Optimizing Data Partitioning at Broadcasting the Data

Optimizing Data Partitioning
at Broadcasting the Data on
Balanced N-ary Tree Formed
P2P Systems
Takashi Yamanoue @ Fukuyama University
ESKM2015, 4th IIAI AAI @ Okayama, Japan, 15Jul, 2015.

Contents
• I. INTRODUCTION
• II. THEORETICAL EQUATION FOR OPTIMIZING THE NUMBER OF DATA
PARTITIONING AT BROADCASTIONG ON A BALANCED N-ARY TREE
• III. SOLAR-CATS
• IV. PERFORMANCE IMPROVEMENT OF SOLAR-CATS
• V. RELATED WORKS
• VI. CONCLUDING REMARKS

I. Introduction
• SOLAR-CATS
– A Teaching tool for large size computer laboratories and small seminar
classes
– Does not need a server because it uses peer to peer (P2P) technology.

• Remote operation
– of an application which is equipped with SOLAR-CATS,
– on every PC in the class from one PC in the class.
• Interactive operation
– of an application by all class members.
– Has a Mutual exclusion function

• Sending of images
– from one display in the class to all other displays quickly
• Annotation
– is also possible and live changes on one display to all other displays can
be continuously distributed.
• The recording and replaying operations
– of applications on SOLAR-CATS.

• The P2P technology of SOLAR-CATS
– a kind of structured P2P system.
– adopt the balanced N-ary tree as the structure.
– TCP connection as the connection.
– When a node receives a broadcast message from one connection, the
node sends the message to all of other connections if the node has
them.

１２
３４
５６
２３
３４４５４５
ｂ＝２, i=4
ｔ＝３

• If traffic of any connection does not affect the traffics of any
other connections in the group
• and when a broadcast message is sent from one node to its all
connections,
• all nodes in the group receive the message in a term of
O(log n) where n is the number of nodes in the group.

• a Theoretical Equation for optimizing the number of
data partitioning at broadcasting on the balanced N-ary tree
shaped P2P system.
• Applied the equation to improve the performance of image
broadcasting function of SOLAR-CATS
• the performance has been improved.
• The SOLAR-CATS has been used in real classes.

II. THEORETICAL EQUATION FOR OPTIMIZING THE NUMBER OF DATA
PARTITIONING AT BROADCASTING ON A BALANCED N-ARY TREE
• We’d like to make it fast
– Broadcasting on a Balanced N-Ary Tree, using TCP.
• We know
– Usually, Messages of
• a small number of a large size > a large number of a small size
– when the data is partitioned into pieces of message and they are sent from one node
to another node.
– For example, Jumbo Frame.

• However,
– when a message is sent from one node to plural nodes,
– without using IP multicast nor frame broadcast for keeping reliability of
the message passing,
– there is the term that one or more of receiving nodes of them is/are
not working.
– When the size of the message becomes larger, the term also becomes
longer. So there should be an optimum number of partitioning.

• We assume
– Bandwidths and latencies of all connections are the same
– Performances of all nodes are the same
– Traffic of any connection does not affect the traffic of other
connections.

• b : the number of branches of the balanced N-ary tree
• N : the number of nodes.
• i: height of the tree.
– if b=2 , N, is given by the equation N=b^i-1 .
• t: partitioning number.
• and messages were sent in turn from the root node.

１２
３４
５６
２３
３４４５４５
ｂ＝２, i=4
ｔ＝1
If no partitioning

１２
３４
５６
２３
３４４５４５

１２
３４
５６
２３
３４４５４５
There are terms,
many nodes
which are not working

１２
３４
５６
２３
３４４５４５
Terms of not working
are reduced.

•T=CBb(t+i-2)+d(t) (1)
–CB: term of transfer one piece of the data
–b: b-ary tree, t: partitioning number
–d(t): the relay time

• CB = 8s/(wt) + dH (2)
–s: whole size(byte) of data
–w: bandwidth(bit/sec)
–dH: the latency between connected nodes.

• d(t) = a + ct (3)
– a,c : constants

• Assigning (2), (3) in to (1),
• T=8sb/w + a + dHb(i-2)+
8sb(i-2)/(wt)+t(dHb+c) (4)
= A + B/t + Ct (5)
– There is the t0 which makes the T minimal
by assigning the t0 into t of the (5)

• A=8sb/w + a + dHb(i-2)
• B=8sb(i-2)/w
• C=dHb+c

• The t0 is the positive solution of the following
–T’=-B/t^2 +C = 0
• So
–t0 = sqrt(B/C)=sqrt(8s(i-2)/{w(dH+c/b)}) (7)

Group Manager
Teacher’s node system
TCP TCP
TCP TCP TCP
TCP
Student’s node
system
Student’s node
system
Student’s node
system
Student’s node
system
Student’s node
system
Student’s node
system
III. SOLAR-CATS
A Computer Assisted Teaching
System
for large size
computer laboratories and
small seminar classes

Writer’s Assistant
Web Browser
Programming Environment
Text Editor
Draw
Applications
Main Controller Command
Transceiver
Event Recorder/
Player
Network

• SOLAR-CATS includes
–PC Screen Image sharing function.
= Sends the screen Image from one PC to other PCs

• Remote operation
• Interactive operation
• Sending of images
• Annotation
• The recording and replaying operations

IV. PERFORMANCE IMPROVEMENT OF SOLAR-CATS
• Environmnent
– Send an Image at one node, 1.3MB, not compressed.
– 100Mbps switch. latency of the switch is 2.3μ sec.
– CPUs : Intel Pentium 1.3GHz or better.
– The sizes of the memory were more than 500MB.
• Before
– The image was partitioned in to 572 pieces.

TABLE I. SENDING TIME OF OLD FUNCTION
Number of Nodes
(with root)
Sending Time(sec.)
(measured 10 times)
Min. Max. Ave.
3 8.84 9.51 9.01
6(i=3) 8.68 10.05 9.33
(t=572)

TABLE I. PARTITIONING NUMBERS AND SENDING TIMES
Number of
Nodes
(with root)
Partitioning
number
(width)
Sending Time(sec)
(measured 10 times)
Min. Max. Avg.
3 572(24)
143(48)
36(96)
21(128)
9(196)
5(256)
4.47
1.72
2.10
1.58
1.65
1.48
8.58
2.54
3.35
2.05
2.35
2.14
6.13
2.09
2.56
1.82
1.99
1.92
7 572(24)
143(48)
36(96)
21(128)
9(196)
5(256)
3.21
2.03
2.21
1.97
1.90
1.99
9.75
4.95
3.09
2.45
2.48
2.08
4.67
3.11
2.64
2.25
2.18
2.01
(4times)
Frozen

• When the number of nodes is three and the b is 2, i.e. i=2, the B
of the equation (5) is 0. So there should be linear relationship
between partitioning numbers and the sending times.
• T=A + B/t + Ct (5)
–B=8sb(i-2)/w

• The fitting line of the (a) of the Figure 4 shows
– partitioning numbers and the sending times is almost linear
– because R^2 is 0.937. … our assumption of (3) is realistic.
• A of (5) is approximately 1.8, C of (5) is approximately 0.007.
• So a of (3) should be about 1.6(sec.). d(t) = a + ct (3)
– The a includes the term of screen capturing and screen rendering. This
value seems realistic.

• When the number of nodes is seven, i.e. i=3, the B of the equation
(5) should be about 0.2 because we know s=1.3MB, b=2, i=3,
w=100Mbps. We assume that we can ignore the B because
partitioning numbers are large enough for the ignorance for
almost every partitioning number in the TABLEII.
T=A + B/t + Ct (5)
B=8sb(i-2)/w

t0 = sqrt(B/C)=sqrt(8s(i-2)/{w(dH+c/b)}) (7)
• The C of the equation (5) is about 0.04 from the best fit line of
the (b) of the Figure 4. The optimal partitioning number t0 of the
equation (7) should be about seven.
Sorry, We could not measure the time when it is. However,
• This number is also realistic because when the partitioning
number was five or nine, average sending times of them in the
TABLE II are almost two seconds and they are the smallest
values in the table.

• Video of an improved SOLAR-CATS. (from 1min.30sec.)

V. RELATED WORKS
• The electronic chalk board by Hirahara and others [3][8] enables
sharing the same image with a large number of users using P2P.
This transmits an image of teacher’s screen to students’
screens uni-directionally. However, SOLAR-CATS enables bi-
directional exchanging of information between the teacher and
students using a real time sharing of operations.

• QuickBoard [4] is a web based WYSIWIS and it can be used for a
large size class which has up to two hundred terminals. However,
it uses a high performance server, and is also uni-directional.
• Wb [2] is a popular tool for real time communication among
remote users using a draw program on a multicast network.
However, wb is not equipped with mutual exclusion function.

• ESM [1], RelayCast [5] and Emma [6] are ALM (Application Level
Multicast) systems; our system is also a kind of ALM. They are
used for exchanging streaming data while SOLAR-CATS is used
for sharing the same operation.

VI. CONCLUDING REMARKS
• A theoretical equation for optimizing the partitioning number at
broadcasting data.
• Improved a computer assisted teaching system
• The equation and the results of our experiment did not conflict
so much with the network programming guide in the
Tanenbaum’s text book [7].
• We are improving the performance of SOLAR-CATS much more.

ACKNOWLEDGMENT
• We thank our students who help us to develop and test SOLAR-
CATS.
• A part of this work was supported by Grant-in-Aid for Scientific
Research of Japan Society for the Promotion of Science,
Fundamental Research(C), 17500041.

• The t0 is the positive solution of the following
–T’=-B/t^2 +C = 0
• So
–t0 = sqrt(B/C)=sqrt(8s(i-2)/{w(dH+c/b)})

Optimizing Data Partitioning at Broadcasting the Data

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