Global state collection Chandy Lamport algorithm does a partial job. Each process collects a fragment of the global state, but these pieces have to be put together. Applications - computing network topology - termination detection - deadlock detection etc

A simple exercise s(i) s(j) i j s(k) s(l) k l Collect global state via all-to-all broadcast At the end, each process will compute a set V, where V= {s(i): 0 ≤ i ≤ N-1 } s(i) s(j) i j s(k) s(l) k l

A simple exercise V.i W.i (i,k) V.k W.k Program broadcast (for process i} define V.i, W.i : set of values; initially V.i={s(i)}, W.i =  andevery channel is empty do V.i ≠ W.i send (V.i \ W.i) to every outgoing channel; W.i := V.i  ¬ empty (k, i) receive X from channel (k, i); V.i := V.i  X od V.i W.i (i,k) V.k W.k

Proof empty (i. k)  W.i  V.k. V.i W.i (i,k) V.k W.k (Upon termination) i: V.i = W.i, and all channels are empty. So, V.i  V.k. On a cyclic path, V.i = V.k must be true. Since s(i) V.i, s(i) V.k V.i W.i (i,k) V.k W.k

Termination detection In a network, processes may be active or passive. Termination implies: (a) every process is passive, (b) all channels are empty, and (c) the global state of the system satisfies the desired postcondition

Termination detection The progress of a computation. Notice how one process engages another process. Eventually All processes turn white - this signals termination.

Dijkstra-Scholten algorithm An initiator initiates termination detection by sending a signal down the edge when it engages another node. At a “suitable time,” the recipient sends an ack back. When the initiator receives ack, it detects termination. Node j engages node k. j k j k j k

Dijkstra-Scholten algorithm Deficit (e) = # of signals on e - # of ack on e For any node, C= deficit along incoming edges D= deficit along outgoing edges edges Invariant 1. (C ≥ 0)  (D ≥ 0) Invariant 2. (C > 0)  (D = 0) 1 2 3 4 5

Dijkstra-Scholten algorithm Invariant 1. (C ≥ 0)  (D ≥ 0) Invariant 2. (C > 0)  (D = 0) The invariants must hold when an interim node sends an ack. So (C-1 ≥ 0)  (C-1> 0 D=0) {follows from INV1 and INV2} = (C>1)  (C≥1  D=0) = (C>1) (C=1  D=0) 1 2 3 4 5

Dijkstra-Scholten algorithm program detect {for an internal node i} initially C=0, D=0, parent = i do m = signal  (C=0)  C:=1; state:= active; parent := sender m = ack  D:= D-1 (C=1 D=0)  state = passive  send ack to parent; C:= 0; parent := i m = signal  (C=1)  send ack to the sender; od 1 2 3 4 5

Dijkstra-Scholten algorithm The engagement edges form a spanning tree. The tree grows and shrinks. Signals are acknowledged on very-first-in-very-last-out basis Termination is detected when the initiator receives ack. 1 2 3 4 5