Is 1 different from 12?

1. Is 1 different from 12? • Eli Gafni • UCLA

2. OR • Is theoretical distributed computing is a brunch of Mathematics or Engineering? • What is theoretical distributed computing ``on one leg?’’ • An algorithm has a geometric representation and a ``code’’ representation. Which should I use?

3. Consensus Task • Two versions: • 0-1: procs input is 0 or 1 • All output o or all output 1 • if output i then some proc input is i • election: proc input is its id • all output the same participating id

4. k-set Cons (Chaudhuri 90) • Two versions • k-value: Proc’s input and output from {0,1,...,k} • At most k distinct output, if i is output then some proc input is i • k-Election: Proc input is its id • All output together at most k participating ids

5. 1=?k • Can any property of cons has it analogue to k-set cons? • TST= 2 procs cons ? • 0-1 cons = election ? • State machine approach to distributed computing ?

6. What about n Processors with m<n consensus? • With 2-cons can do Test-and-Set

7. TST (Test-and-Set) • TST: One proc output 0 all the rest 1 (2,1) • k-TST: At most k at least one proc output 0 all the rest 1 (k+1,k) • TST implementable for 2 proc cons =(2,1) • k-TST implementable from (k+1,k) set cons. 1 2 3 n

8. But what about m=3 • (3,1), (k+2,k) • ...[Afek, Weisberger]

9. Is View of k-set a la Chahudhuri is the most convenient? • What will be the analogue of binary consensus? • With consensus b.c.=election, how do I use at most k values to elect at most k leaders?

10. All of DC ``on one foot!’’ • Geometric approach • Agreement:

11. Digression: DC= -Agreement MWMR Registers While total weight < W make next 0 into 1 at your corner Return CG 1 1 1 0 0 Center of Gravity 0 1 1 0 0 1 0 0 0 0

12. Algorithmically a • All processors return vertices from same triangle restricted to the face of participation b c d...

13. How to converge on ``abstract’’ subdivided simplex • Add ``weight’’ until the ``longest path’’ • If not max add 1 and step one in shortest path to max • Cannot converge without ``adding’’ unless n=2.

14. ? Ahm...but 2 procs can do 2 0-1 independent cons with at least one output! • Can do - agreement at least one agreement no proc will output ? • But can do - agreement for any number of processors if I can group them in two groups • With 2-set cons can solve at least one of 2 independent cons! 0 1 ? 1 0 2 3 3 2

15. o-1 = election (k?) • What is the meaning of 0-1 when it comes to k-set???

16. In General • 3 procs • Cons i=1,2,3 • Say cons 2 • -agreement on the line perpendicular to the base • Only 2 lines intersect at a point 0(2) 0 1 1(2) 0(2)

17. What if 2-set and 3 ind cons? • 2 out of the 3 cons will return decision • With k set cons and m>k ind cons then m-k+1 cons will return Text

18. How to do Multi-Valued 123 123 123 123 123 123 123 123 123 123

19. Algorithmically • 2 set = 2 ind cons at least on return • Choose two vectors (a1,a2) (b1,b2) • post vector proposal for cons on 1st entry • if alone return for first entry • else proceed return b2 for 2nd entry

20. Algorithmically • 2 set cons with 3 ind cons return from at least 2 con’s • will have 2 3-entry vectors (a1,a2,a3) (b1,b2,b3) • post for 1st, if alone return from 1st, post chosen value, proceed to 2nd • if not alone proceed with B vector to second • ... • Return all posted values

21. How to do election from set binary-cons • Will see ``prefix’’ of id for jth cons and propose a bit continuation for a leader name • But how I know who return what from were: While I assume nobody returned in fact ``she just was late’’

22. How to do Multi-Valued • If take small enough if any proc commit all commit or adopt same commit 0 commit 1 adopt 1 adopt 0

23. 1-State-Machine • Want to emulate: • Repeat: • All send command to the ``center’’ • Center chooses one and send it to all • Implement in rounds in SM using consensus in each round

24. 2-State-Machine • Want to emaulate • Repeat synchronously • All send to ``center1’’ and ``center2’’ at least one of which is omission non-faulty at the round • Center(s) choose and send back • Can emulate with 2-set cons?

25. Solution 2-State-Machine with 2-set cons • Do 2 independents cons • If get ``adopt v’’ at cons[i] the next command to cons[i] is the adopted value • If the adopted value was commited all will propose same and commit • If collid with the new value just commit the adopted

26. ? Too Numerous Ramifications • Subsumes BG • What is BG? • If cannot answer 1st-cons both participate, will proceed to second, if collide in 2nd-cons, see exact positions in 1st-cons and can decide it ? 1 2

27. Conclusions • It is........ MATH.....