Towards optimal synchronous counting
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1 Towards optimal synchronous counting Christoph Lenzen Joel Rybicki Jukka Suomela MPI for Informatics MPI for Informatics Aalto University Aalto University PODC 5 July 3
2 Focus on fault-tolerance Fault-tolerant co-ordination primitives: permanent failures (Byzantine faults) transient failures (self-stabilisation)
3 Focus on fault-tolerance Fault-tolerant co-ordination primitives: permanent failures (Byzantine faults) transient failures (self-stabilisation) Find solutions that are fast to recover space and communication-efficient
4 Our contribution A deterministic round counter with high resilience optimal recovery time low space/message complexity
5 Model of computing n state machines Synchronous rounds:. broadcast. receive 3. update state 3 4
6 Model of computing n state machines Synchronous rounds:. broadcast. receive 3. update state 3 4
7 Model of computing n state machines Synchronous rounds:. broadcast. receive 3. update state 3 4
8 Model of computing n state machines Synchronous rounds:. broadcast. receive 3. update state 3 4 Algorithm A maps a vector of states to a new state!
9 Complexity measures Time complexity: #rounds recovery time Space complexity: log #states complexity of the circuit number of bits broadcast per node
10 On failures our adversary
11 Transient failures n state machines arbitrary initial states chosen by adversary! = self-stabilisation 3 4
12 Byzantine failures n state machines f Byzantine failures 3 4
13 Byzantine failures n state machines f Byzantine failures 3 4
14 Byzantine failures n state machines f Byzantine failures 3 4
15 Byzantine failures n state machines f Byzantine failures 3 4
16 Byzantine failures n state machines f Byzantine failures 3 4 Non-faulty nodes can observe different states for the system!
17 Counting mod c 3-counting increment counter + mod c 3 4 Global pulse
18 Synchronous counting Counting 3 4 Global pulse
19 Synchronous counting Stabilisation Counting 3 4 Global pulse
20 Synchronous counting Stabilisation Counting! 3????????? 4 Global pulse
21 A related problem: Consensus Input: private bit Output: agreement
22 A related problem: Consensus Input: private bit Output: agreement
23 A related problem: Consensus Input: private bit Output: agreement
24 A related problem: Consensus Input: private bit Output: agreement
25 A related problem: Consensus Input: private bit Output: agreement
26 Reduction from consensus Given a -counting algorithm A that stabilises in t rounds consensus solvable in t rounds Dolev et al. (3)
27 Consensus lower bounds* Resilience f<n/3 Pease et al. (98) Time At least f rounds to reach agreement Fischer & Lynch (98) *deterministic
28 Prior work on -counting Resilience Time State bits D. Dolev & Hoch 7 * f<n/3 O(f) O(f log f) *deterministic
29 Prior work on -counting Resilience Time State bits D. Dolev & Hoch 7 * f<n/3 O(f) O(f log f) S. Dolev & Welch 4 f<n/3 (n f) O() *deterministic
30 Prior work on -counting Resilience Time State bits D. Dolev & Hoch 7 * f<n/3 O(f) O(f log f) S. Dolev & Welch 4 f<n/3 (n f) O() Ben-Or et al. 8 f<n/3 O() n O() *deterministic
31 Current state Resilience Time State bits D. Dolev & Hoch 7 * f<n/3 O(f) O(f log f) S. Dolev & Welch 4 f<n/3 (n f) O() Ben-Or et al. 8 f<n/3 O() n O() Our work* f = n o() O(f) o(log f) *deterministic
32 Counting vs consensus? Counting Consensus
33 Counting vs consensus Counting Consensus Solve consensus to agree on counters
34 Counting vs consensus Counting Consensus Use a c-counter as a round counter; execute a consensus algorithm
35 Counting vs consensus? Counting Consensus
36 Counting vs consensus! Counting Consensus Solution: counters that work once in a while
37 Counting once in a while?????? 3 4??!??? Counting Arbitrary Counting Arbitrary
38 Counting once in a while Use proper counters with low resilience, to count once in a while with high resilience! Counting Counting once in a while low resilience high resilience
39 Clock for consensus If we can count once in a while with high resilience.. then we can execute a highly-resilient consensus algorithm (phase king) Berman et al. (989)
40 Agree on a new counter Use consensus protocol with high resilience, to agree on a new proper counter!
41 High-level idea Counting low resilience
42 High-level idea Counting low resilience Counting once in a while high resilience
43 High-level idea Counting low resilience Counting once in a while high resilience Consensus high resilience
44 High-level idea Counting low resilience Counting once in a while high resilience Consensus high resilience Counting high resilience
45 Rinse and repeat low resilience high resilience higher resilience
46 The boosting theorem More formally
47 The boosting theorem Algorithm A n f nodes faults
48 The boosting theorem Algorithm A n f nodes faults Algorithm N = kn B F (f + )k/
49 The boosting theorem n f Stabilisation time: Space complexity: N = kn F (f + )k/ T (B) =T (A)+O(Fk k ) S(B) =S(A)+O(log C) C = new counter range
50 Boosting resilience Boost resilience recursively while keeping time and space complexity small enough
51 Main result Resilience f = n o() Stabilisation time O(f) Space complexity O(log f/log log f)
52 Main result Resilience f = n o() Stabilisation time O(f) Space complexity O(log f/log log f) Thanks!
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