1. 1 © P. Kouznetsov A Note on Set Agreement with Omission Failures Rachid Guerraoui, Petr Kouznetsov, Bastian Pochon Distributed Programming Laboratory Swiss Federal Institute of Technology in Lausanne (EPFL) GETCO 2002, Toulouse, France

2. 2 Contribution  We consider the k-set agreement problem in a synchronous system with send-omission failures (up to f processes can fail).  We show that  f/k  +1 rounds are necessary to solve the problem and present the algorithm that matches the lower bound.  The lower bound proof develops the ideas of applying algebraic topology to distributed computing [HS93, BG93, HRT98,…]

3. 3 Related work  Asynchronous models: There is no k-resilient solution to k-set agreement in an asynchronous system of k+1 processes [BG93,HS93]  Synchronous crash-stop models: k-set agreement requires exactly l  f/k  +1 rounds, if  f/k  k  n-k l  f/k  rounds, if  f/k  k>n-k [HRT98,CHLT00]  Synchronous send-omission model: [Gaf98]: First  f/k  rounds of the model can be implemented from asynchronous (atomic snapshot) shared memory: [BG93] and [AAD+93] give  f/k  +1 lower bound

4. 4 Roadmap  Synchronous model with omissions  Problem of set agreement  Topological notions  The lower bound  The algorithm

5. 5 Model  n+1 processes p0,…,pn  Synchronized rounds: in each round r, every process pi : sends its local state to everyone; receives messages from other processes updates its local state  Send-omission failures might occur: in a given round, messages sent by pi to a subset of processes can be lost  At most f<n+1 processes can fail by send-omission p0 p1 p2 r=1r=2 n=2 f=2

6. 6 Problem : k-set agreement [Cha91] Processes propose initial values and are required to: 1.choose a decision value after a finite number of steps (termination) 2.choose as a decision value some process’s input value (validity) 3.collectively choose no more than k distinct decision values (agreement) k=1 : consensus -– processes eventually agree on a single proposed value Conjecture: k-set agreement is not solvable in our model in  f/k  rounds.

7. 7 Simplexes and complexes 1.A global state of the system is represented as an n-dimensional simplex S=(s0,…,sn), where s i defines local state of process pi 2.The result of applying a protocol (a set of model executions) P to an initial state S is represented as a protocol complex P(S): a set of simplexes corresponding to a set of global states of the system reachable by applying executions from P to S q p r S P(S) p,{p,q,r} q,{p,q,r}r,{p,q,r} q,{q,r}r,{q,r} p fails

8. 8 Connectivity A complex C is k-connected iff every continuous map of the k- sphere to C can be extended to a continuous map of the (k+1)-disk to C. (There are no «holes» of dimension k+1) 1. C1=({p,q},{q,r},{p,r),{p},{q},{r},{  }) 0-connected (graph connected), but not 1-connected (simply connected) 2. C2=({p,q,r},{p,q},{q,r},{p,r),{p},{q},{r},{  }) Both 0- and 1-connected p qr p qr

9. 9 Connectivity: continued 1.Non-empty complex is (-1)-connected 2.Any complex is k-connected for k+1<0 3.If K and L are k-connected and K  L is (k-1)- connected, then K  L is k-connected

10. 10 Pseudospheres:definition A complex  (S n ;U0,…,Un), where S n =(s0,…,sn) is defined as a set of simplexes {,…, }, where u i  U i, i=0..n, closed under containment. (If U0=…=Un=U we simply write  (S;U))

11. 11 Pseudospheres: examples  Simplex: S n  (S n ;U),  U  =1  Binary consensus:  (S n ;U),  U  =2 n=2: S n =(p,q,r) U=(0,1) (n=2) p qr

12. 12 Auxiliary lemma Lemma 1 For any P, S n, and constant c, such that, for any S m  S n, P( S m ) is (m-c-1)-connected, and a finite matrix of finite sets { A ij },i=0..l, j=0..n such that, for any j=0..n, l  0,  i=0..l A ij  , the complex P(  i  (S n ; A i0,…, A in )) is (n-c-1)-connected.

13. 13 Proof of Lemma 1 Reuse of arguments from [HRT98]: 1.For any non-empty sets U 0,…U n, P(  (S m ; U 0,…U n )) is (n-c-1)-connected By induction, starting from  Uj  =1, j=0..n (pre-condition) 2.For any l  0 and sequence { A ij }, such that  i=0..l A ij  , P(  i=0..l  (S n ; A i0,…, A in )) is (n-c-1)-connected By induction, starting from l=0 (case 1)

14. 14 Connectivity and set agreement  Theorem 1. If for every  (S n ;V), where V is non- empty, P(  (S n ;V)) is (k-1)-connected, then P cannot solve k-set agreement. [HRT98] (There is no map of each vertex of the protocol complex to a decision value, such that the properties of the problem are satisfied) Sperner’s lemma: For any map F:  (S n )  S n, that sends each vertex of a subdivision  (S n ) to a vertex of its carrier, there is (t0,…,tn) in S n, such that all F(ti) are distinct. n=2; k=2 There is no coloring scheme, such that each simplex has at most k different colors

15. 15 Lower bound: strategy  Main step: define a set R of 1-round executions of our model, such that preconditions of Theorem 1 are satisfied for t rounds of R: R t (  (S n ;V)) is (k-1)-connected for t   f/k   no decision map exists for k-SA (Intuition: R defines a set of 1-round executions in which at most k processes fail by omission [HRT98])  Conclusion: R  f/k  does not solve k-set agreement  there is no algorithm to solve k-set agreement in  f/k  rounds

16. 16 Lower bound: one round All executions in which at most k processes fail in a round: R(S m )    K  k  (S m ;2 K-{p0},…, 2 K-{pn} ) (m  n-k)

17. 17 Lower bound: multiple rounds Induction argument:  t=1: by Lemma 1, for any m, R(S m ) is (m-(n-k)-1)-connected  1<t   f/k  : assume that, for any m, R t-1 (S m ) is (m-(n-k)-1)- connected R t (S m )= R t-1 (R(S m ))   R t-1 (   K  k  (S m ;2 K-{p0},…, 2 K-{pn} )) (*) By Lemma 1, (*) is (m-(n-k)-1)-connected.

18. 18 Lower bound: final step 1.For any m and t   f/k , R t (S m ) is (m-(n-k)-1)- connected. 2.By Lemma 1, for any non-empty V, R t (  (S n ;V)) is (k-1)-connected. 3.By Theorem 1, R  f/k  cannot solve k-set agreement. Thus, no algorithm can solve the problem in  f/k  rounds.

19. 19 An optimal algorithm  Process pi: est_i := initial proposal for t=0..  f/k  do if (tk<=i<(t+1)k) then send est_i to all receive messages from other processes if some est_ j is received then est_i:=est_ j end for decide est_i Since (  f/k  +1)k>f, there is a round t in 0..  f/k  in which some process that never loses messages emits its message and every process updates its estimate. Not more than k distinct values can be adopted in round t.

20. 20 Concluding remarks  Contributions A «new» tight lower bound result. The proof is self-contained and simple.  Open issues Partially synchronous (eventually synchronous) lower bounds? Lower bounds for early deciding algorithms (in terms of «real» number of failures)?