Group-Solvability The Ultimate Wait-Freedom

1. Group-SolvabilityThe Ultimate Wait-Freedom Eli Gafni UCLA DISC 2004 10/4/04

2. Outline • Motivation • Group Solvability Solvability • Non-Trivial Group-Solvable Task • The Main Result • Conclusion

3. Motivationa great solution in search of a fitting problem :) • Clients-Servers Model • Clients announce input to server via SWMR-SM • Servers compute and deposit correct result for client pj in a MWMR register Cj (initial ) • Server do not work on behalf of any particular client and the number of servers may be unbounded • If all but one server fail-stop all Cj are eventually not Is there a non-trivial task r/w solvable in this model

4. Motivation Cont’ed • Suppose we restrict the requirement and a server just works on behalf of a single client • The set of server working on behalf of the same client is a group • Servers of the same group step on each other writing an output • No matter what output is chosen it is ok • Conclusion: the tuple of outputs created by a chice of any single rep for a each group constitutes a valid output tuple!

5. Motivation Cont’ed • Example- 3 processor renaming (3,5) • Input/output: a client pi appears i=0,1,2 output unique slot in {1,…,5} • Servers: qi,1 ,qi,2 working for pi i=0,1,2 • Can be viewed as the following task over servers: • 6 processors task • A processor outputs a slot in {1,…,5} • qi,*, q j,* i j output different slots

6. Motivation Cont’ed • Since no apriori bound on the number of servers - of particular interest each group size is infinite (unbounded). • A Task Tn is r/w group-solvable if the task with each group infinite is solvable • Motivation in the paper for group-renaming • Each member of the group in possession of the same info to be posted • As long as the whole group does not fail at least one posting will happen • Minimal number of MWMR posting boards needed?

7. Group Solvability Solvability 3-proc convergence task 3-proc in each group: 1 2 3 1 2 3 1 2 3 3 simulators each simulating In order: P 1 2 3 P 1 2 3 P 1 2 3 A simulator determines a simplex and outputs the color-tower By compatibilty of outputs only two adjacent simplexes possible 3 simulators solve 2-set election!

8. Is there Anything Interesting which is Group-Solvable? Yes: Renaming R(n,(n+1)n/2) n MWMR registers C0,…,Cn-1 initialized to pi,* : Ci :=1 k := |Si := snap{j|Cj=1}| r := rank i in Si return k(k-1)/2 + r Each snap of size k has k consecutive dedicated registers that come after all the dedecated registers fo j<k and before all the registers for j>k.

9. Same Result via “Splitters” Splitter: Ci i=0,…,n-1 MWMR registers initially pi,* Ci:=1 Si:=Scan{j| Cj=1} if |Si|=1 then return “stay” else if i<max(Si) then return “left” else return “right” Max does not go “left” min not “right” Donot try X,Y Group-splitter - by AEG03 does not exits

10. Same Result via “Splitters” Cont’ed 1 2 3 5 6 4 10 8 9 7 . . . . . AR95

11. Contemplation: • Renaming with n infinite groups can be done in finite number of slots. • If we grow up the size of the groups 1,2,… at what integer the growth of the number of slots stops?

12. The Main Result: Theorem: A task Tn-1 on n processors p0,…,pn-1 is group-solvable iff Tn-1 is solvable for groups of size n-1. Proof proceed by infinite number of processors simulating the n-1 group size algorithm. The simulation uses about bn^3 MWMR registers and simulator make take a step on behalf of any processor. Corollary: 3-proc convergence is not solvable even for groups of size 2.

13. Conclusions • New Clients-Servers Model • Eg BG simulationcan be thought of as client server with guarantee of receiving c-(s-1) results • Lower bound on Group-Solvable renaming and seamless algorithm as groups grow • Imm Snaps is not Group Solvable but “there exists” Group-Solvable