, sopt ) • TheoremF(L,n,s) = sL + rM(p+1) + (s-r)M(p) where n = ps + r AofA 2004

  30. Shape of F(L,n,s) sopt = 182 AofA 2004

  31. Convexity of F(L,n,s) • Define Fk(s)< n/s < Fk(s)+1 • Theorem • Fk(s+1)+2< L + 2 implies F(L,n,s) > F(L,n,s+1) • Fk(s)+2> L + 2 implies F(L,n,s) < F(L,n,s+1) AofA 2004

  32. Finding sopt • Direct approach • Compute h, such that Fh+1 < L+2 < Fh+2 • sopt = n / Fh or n / Fh + 1 • Example • n = 10,000 and L = 100 • h = 10 and Fh = 55 • n / Fh = 181 AofA 2004

  33. Conclusions • Optimal stream merging can be solved efficiently. • Optimal fully loaded stream merging has an elegant closed form solution. AofA 2004

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