1. R/W Reductions Eli Gafni UCLA ICDCN06 12/30

2. Outline Tasks and r/w impossible task: –2 cons –3 cons NP-completeness R/W reduction “Weakest Unsolvable Task” Thesis Reductions Conclusions/Speculations

3. 2-cons Two procs P0 P1 P0 alone output 0 P1 alone output 1 P1 and P2 both output either 0 or 1

4. 2-cons impossible in r/w The protocol complex is a line The output is disconnected

5. 3-cons (election) Procs P0 P1 P2 Pi alone output i All procs output same i

6. 3-cons impossible in r/w If 3-cons was possible than 2-cons would be possible, contradiction. We proved 3-cons impossible by reduction FROM 2-cons

7. Analogy with NP-completeness In my Algorithm class students just get the fact that SAT is NP-complete. Showing that a problem is NP-complete does not require understanding TMs Showing a Task is r/w impossible should not require knowing Algebraic Topology!

8. The Quest: An Analog of SAT In our case it should be the Weakest (W) impossible task In some sense topology implies no such Weakest exists (upcoming paper: it does for any class of real interest)

9. R/W reducibility Cont’ed Reducibilty induces a directed graph over tasks A strongly connected component are tasks which are r/w equivalent Wishful “Weakest-Thesis:” –There exist a task WEAKEST(n): WEAKEST(n) is r/w unsolvable. WEAKEST(n) is reducible to any task which is unsolvable when restricted to participating set of at most n procs.

10. R/W reducibility Cont’ed If Weakest-Thesis hold then –“all Maurice can do, we can do better.” –Can think Java and not worry about not knowing Basic. –Can go back to thinking “distributed” rather than “topology.” Plausibilty All known unsolvable tasks are reducible to SB(n,2n-1) (Symmetry Breaking) SB(n,2n-1): p 0,…,p 2n-2 procs output 0 or 1 |P|=n not all 0’s and not all 1’s

11. Have to talk about Task Implementation 4 procs P0 P1 P2 P4 of which I’ll wake at most 2 If alone can output any 0 or 1 If both same parity, one of them output 0 and the other 1, if different parity, then both output same. (Vassos,Lo) Cannot do anything above r/w

12. But any implementation will do If implementable by r/w then a processor in solo execution is apriori decided whether will output 0 or 1. 2 out of the 3 have same solo say 0 –If same parity, 0 then win (since the other changes to 1) – If no same parity, then there is one parity 0 and other parity 1, the one that changes wins

13. Instead of Task, any implementation that solves the Task If the task is on n-procs then commit to some r/w protocol for the max size participating set for which is solvable For n get the output from the ``oracle.’’

14. SB(n,2n-1) equivalent to ``comparison’’ If SB(n,2n-1) know how to rename n ``comparison’’ into 1 to 2n-1 If ``comparison’’ then can glue together, I.e. the algorithm for 1,2,3 and 1,2,4 when 1,2 work alone no need to worry whether they are going to work with 3 or 4. Invoke object only in ``middle’’ (show example with SB11)

15. A weakest task for 3 procs

16. Family of weakest No task with 0,1 mapping symmetric on the boundary can be r/w colored to avoid all 0’s or all 1’s

17. R/W Reducibility Between Tasks Task A r/w reducible to B: A R/W B

18. Reductions: (3,2) -tst = always at least one proc outsputs 0, not all output 0. Impossible: Will solve SB11 –Do IS –At level 3 invoke tst, loser goes down –If remain at level 3 do 2-IS to get 0 if alone or 1 if see another

19. (3,2)-TST=(3,2)-ELECTION TST implies election: Register in SM Apply to tst If 0 elect yourself If 1, write ``I got 1’’ in SM Scan, elect registerd proc who has not announced 1 The proc who wrote ``got 1’’ first will not be chosen

20. Lemma: (3,2)-Election = (3,2) Strong Election Strong election = if elected by any then you elect yourself Election implies strong: –Phase 1: write (i, elected j) Scan, if someone elected you elect yourself, if choose else and see someone choose himself choose him –Phase 2 (separate memory): write(I,elected j) Scan, if someone elected you elect yourself, if choose else and see someone choose himself choose him

21. Tst=Strong Election (cont’) Strong election implies TST: –If elect yourself output 0 –Else 1

22. Conclusions: Weakest Thesis has ``experimental’’ support. Not true if ``any’’ task is considered Find the ``family’’ for which it is the weakest Missing; Internal reduction 3 SB implies 4 SB