The algorithm. For any nonempty set of vertices X ⊆ V , we choose a vertex ψ(X) from the subgraph H induced by (X) as follows: If the center of H contains a vertex that is non-simplicial in H, then let ψ(X) be any such vertex. Otherwise, let ψ(X) be any vertex in the center of H. By definition, ψ(X) has minimum eccentricity in the subgraph of G induced by (X). Since ψ(X) is a vertex in the convex hull of X, it is on some shortest path between two vertices in X. Let xi(0) be the input of process pi and let T∗ = |log3/2 diam(G)| + 1. The processes com- municate using a sequence S0, . . . , ST of single-writer snapshot objects, where T = max{|V |, T∗}. In each iteration t = 0, . . . , T , each process pi: – performs update on the ith component of the snapshot object St, setting it to the vertex xi(t), – performs scan on the snapshot object St, – defines Xi(t) be the set of vertices returned by its scan, and – sets xi(t + 1) = ψ(Xi(t)). Once pi has computed xi(T + 1), the process outputs this vertex and terminates. S
The algorithm. The basic idea of the algorithm is that we avoid the recursive majority voting of EIG by running a vote in each of f + 1 phases through a phase king, some process chosen in advance to run the phase. Since the number of phases exceeds the number of faults, we eventually get a non-faulty phase king. The algorithm is structured so that one non-faulty phase king is enough to generate agreement and subsequent faulty phase kings can’t undo the agreement. / Pseudocode appears in Algorithm 10.2. Each processes i maintains an array prefi[j], where j ranges over all process ids. There are also utility values majority, kingMajority and multiplicity for each process that are used to keep track of what it hears from the other processes. Initially, prefi[i] is just i’s input and prefi[j] = 0 for j = i. The idea of the algorithm is that in each phase, everybody announces their current preference (initially the inputs). If the majority of these preferences is large enough (e.g., all inputs are the same), everybody adopts the majority preference. Otherwise everybody adopts the preference of the phase king. The majority rule means that once the processes agree, they continue to agree despite bad phase kings. The phase king rule allows a good phase king to end disagreement. By choosing a different king in each phase, after f +1 phases, some king must be good. This intuitive description is justified below.
The algorithm. HRL warrants that to the best of its actual knowledge, the Algorithm does not infringe on any United States patent by anything, material, design, composition, or processing. HRL warrants it has not, prior to the Effective Date, entered into any agreement or otherwise granted any now existing, or agreed to grant any future, license, right or privilege relating to the Algorithm that conflicts in any way with this Agreement. HRL further warrants that the Algorithm is the product of its own research and invention and was created as a work for hire for EDI by HRL's staff and personnel. No third party has any claim, title or right to the Algorithm or any protected component parts.
