Transcript Application Layer Multicast

Application Layer Multicast
Overlay Networks
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Treat the Internet as a
physical network
Build you own network on
top of it
In theory: all-to-all
connectivity
Challenge: make it efficient
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Example: IP Multicast
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Multicast is an efficient delivery mechanism for
Group-Communication applications.
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IP Multicast
- Relay on network layer to replicate and deliver data
packets to receivers.
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Problems with IP Multicast
- Scales poorly with number of groups
- Supporting higher level functionality is difficult
- Deployment is difficult and slow
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Application-Layer Multicast
An alternative: provide multicast functionality above the IP
layer i.e., Application-Layer or Overlay Multicast.
- End-Hosts perform packet duplication and routing.
- End-Hosts construct overlay network on unicast infrastructure
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Efficiency of Multicast Trees
Challenge: construct efficient overlay trees
Application centric performance metrics: delay, throughput
Todays focus: delay optimized multicast trees
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Naive Solution, Shortest Path Tree (SPT)
 Does not limit node load
 Creates bottlenecks (BW and CPU limits)
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Efficiency of Multicast Trees (cont.)
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Suggestions for Application-Layer multicast
- Mesh First - Narada [CRZ00]
- Tree First – Yoid [F99]
- Minimal diameter trees [ST02],[BKKBS03]
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Common Properties
 Build ‘minimal’ diameter (height) trees with degree constrains
 Low diameter  Low delay
 Low degree  Low stress and bandwidth utilization
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Efficiency of Multicast Trees (cont.)
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Disadvantages:
- Arbitrary selection of degrees
• Requires trial and error
- Ignores serialized message distribution
• Scalability problem (known problem [CRZ00])
H1
Non-Optimal Tree
Optimal Tree
Low-speed Link
H1
R1
H2
R2
H1
H3
H2
H3
H3
H2
High-speed Links
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Todays Talk
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New overlay network model
 Map the node load to delay penalty
 Quantifies multicast performance using a single delay metric.
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Novel approximation and heuristic algorithms
 Generate delay optimized trees that intrinsically balance short latency
with small degree
 Suitable for delay-sensitive applications (audio-confreencing, CDN,
etc)
 Provides scalable solution
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Performance analysis of structured overlay topologies
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The Overlay Network Model
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A complete directed graph G=(V,E)
Communication cost function c : E  R+
Processing cost function p : V  R+
Sequential communication, p(v) is the minimum
time interval between consecutive message
transmissions of host v
Processing
Delay
Message
Initiation
Sender host is
free to handle
other
operations
Communication
Delay
Message
arrival
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The Minimal Delay Multicast (MDM)
Problem
Given a directed complete graph G=(V,E); a multicast group
M  V; a source host sM; a processing cost p(v), vV; and a
communication cost c(e), eE;
Find
a scheme that minimizes the delay required to disseminate a
message from s to all the other hosts in M. Only the hosts in
M are allowed to participate in the distribution
We assume that M≡V
 Group size n = |V|
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The Ordered Tree Solution
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Optimal solution is represented by an ordered tree T
which spans V, rooted at s.
The i-th outgoing edge of node u corresponds to the i-th
transmission from host u
Notations
Reception delay of v, tT(v)
Tree cost, maxvV {tT(v)}
By Def. tT(s) =0
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S
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V1
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V3
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V5
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V2
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V6
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Optimal Multicast
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Given a multicast tree T=(V,E) we calculate the
optimal ordering using a recursive computation,
working bottom-up.
Idea: The i-th delivery goes to the i-th largest cost
subtree
max 1i  k m(r (v, i )  i  p (v) if v has k children
M (v )  
0
if v is a leaf
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(u , v)  E : m(v)  M (v)  c(u , v)
r (v, i ) is the child of v with the i - th largest m(.) quantity
M(v) is the optimal cost of the subtree rooted at v
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Optimal Multicast – cont.
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We neglect the ordering and
concentrate on finding optimal tree
Optimal solution is a ‘non-lazy’
multicast scheme.
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1
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V1
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V4
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V2
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V3
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S
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V5
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V6
The optimal multicast problem is NP-Complete
- Reduction from the telephone broadcast problem
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Telephone Broadcast (TB) Problem
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TB assumes a synchronous communication graph
- In each communication round a node may send a message
to at most one other node.
- The problem is to notify the entire graph in minimal number
of rounds.
- Studied extensively, known to be NP-Complete [GJ79]
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Reduction from TB problem
- construct an overlay configuration with unit processing
costs; zero communication costs for all the edges in E,
and cost n for the rest.
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Computation Models
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Homogenous models
- Active Networks [RS01]
• Multicasting algorithm for line topologies
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The heterogeneous postal model [BGNSS01]
- Assumes undirected and complete graph G=(V,E)
- Communication latency function λ
- Switching (sending) time function s
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Postal Approximation
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log k approximation,
k = size of multicast group
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Inappropriate for overlay networks
- Incorporates the sending time at the communication latency,
whereas the overlay model incorporates this quantity at the
processing delay.
-  sv  λu,v, v,uV;
Restricted cost model
-  Delay of the I-th message from u to v:
• su*(i-1)+ λu,v
• p(u)*i + c(u,v)
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Postal Model
Overlay Model
Doesn't support asymmetric costs
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Our Approximation Approach
Devise an approximation algorithm based on the
postal approximation scheme
1. Define unrestricted cost model
Generalized Heterogeneous Postal Model (GHP)
2. Adapt the postal approximation to support the GHP
model. New approximation ratio:
O(log n)  max{1, }
  max (u ,v )E {su u ,v }
3. Use a cost preserving transformation and invoke the
adapted GHP approximation.
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Postal Approximation Overview
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Multicast problem: Find minimum time delivery from source r to
a group of terminals U V. All the nodes in V may participate in
the multicast.
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Basic Idea
- OPT ≥ 0.5 (LT*+∆T*)
• T* - optimal multicast tree, spans U
• LT* - maximum distance from r to any node in U
• ∆T* - maximum generalized degree, the generalized degree of
vU is its degree in T* multiplied by sv.
• OPT - cost of the optimal solution
- Find a multicast tree T which minimizes the quantity LT+∆T
• NP-Complete problem
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Postal Approximation
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Iterative computation of the tree
At the i-th round use the Core proc. to compute
1. A core subset Ui  Ui-1 of size at most 0.75|Ui-1|, rUi
2. A dissemination scheme from Ui to Ui-1, s.t the
obtained time is linear in the optimal multicast time
from r to Ui-1
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log |U| rounds  log |U| approximation factor
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Core(U’) Procedure
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Find a set of |U’| paths, one for each terminal, where the
path length and congestion (the generalized degree
induced by the paths) is linear in LT*+∆T*
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Use a multicommodity flow linear program to find a set of fractional
paths that minimize the sum of two quantities:
1. L - the average length of the paths
2. ∆ - the congestion of the paths
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The program ensures that L+ ∆  LT*+∆T*
Each flow path has length at most 4L
The total congestion is at most 6∆
Round the set of fractional paths into integral paths
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Core(U’) Procedure-Cont.
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Transform the set of paths into a set of spider
graphs
- Each spider contains two nodes from U’
- The spiders span at least half of the nodes in U’
- The diameter and generalized degree of each spider is
linear in LT*+∆T*
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Include in the core an arbitrary node vU’ from
each spider and all the nodes not contained in
any spider.
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GHP Adaptation
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GHP support requires modification to the rounding mechanism
only
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Rounding Theorem [KLRTVV87]
- Given a real matrix A, real vectors b,y, s.t Ay=b, a real value t ≥0 s.t
in every column of A
• The sum of all positive entries  t
• The sum of all negative entries ≥ -t
Then, we can compute an integer vector y, s.t for every i
yi   yi  or yi   yi  and Ay  b , where bi  bi  t
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GHP Rounding matrix
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Notations:
-
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{Pi} is the set of fractional flow paths
E(Pi) and V(Pi) is the set of edges and nodes in Pi
f(Pi) is the amount of flow pushed on path I
Pj is the set of all paths from {Pi} which carry the j-th commodity
We use the following matrix for rounding the fractional paths:
for each v
for all j
 f ( P )  6
 4 L '  f ( P )  4 L '
sv
i:vV(Pi )
i
 '  max{1, }
i
i:Pi P
j
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GHP Rounding matrix-Cont.
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The sum of the positive entries is at most 4Lα’
The sum of the negative entries is at most -4Lα’
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Applying the rounding theorem, we get a set of integer paths s.t
the length of each path  4Lα’, and the congestion  4Lα’+6∆
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Therefore, the length and the congestion are linear in (LT*+∆T*) α’
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MDM Approximation Algorithm
Input
G=(V,E)
edge costs c(u,v), node costs p(u), v,uV
source host rV
Algorithm Approx-MDM
1. Construct a GHP configuration (G,s,λ) s.t
sv=p(v), vV; λu,v=0.5(p(u)+p(v))+c(u,v), eE;
2. Invoke the GHP approximation, assuming
multicast group V, and source rV.
3. Return the computed tree
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An Example
p1=s1
p1+p3
λ1 =
+ c1
2
s 3= p 3
p1+p2
c2 +
=λ2
2
p3+p2
c3+
=λ3
2
p2 =s2
Path cost =c1+ c3 +p3+ 0.5p1+0.5p2
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Approximation Ratio
Theorem 1:
The approximation ratio of Approx-MDM is
(OPT+pmax-pmin) O(log n)
Cost of
optimal tree
Maximal
processing cost
Minimal
processing cost
Corollary:
The approximation ratio for an overlay network with
homogenous processing costs p(v)=p, vV
OPT • O(log n)
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Approximation ratio-Cont.
Proof:
GHP Notations
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tGHPT(v) - reception delay of v
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1.
2.
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OPTGHP - cost of optimal solution
tGHPT(v) = tT(v)+ 0.5(p(v)-p(s))
, for any T, vV
OPTGHP  OPT+ 0.5(pmax- pmin)
α  2 (α is the switching to communication ratio)
The approximation ratio of the GHP approximation is at most
OPTGHP •α •log n
 The theorem follows.
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Heuristic Algorithm
Approx-MDM suffers from high polynomial running time of
Θ(n7).
An alternative: a greedy algorithm
Algorithm Heuristic-MDM
Init: Add s to an empty tree
1. Compute the minimal reception delay of each non-notified
host
2. Select the non-notified host with maximal reception delay
3. Add this host and the minimal latency path to the
constructed tree
Repeat 1-3 till all hosts are notified
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Heuristic Algorithm - Cont.
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Minimal latency path is computed using AllPairs Shortest-Path (Floyd-Warshall) with
weight matrix W=(wvi,vj) defined as:
wvi ,v j
 p(vi )  c(vi , v j ) if vi  v j

0
o.w
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Time complexity Θ(n3)
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Supports arbitrary directed graphs
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Homogenous Cost Overlays
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Homogenous overlay network
p(v)=p, vV; c(e)=c, eE
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Any ‘non-lazy’ algorithm provides optimal
solutions (e.g., Heuristic algorithm)
Notations
N(t), maximal number of hosts reached in time t.
Rτ(t), maximal number of messages received in the
interval (t-τ,t]
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Convergence Rate in
Homogenous Overlay Networks
Lemma 3:
Given an homogenous cost overlay
N (t  c)  R p (t  p), for any t  c
Proof
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There is one-to-one correspondence between the number
of messages received in the interval (t,t+p] and the number
of hosts which initiated processing in the interval (t-c-p,t-c]
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Two types of hosts in the interval (t-c-p,t-c]
- X, newly notified hosts
- Y, hosts notified before t-c-p
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Rp(t+p)= |X|+|Y|=|X|+N(t-c-p)=N(t-c)
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Convergence Rate in
Homogenous Overlay Networks
Theorem 3:
Given an homogenous cost overlay
 N (t  p)  N (t  p  c) if t  p  c
N (t )  
1
o.w
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Corollary
Any ‘Non-Lazy’ algorithm achieves logarithmic
multicast delay, with the following bounds:
2
 t 


 p c 
 N (t )  2
t 
 
 p
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Simulation Results
The multicast delay for a clique topology with random
costs from [1,10]
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Max. latency is the weight of the longest path in the SPT
Problem is NP-Hard  Use Max. latency as lower bound
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Clique Topologies – Cont.
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The multicast delay for a clique topology with random
communication costs from [1,10] and unit processing costs
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SPT has almost linear growth rate for large sizes
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Power-Law Topologies
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The multicast delay assuming random costs from [1,10]
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Simulation based on power-law graphs (Notre-Dame Model
[AB00]), average degree is 4.38.
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Some Conclusions
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Approximation Algorithm
- High Computation Θ(n7)
• Limited to small groups
- Logarithmic height trees
• Performance degradation in networks with
dominating communication costs.
- Support only complete undirected graphs
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SPT
- Suitable for small scale groups with dominating
communication costs
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Simulations Summary
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On average the heuristic algorithm has a similar or better
performance than the approximation algorithm
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Heuristic trees are scalable for large group sizes
- Near optimal result
- Logarithmic like growth rate
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Remarks
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We can reduce the approximation ratio of the
Approx-MDM algorithm to a pure logarithmic
factor
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Graphs in which the triangle equality holds may
have better bounds
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