Multicast Trees: Structure and Size Estimation

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Multicast Trees Structure and Size Estimation

• Danny Dolev1, Ossi Mokryn1,2, Yuval Shavitt2
• 1 School of EE and CS, Hebrew University
• 2 Dept. of EE -- Systems, Tel-Aviv University

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Why is it interesting?

• Structure
• Accurate simulations for research/proof of
  concept
• Server locations/feedback suppression/congestion
  control
• Size estimation
• RTCP, congestion, feedback, decisions, first
  order evaluation

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Overview

• Background
• Power laws in internet topologies
• Multicast trees
• Multicast tree structure our findings
• Fast estimation of multicast tree client
  population based on tree characteristics

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Power Laws

• Of the form

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Multicast Trees - Overview

• Structure depends on the protocol used for
  constructing the tree (CBT, PIM etc.).
• Shortest Path Tree acceptable method
• A uniform client distribution was shown to be a
  valid assumption Shenker et al Almeroth et al.
• Previous findings low average internal degree,
  high frequency of relay nodes, maximal height
  of 23. Almeroth,Chalmers INFOCOM01

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Generating Multicast Trees

• We generate topologies using the Notre Dame
  scale free algorithm.
• We first choose a root and clients according to
  degree specifications.
• A shortest path tree from the root to the clients
  is cut from the topology. Each such tree is
  generated 14 times (different random seeds).
• Summary four types of trees, ten client group
  sizes, 14 instances per such tree.

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• Sparse

• Less Sparse

• Connected

• Internet Like

• Competition..

• The Real World

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Our Main Empirical Findings

• Trees obey two power laws
• Degree-rank for the tree nodes.
• Size-rank for the sub-trees.
• Conforms with earlier findings of a majority of
  relay nodes. Chalmers,AlmerothPansiot,Grad
• Scale free characteristics

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Our Main Findings Cont.

• Distance distribution of nodes resembles a Gamma
  law also Cheswick et al..
• High degree nodes tend to reside in several
  adjacent rings
• They form a core also ISI-ATT Infocom02
• The rest of the nodes are usually within 5-9
  hops from this core

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Real Internet Data

• The same results were obtained on real data.
• KRS00Given Bell-Labs as the root, a list of
  clients was obtained, and a traceroute was
  performed for each client technical conference
  session.
• Cheswick et al Lucent Internet Mapping Project
  data as the topology 113000 nodes.
• Govindan et al Scan project 228000 nodes.

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Bell-Labs
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Lucent
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Scan
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Client Group Size Estimation

• High degree nodes have special characteristics
• Create a core of adjacent rings
• Relatively rare (due to power law)
• We found a linear ratio between the number of
  high degree nodes in the tree and the number of
  clients. The ratio determined a predictor.

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Degrees for
Degrees
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Degrees for
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Why HDC6 Equals 16? (?3)
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THE TREE ROOT
HIGH DEGREE NODE

• x

RECEIVER
INTERNET ROUND TRIP DELAY

• x

• x

• x

• x

• x

• x

ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE
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THE TREE ROOT
HIGH DEGREE NODE

• x

RECEIVER
INTERNET ROUND TRIP DELAY

• x

• x

• x

• x

• x

• x

ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Current Solutions
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THE TREE ROOT
HIGH DEGREE NODE

• x

RECEIVER
INTERNET ROUND TRIP DELAY

• x

• x

• x

• x

• x

• x

ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Current Solutions
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THE TREE ROOT
HIGH DEGREE NODE

• x

RECEIVER
INTERNET ROUND TRIP DELAY

• x

• x

• x

• x

• x

• x

ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Fast Algorithm
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THE TREE ROOT
HIGH DEGREE NODE

• x

RECEIVER
INTERNET ROUND TRIP DELAY

• x

• x

• x

• x

• x

• x

ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Fast Algorithm
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Fast Algorithm Characteristics

• Main sampling period (Td1) is used to reach the
  core, and collect its data.
• Iterative sampling period (Td2) is used to reach
  the next hop and collect its data
• Termination condition depends on required
  estimation error
• This is the first use of the power law
  characteristics of the underlying topology to
  improve upper level algorithms

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Fast Algorithm Delay
The delay Lets define
gamma random variable with
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Fast Algorithm Delay (cont.)
The total delay of gathering information from h
hops where is the probability of
a high degree node to be at distance i from the
root.
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Fast Algorithm Delay (cont.)
We should choose and so that the
majority of replies arrive. Define Rc
estimated core radius Re averaged distance to
edge client Enables the algorithm to terminate
faster than the Internet round trip delay.
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How Robust Is The Result?

• Hubs-to-clients ratio HBC6 16
• Specific trees may exhibit a slightly different
  HBC6 ratio (a.k.a. predictor).
• Most predictors are within 10 error from 16,
  statistical error can get to 30.
• Instances of a tree with the same root behave
  similarly (predictor error within 4).

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Summary

• Our findings
• Trees obey
• rank-degree power law.
• sub-tree size power law.
• High degree nodes form a core.
• A linear ratio between high degree nodes and the
  client population.
• Based on the above we devised the Fast
  Algorithm, that estimates client population in
  less than Internet RTT.