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Multicast Trees: Structure and Size Estimation

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Title: Multicast Trees: Structure and Size Estimation


1
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

2
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

3
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

4
Power Laws
  • Of the form

5
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

6
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.

7
  • Sparse
  • Less Sparse
  • Connected
  • Internet Like
  • Competition..
  • The Real World

8
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.

15
Bell-Labs
16
Lucent
17
Scan
18
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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21
Degrees for
Degrees
22
Degrees for
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Why HDC6 Equals 16? (?3)
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26
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
27
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
28
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
29
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
30
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
31
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

32
Fast Algorithm Delay
The delay Lets define
gamma random variable with
33
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.
34
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.
35
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).

36
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.
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