Distributed Algorithms (22903)

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

2. Model P1:17 P3:46 P0:4 P2:-6 • 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.

3. Global computation P1:17 P3:46 P0:4 p2 P2:-6 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,v1, …, vn, such that the following holds: i {1,…. n}  ui: F(v1,…, vi,…vn) ≠ F(v1,…, ui,…vn) To compute a global sensitive function, we need to see ALL inputs.

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,v1, …, vn, such that the following holds: i {1,…. n}  ui: F(v1,…, vi,…vn) ≠ F(v1,…, ui,…vn) Max, sum, xor, … Examples:

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. Lower Bound on Uniform Algorithms Theorem For every n, mO(n2), 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. Lower Bound Proofs on Global Computation

8. The graph G(n,m) p0 k=√m tl1 tl2 tl3 tlk br1 br2 br3 brk CutR CutL bl1 bl2 bl3 blk br1 br2 br3 brk e1 e2 e3 en-1 p1 p2 p3 p4 Pn-1 Pn Path

9. BP0 UV0 Phase 0 p0 k=√m tl1 tl2 tl3 tlk br1 br2 br3 brk CutR CutL bl1 bl2 bl3 blk br1 br2 br3 brk e3n/4 en/4 e1 en-1 p3n/4 p3n/4+1 pn/4+1 pn/4 p1 p2 Pn-1 Pn

10. Bpi+1={el, em} Bpi+1={em, er} el em em er Phase evolution BPi={el, er} el em er UVi