1. Message Complexity Analysis of Shavit-Francez Termination Detection Algorithm
By:- Rizwan Malik
For Advanced OS Course 2007
Instructor: Dr. Mikhail Nesterenko

2. Overview Introduction. Objective. Enviornment. Observations. Results.
Coding Logic.
Conclusion & Future Work.
References.

3. Introduction
Shavit-Francez Algorithm is used for Explicit Termination of De-centralized Distributed Systems(Systems with Multiple Initiators).
Every initiator maintains its own computation tree and when its own computation tree collapses it may participate in the computation tree of other initiators.
All processes participate in a Wave to detect the the termination of the Computation.

4. Objective
To analyze the message complexity of Shavit-Francez Algorithm.
To determine how to reduce the message complexity by analyzing different Waves.

5. Enviornment Randomly Generated Trees were used for this project.
The Basic Computation was generated randomly.
Message Queue delays were introduced.
For the Wave, Echo and Tree algorithm were used.
Each data point value for the graph is based on an average of 10 values.

6. Observations
For a Basic computation with a Message complexity of M, the algorithm further introduces an increase of M(signals introduced by S-F Algo) + W(tokens generated by the Wave)+A(Stop messages sent by the Announce routine) messages.
The Shavit-Francez Algorithm always runs with worst case complexity.
Selection of the Wave Algorithm impacts the overall message complexity.

7. Results
Term with Echo
Term with Tree 10
70.2
62.2 20
144.2
126.2 30
207.6
179.6 40
271.2
233.2 50
334.6
286.6 60
404
346 70
468.2
400.2 80
525
447 90
595
507
100
655.2
557.2
200
1261.4
1063.4
300
1871.6
1573.6
The SF Algorithm with tree based wave has better message complexity than Echo based Wave.
Message Complexity – Echo=2E,Tree=N,Announce=2E.

8. Results(Continued)
M(BC+Signals+Echo+Announce)
W(BC+Signals+Tree) 10
74.4
56.4 20
137.6
99.6 30
210.8
152.8 40
278.8
200.8 50
342
244 60
401.2
283.2 70
464
326 80
530.4
372.4 90
589.6
411.6
100
650.4
452.4
200
1272.8
874.8
300
1872
1274
Since decide event takes place at all processes in a Tree based wave, if we halt every process after it decides, we can achieve a better message complexity for the algorithm.However, this observation is limited to tree graphs.

9. Result(Continued..) Fig1
No. Of Nodes
Edge Count
Basic Messages
Echo Tokens
Tree Tokens
Signals
Announce Stop Messages 10 9
17.1 18 20 19
34.1 38 30 29
45.8 58 40 39
57.6 78 50 49
69.3 98 60 59 84
118 70 69
96.1
138 80 79
104.5
158 90 89
119.5
178
100 99
129.6
198
200
199
232.7
398
300
299
337.8
598
Fig1
Fig1– Data for Graph 1 Mcomplexity for Tree=N.Fig2- Mcomplexity = 2N-2=2E(for trees).
No. Of Nodes
Edge Count
Basic Messages
Echo Tokens
Tree Tokens
Signals
Announce Stop Messages 10 9
19.2 18 20 19
30.8 38 30 29
47.4 58 40 39
61.4 78 50 49 73 98 60 59
82.6
118 70 69 94
138 80 79
107.2
158 90 89
116.8
178
100 99
127.2
198
200
199
238.4
398
300
299
338
598
Fig2

10. Coding Logic Randomly Generated Computation:- Queue Delay:-
One message sent/received at a time.
Leaves are initiators.
A process can, at the maximum, send 5 messages.
Queue Delay:-
Delay simulated by Placing the message at index generated randomly between values 0 and Queue length.
Random Trees:-
Matrix(Double indexed array) of size NxN --- N is a user input value.
Value of pair(I,J)=1 denotes the presence of an Edge between I & J.
Recursively checks for cycles before adding an Edge to the tree.
Probability of an Node joining the tree is 50%.

11. Conclusion & Future Work
The Shavit-Francez component of the Algorithm always runs with worst-case message complexity i.e. it always generates number of messages equal to number of messages generated by the Basic algorithm.
The choice of wave algorithm to be run can effectively reduce the message complexity of the Algorithm as a whole.
For Tree networks the wave with best message complexity is Tree Algorithm based wave.
Future Work:-
Perform the same analysis on Arbitrary Graphs.
Find the optimal wave for different graph types.
To work towards genralize the technique of converting algorithm that work on diffusing Algorithms to those that work on non-diffusing Algorithms as suggested by N. Shavit, N. Francez in their paper titled “A New Approach to Detection of Locally Indicative Stability“.

12. References
Class notes for Advances Operating Systems 2007 by Dr. Mikhail Nesterenko.
Introduction to Distributed Algorithms By Gerard Tel.
Nir Shavit, Nissim Francez: A New Approach to Detection of Locally Indicative Stability.