Group-Solvability The Ultimate Wait-Freedom Eli Gafni UCLA DISC /4/04

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

Motivation a 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 p j in a MWMR register C j (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 C j are eventually not Is there a non-trivial task r/w solvable in this model

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!

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

Motivation Cont’ed Since no apriori bound on the number of servers - of particular interest each group size is infinite (unbounded). A Task T n 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?

Group Solvability Solvability 3-proc convergence task 3-proc in each group: simulators each simulating In order: P123 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!

Is there Anything Interesting which is Group-Solvable? Yes: Renaming R(n,(n+1)n/2) n MWMR registers C 0,…,C n-1 initialized to p i,* : Ci :=1 k := |S i := snap{j|C j =1}| r := rank i in S i 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.

Same Result via “Splitters” Splitter: C i i=0,…,n-1 MWMR registers initially p i,* Ci:=1 Si:=Scan{j| Cj=1} if |S i |=1 then return “stay” else if i<max(S i ) 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

Same Result via “Splitters” Cont’ed AR

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?

The Main Result: Theorem: A task T n-1 on n processors p 0,…,p n-1 is group-solvable iff T n-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.

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