1. Distributed Algorithms (22903) Lecturer: Danny Hendler Global Computation in the presence of Link Failures

2. 2 Model P0: 4P0: 4 Processes are represented by graph nodes, each node stores an input value Bi-directional communication links Asynchronous Links may fail-stop but connectivity is assured (safe network) Failures cannot be detected. We let n, m respectively denote the number of nodes and links. P 1 : 17 P 2 : -6 P 3 : 46

3. 3 Global computation p2p2 We need to compute a global sensitive function of process inputs P0: 4P0: 4 P 1 : 17 P 2 : -6 P 3 : 46 Definition An n-variate function F is global sensitive, if there is an n-tuple, v 1, …, v n, such that the following holds:  i  {1,…. n}  u i : F(v 1,…, v i,…v n ) ≠ F(v 1,…, u i,…v n ) To compute a global sensitive function, we need to see ALL inputs.

4. 4 Global computation We need to compute a global sensitive function of process inputs Definition An n-variate function F is global sensitive, if there is an n-tuple, v 1, …, v n, such that the following holds:  i  {1,…. n}  u i : F(v 1,…, v i,…v n ) ≠ F(v 1,…, u i,…v n ) Examples: Max, sum, xor, …

5. 5 Global computation algorithm Every process broadcasts its input to all other processes. Worst-case message complexity: Ω(mn) Can we do better (in all networks)?

6. 6 Lower Bound on Uniform Algorithms Theorem For every n, m  O(n 2 ), there exists a safe network with θ(n) nodes and θ(m) links, on which the worst-case message complexity of any global computation is Ω(m log n).

7. 7 Lower Bound Proofs on Global Computation

8. 8 The graph G(n,m) p1p1 p2p2 p3p3 p4p4 P n-1 PnPn p0p0 bl 1 bl 2 bl 3 bl k tl 1 tl 2 tl 3 tl k br 1 br 2 br 3 br k br 1 br 2 br 3 br k k= √ m Cut L Cut R Path e1e1 e2e2 e3e3 e n-1

9. 9 Phase 0 p1p1 p2p2 P n-1 PnPn p0p0 bl 1 bl 2 bl 3 bl k tl 1 tl 2 tl 3 tl k br 1 br 2 br 3 br k br 1 br 2 br 3 br k k= √ m Cut L Cut R e1e1 e n-1 p n/4 p n/4+1 p 3n/4 p 3n/4+1 e n/4 e 3n/4 BP 0 UV 0

10. 10 elel emem erer BP i ={e l, e r } UV i elel emem Bp i+1 ={e l, e m } emem erer Bp i+1 ={e m, e r } Phase evolution