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

Demand Amotz Bar-Noy

CUNY and Brooklyn College

Richard LadnerUniversity of Washington

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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

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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.

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Stream Merging System -Receive Two

Server

Client

BufferPlayer

MulticastChannels

old stream

new stream

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Stream Merging System

Server

BufferPlayer

MulticastChannels

Client

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Stream Merging System

Server

BufferPlayer

MulticastChannels

Client

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Stream Merging System

Server

BufferPlayer

MulticastChannels

Client

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Stream Merging System

Server

BufferPlayer

MulticastChannels

Client

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Batching

clientstartup delay guaranteestream

65

4

7

1

2

3

65

4

7

1

2

3

Cost = 14

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Stream Merging with Batchingclientstartup delay guaranteestream

A B

1

2

3

65

4

7

1

2

3Cost = 10

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Delay vs. Bandwidth Tradeoff2 hour movie 720 arrivals per movie

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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

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Outline

• Model – Merge cost vs. Full cost

• Optimal merge cost• Optimal full cost• Fully loaded arrivals

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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)

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Length of Streams

• The length of streams z(x))2(xxp(x)length(x)

A B C D

x = B

z(x) = D

p(x) = A

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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

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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)

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O(n2) Optimal Algorithm

• SetupArrivals 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)

optimal

M(i,j) = minr(i,j-1) < k < r(i+1,j){M(i,k-1) + M(k,j) + (tk - ti) + 2(tj - tk)}

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New Algorithm Behavior

O(n2) time

M(i,j)

r(i,j-1)

r(i+1,j)

r(i,j-1) r(i+1,j)

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New Algorithm Behavior

O(n2) time

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Optimal Full Cost

L = 8

Optimal Full CostOptimal MergeCost + L

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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.

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Complexity of Full Cost Algorithm

Phase 1: Precompute M(i,i) to M(i,mi)for all i.

Phase 2: Compute G(n+1) to G(1)

n

1i

n

1iii nm/n)mn(m

time using monotonicity

n

1ii )mO(

time from the recurrence for G

n

1ii )mO(

where m is the average number of arrivals in an interval of length L that starts with an arrival

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Fully Loaded Arrivals Merge Cost

Streams at 0,1,2,3,4,…,n-1

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Recurrence

• Merge recurrence for fully loaded arrivals

• Examples

2}j2nj)M(n{M(j)minM(n)0M(1)

1nj1

n 1 2 3 4 5 6 7 8 9 10 11 12 13

M(n) 0 1 3 6 9 13 17 21 26 31 36 41 46

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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

1kk2k FnF2,F1)n(kM(n)

2}j2nj)M(n{M(j)minM(n)0M(1)

1nj1

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Induction Hypothesis Design

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},

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},

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},

2}j2nj)M(n{M(j)argminRM(n)1nj1

3

3

2

5

5

3

8

8

5

13

13

8

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Optimal Full Cost

• L = 24 and n = 39• Cost = 210

Full steam

merged steam

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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

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Shape of F(L,n,s)

sopt = 182

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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)

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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

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Conclusions

• Optimal stream merging can be solved efficiently.

• Optimal fully loaded stream merging has an elegant closed form solution.