1. chapter 5.1 - Resynchsonous Stabilizer 5.1- 1 Chapter 5.1 Resynchsonous Stabilizer Self-Stabilization Shlomi Dolev MIT Press, 2000 Draft of Jan 2004, Shlomi Dolev, All Rights Reserved ©

2. chapter 5.1 - Resynchsonous Stabilizer 5.1- 2 The Problem  Converting non-stabilizing algorithms to self- stabilizing algorithms  We will deal with a fixed input, fixed output algorithms  The main problem is that we do not know in what state does the algorithm begin, so we cannot perform initiation

3. chapter 5.1 - Resynchsonous Stabilizer 5.1- 3 Definitions Denote :  AL – A non stabilizing algorithm  SA – The stabilizing version of AL  Synchronous execution time of AL – the number of steps until the algorithm produces the desired output, in a synchronous system (any asynchronous algorithm works also in a synchronous system)

4. chapter 5.1 - Resynchsonous Stabilizer 5.1- 4 A First Solution  Every processor maintains the list of states it had been in during the algorithm  At every step the processors send to their neighbors their lists  A processor that receives the tables of it’s neighbors checks it's table. If there is an error, fix it. Else compute the next state

5. chapter 5.1 - Resynchsonous Stabilizer 5.1- 5 Example - Leader Election 3 10 … 7 5 9 … 1000 105 3 … 20 5 7 … 3 … 10 1000 … 5 20 … 7 1000 … 10 … 3 7 5 5 1000 … 7 7 … 3 10 1000 … 7 7 10 … 5 … 3 … …

6. chapter 5.1 - Resynchsonous Stabilizer 5.1- 6 Why Does it Work?  The first state of AL is defined only by the initiation of the processors  After the first cycle, all the processors have the right first state  Let us assume that after n cycles all processors have n correct entries in their tables  In the next cycle every processor will get the tables of its neighbors and compute its n+1 state correctly

7. chapter 5.1 - Resynchsonous Stabilizer 5.1- 7 Perhaps it is too Expensive  The shown approach produces an algorithm that has the same time performance of the non- stabilizing algorithm  Space complexity is a different story. The processors hold in memory the computing sequence of all their neighbors  The messages are very big, and on every cycle messages are sent on every edge

8. chapter 5.1 - Resynchsonous Stabilizer 5.1- 8 A New Hope  A possible way of improving the synchronizer is to hold a floating output  If AL has execution time of t, the system includes a stabilizing counting algorithm which has a wraparound larger than t  Every time the counter reaches 0, the system is initialized. The floating values are not initialized.

9. chapter 5.1 - Resynchsonous Stabilizer 5.1- 9 Example - Consensus Algorithm  In the consensus algorithm, the algorithm should return an answer after d cycles  Every d cycles, the output is copied to the floating output, and the input is copied to the output  After the first round, the floating output will remain constant

10. chapter 5.1 - Resynchsonous Stabilizer 5.1- 10 Randomized Algorithms  When we try to implement the above method to random algorithms we encounter two difficulties:  A randomized algorithm does not have an upper bound to it’s execution  The output of such algorithms may change from one execution to another. The floating value can be changed infinitely often

11. chapter 5.1 - Resynchsonous Stabilizer 5.1- 11 The Solution  Check the old output validity, if invalid  Execute the algorithm  Every t cycles, check if the algorithm is stabilized. if it is not, reset the system  In some executions of the algorithm it will stabilize (with probability 1)  When stabilized update the output

12. chapter 5.1 - Resynchsonous Stabilizer 5.1- 12 Example – Leader Election  In the uniform system leader election algorithm, every processor chooses a random ID, and sends it to it’s neighbors  If a processor receives an ID larger than it’s own, it resigns from the “race”, and will not participate in the next round  If no message is transmitted through a whole round, all processors are initiated