10. An intern in the Independent Party campaign designs an independent SWIM/ping-based failure detection protocol, for an asynchronous distributed system, that works as follows. Assume there are N=M*K*R processes in the

system (M, K, R, are positive integers, each > 2). Arrange these N processes in a MxKxR 3-dimensional matrix (tesseract), with M processes in each column, and K processes in each row, and R processes in the 3rd dimension (aisles). All processes maintain a full membership list, however pinging is partial. Each process Pijk (in i-th row and j-th column and k-th aisle) periodically (every T time units) marks a subset of its membership list as its Monitoring Set. The monitoring set of a given process, once selected, does not change. The monitoring set of Pijk contains: i) all the processes in in its own column Pjk, ii) all the other processes in its own row Prk, and ii) all the processes in in its own aisle Pij. At this point, there are two options available to you: Option 1 - Each process sends heartbeats to its monitoring set members. Option 2 - Each process periodically pings all its monitoring set members; pings are responded to by acks, just like in the SWIM protocol (but there are no indirect pings or indirect acks.). Failure detection timeouts work as usual: Option 1 has the heartbeat receiver timeout waiting for a heartbeat, while Option 2 has the pinging process (pinger) time out. The suspected process is immediately marked as failed. This is run in an asynchronous distributed system. a. How many failures does Option 1 take to violate completeness? That is, find the value L so that if there are (L-1) simultaneous failures, all of them will be detected, but if there are L simultaneous failures then not all of them may be detected. b. Answer the same above question for Option 2. c. An opposition party candidate claims that for K-R=2, both Option 1 and Option 2 provide completeness for all scenarios with up to (and including) 9 simultaneous failures. You gently respond that they are wrong and that it also depends on M. What are all the values of M (given K=R-2) for which your opponent's claim above is true? Justify your answer clearly.