Title: Multicast Trees: Structure and Size Estimation
1Multicast 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
2Why 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
3Overview
- Background
- Power laws in internet topologies
- Multicast trees
- Multicast tree structure our findings
- Fast estimation of multicast tree client
population based on tree characteristics
4Power Laws
5Multicast 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
6Generating 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 8Our 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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12Our 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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14Real 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.
15Bell-Labs
16Lucent
17Scan
18Client 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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21Degrees for
Degrees
22Degrees for
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24Why HDC6 Equals 16? (?3)
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26THE TREE ROOT
HIGH DEGREE NODE
RECEIVER
INTERNET ROUND TRIP DELAY
ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE
27THE TREE ROOT
HIGH DEGREE NODE
RECEIVER
INTERNET ROUND TRIP DELAY
ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Current Solutions
28THE TREE ROOT
HIGH DEGREE NODE
RECEIVER
INTERNET ROUND TRIP DELAY
ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Current Solutions
29THE TREE ROOT
HIGH DEGREE NODE
RECEIVER
INTERNET ROUND TRIP DELAY
ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Fast Algorithm
30THE TREE ROOT
HIGH DEGREE NODE
RECEIVER
INTERNET ROUND TRIP DELAY
ESTIMATING THE NUMBER OF RECEIVERS IN AN INTERNET
MULTICAST TREE Fast Algorithm
31Fast 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
32Fast Algorithm Delay
The delay Lets define
gamma random variable with
33Fast 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.
34Fast 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.
35How 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).
36Summary
- 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.