Designing overlay multicast networks for streaming

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Designing overlay multicast networks for streaming

• Konstantin Andreev
• Bruce Maggs
• Adam Meyerson
• Jevan Saks
• Ramesh Sitaraman

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And now a word from our founder
Governor Sanford sic has created a useless
rival to the State University, as you and I saw
when last in San Francisco. I could have no part
in such a thing. -- Andrew Carnegie in a letter
to Andrew White, Ambassador to Berlin, 1901.
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Delivering streaming media

• Video quality is heavily impacted by packet
  losses on individual links
• Feed from live event must be distributed to
  streaming servers

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Delivering Streams the Conventional Way
Live Events
Internet
Encoder
Production Signal Acquisition
Media Player or client
Streaming Server
On-Demand Clips
On-Demand Clips
END-USER
CONTENT PROVIDER
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Akamais Live Streaming Architecture
Reflectors
Entry Point
Encoder
Edge Servers (Sinks)
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Goals when building the overlay multicast network

• Minimize cost
• Obey fan-out and capacity constraints
• Satisfy reliability requirements

Sources - S
Reflectors - R
Sinks - D
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Formal definition

• 3-level network reliability min-cost
  multicommodity flow problem
• Tripartite digraph VS R
  D
• Costs on the edges
  (depending on the commodity) cu Eu ! Ru
• Cost of using a reflector r R ! R
• Fanout constraint on reflectors F R ! R
• Failure rate on the edges
• 
  p E ! 0,1
• Success thresholds for each sink, commodity
• pair
  Fu Du ! 0,1u
• Here u is the number of commodities

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Important Caveats

1. Assume losses on different edges are independent.
   Not true in practice! (But will return to this
   issue.)
2. Only model and optimize packet loss rates.
   Burstiness of losses matters a lot in practice.

On the positive side dont assume anything
about distribution of losses on a single edge.
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Flow combination rules

• The fraction of packets not arriving is
• p1p2-p1p2
• The fraction of packets not arriving is
• p1p2

p1 p2 p1

p2
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Unusual aspects of our flow problem

• No preservation of flow
• 1
  unit of flow
• 1 unit of flow
• 
  
• 1 unit of flow
• Nonlinear flow combination rules
• 1-p1 1-p2
  Total flow
• 
  (1-p1)(1-p2)

1 unit of flow
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Problem transformation
Konstantin Andree

• Shortened notation
• pijk pkipij-pkipij
• wijk - log pijk
• Path recombination
• pijk x pmjk
• wijk wmjk

pki
k
i
pij
j
k
i
m
pijk
pmjk
j
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Our results

• Approximation algorithm solution which
• has cost within a O(log n) factor of optimal
• violates fan-out constraints by at most O(1)
  factor
• violates weight constraints by at most O(1)
  factor
• Note weights are logs of probabilities
• E.g. We want success rate .999 and we violate the
  weight constraint by a factor of at most 3. That
  implies success rate of .9 or better.

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Previous related work

• The Network Reliability Problem
• In a network with a given source s and sink t
  where every edge has a probability of failure
  associated with it, compute what is the
  probability of a path connecting s to t.

• We can encode Set Cover, thus no approximation
  better than log n on the cost

• In general networks this problem is P-complete.
• There is an FPRAS that approximates it with in
  1e
• In 3-level networks one can compute the exact
  reliability in polynomial time.

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Constructive Criticism
The introduction is the best part of this
paper. -- FOCS Program Committee
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IP formulation/LP relaxation

• s.t.
• yik zi 8 i 2 R, k 2 S
• xijk yik 8 i 2 R, j 2 D,
  k 2 S
• åk 2 S, j 2 D xijk Fizi 8 i 2 R
• åj 2 D xijk Fi yik 8 i 2 R, k 2 S
• åi2 R xijk wijk Wjk 8 j 2 D, k 2 S
• xijk 2 0,1, yjk 2 0,1, zi 2 0,1

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Phases of our solution

• If we use straight LP rounding we get both log
  n blow up in cost and w.h.p. log n factor
  violation of the constraints
• Relaxation followed by partial randomized
  rounding (round zi and yik)
• Modified GAP rounding of the remaining fractional
  variables similar to Shmoys and Tardos

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Words of Encouragement
These are all standard techniques. -- FOCS
Program Committee
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Modified GAP approximation
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GAP approximation analysis

• There exists an optimal flow with values only 0,
  ½ and 1 (Shmoys and Tardos).
• We double this solution
• We violate the fanout and capacity constraints by
  at most a factor of 2

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Running time

• Let S of streams
• R of reflectors
• D of (stream, sink) pairs
• The algorithm running time is dominated by
  solving an LP on O(S.R.D) variables
• Polynomial time.

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Extensions
Reflectors
Entry Point
UU Net
Encoder
Edge Server (Sink)
BMM Net
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Extensions constraints

• Color constraints
• where RR1 R2 Rm
• (Ri is the set of reflectors on the ith ISP)
• Here different colors represent different ISPs
• There is no added value in serving to a fixed
  sink from two reflectors of the same color.

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Flow problem with additional set constraints

• Additional set capacity constraint
• E.g. AB, PQ has capacity 3
• There is an LP/IP gap
• E.g. Fractional flow is 3.5, best integral flow
  is 3

2
2
2
1
S
T
2
2
P
Q
2
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Flow problem solution

• Theorem Let A be a real valued r s matrix and
  y be an s-vector. Assume that in every column of
  A
• the sum of all positive entries is at most t
• the sum of all negative entries is at least t
• Then we can round the solution to Ayb component
  wise up or down (say y is the rounded integral
  vector) in such a way that Ayb where bi-biltt

• Because of the bounded depth in our case t 7
• The running time is at most O(R3.D3)

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Keys to High Course Evaluations

• Absence of mathematical content
• Attractiveness of male instructors
• Expected grade in course
• Conformity to traditional teaching methods

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An Example
(Thanks to graphviz)
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Comparison
Approx Solution (Cost 62)
IP Solution (Cost 59)
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How?
Use C, of course!
Multicast Network Configuration Data
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Timing Comparison (Log-plot)
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Constraint Violations

• Preliminary results
• Fanout constraints
• Average violation overloaded by 25
• Weight constraints
• Violated less than 50 of the time
• On average, under-supplied by 10

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Cost (Objective) Comparison
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Approxhack2

• Instead of Modified GAP, use IP solver
• Separates effects of Randomized Rounding
  andModified GAP

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Cost (Objective) Comparison
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Applied to Real-Word Data

• Two-months of streaming logs
• Fan-out constraints
• Costs
• Packet loss experiments

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If our research is successful