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

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

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.

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

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.

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.

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.

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.

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

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%.

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“.

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.